AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 3.
More info at http://summerschool.ssa.org.ua
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.
More info at http://summerschool.ssa.org.ua
The document describes an alternating minimization algorithm for solving a shifted speckle reduction variational model. It summarizes previous speckle reduction methods including convex variational models with total variation regularization and augmented Lagrangian based algorithms. It then proposes a shifted variational model with total variation and an alternating minimization algorithm to solve it. Numerical experiments are presented to evaluate the proposed method.
This document provides definitions and notations for 2-D systems and matrices. It defines how continuous and sampled 2-D signals like images are represented. It introduces some common 2-D functions used in signal processing like the Dirac delta, rectangle, and sinc functions. It describes how 2-D linear systems can be represented by matrices and discusses properties of the 2-D Fourier transform including the frequency response and eigenfunctions. It also introduces concepts of Toeplitz and circulant matrices and provides an example of convolving periodic sequences using circulant matrices. Finally, it defines orthogonal and unitary matrices.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
Adiabatic Theorem for Discrete Time Evolutiontanaka-atushi
The document summarizes the proof of an adiabatic theorem for discrete time evolution described by quantum maps. It extends Kato's proof of the adiabatic theorem to the discrete setting by using an interaction picture and showing that the main estimation involves terms that decay as O(N^-1) through destructive interference, proving the theorem. The proof technique involves introducing a geometric evolution operator and using a discrete version of integration by parts.
The document discusses finite speed approximations to the Navier-Stokes equations. It introduces relaxation approximations and a damped wave equation approximation as two methods to derive finite speed approximations. It then discusses a vector BGK approximation, which takes a kinetic approach using a system of hyperbolic equations that approximates the Boltzmann equation and can be shown to converge to the incompressible Navier-Stokes equations in the diffusive limit. The document provides details on the vector BGK model, including compatibility conditions, conservation laws, and proofs of consistency, stability, and existence of global solutions.
New Mathematical Tools for the Financial SectorSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua
The document summarizes several results on metric embeddings. It begins by defining metric embeddings and distortion. It then states three theorems:
1) There is a randomized polynomial-time algorithm that embeds any metric space into a tree metric with expected distortion O(log n).
2) Any n-point 2-metric can be embedded into R^O(log n) with distortion 1+ε.
3) There is an algorithm that embeds any metric space into l_1 with distortion O(log k) such that it preserves distances between k given terminal pairs up to a factor of O(log k).
The document then discusses properties and algorithms for embeddings into l_1, l
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.
More info at http://summerschool.ssa.org.ua
The document describes an alternating minimization algorithm for solving a shifted speckle reduction variational model. It summarizes previous speckle reduction methods including convex variational models with total variation regularization and augmented Lagrangian based algorithms. It then proposes a shifted variational model with total variation and an alternating minimization algorithm to solve it. Numerical experiments are presented to evaluate the proposed method.
This document provides definitions and notations for 2-D systems and matrices. It defines how continuous and sampled 2-D signals like images are represented. It introduces some common 2-D functions used in signal processing like the Dirac delta, rectangle, and sinc functions. It describes how 2-D linear systems can be represented by matrices and discusses properties of the 2-D Fourier transform including the frequency response and eigenfunctions. It also introduces concepts of Toeplitz and circulant matrices and provides an example of convolving periodic sequences using circulant matrices. Finally, it defines orthogonal and unitary matrices.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
Adiabatic Theorem for Discrete Time Evolutiontanaka-atushi
The document summarizes the proof of an adiabatic theorem for discrete time evolution described by quantum maps. It extends Kato's proof of the adiabatic theorem to the discrete setting by using an interaction picture and showing that the main estimation involves terms that decay as O(N^-1) through destructive interference, proving the theorem. The proof technique involves introducing a geometric evolution operator and using a discrete version of integration by parts.
The document discusses finite speed approximations to the Navier-Stokes equations. It introduces relaxation approximations and a damped wave equation approximation as two methods to derive finite speed approximations. It then discusses a vector BGK approximation, which takes a kinetic approach using a system of hyperbolic equations that approximates the Boltzmann equation and can be shown to converge to the incompressible Navier-Stokes equations in the diffusive limit. The document provides details on the vector BGK model, including compatibility conditions, conservation laws, and proofs of consistency, stability, and existence of global solutions.
New Mathematical Tools for the Financial SectorSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua
The document summarizes several results on metric embeddings. It begins by defining metric embeddings and distortion. It then states three theorems:
1) There is a randomized polynomial-time algorithm that embeds any metric space into a tree metric with expected distortion O(log n).
2) Any n-point 2-metric can be embedded into R^O(log n) with distortion 1+ε.
3) There is an algorithm that embeds any metric space into l_1 with distortion O(log k) such that it preserves distances between k given terminal pairs up to a factor of O(log k).
The document then discusses properties and algorithms for embeddings into l_1, l
This document summarizes Andrew Hone's talk on reductions of the discrete Hirota (discrete KP) equation. Plane wave reductions of the discrete Hirota equation yield Somos-type recurrence relations. Reductions of the discrete Hirota Lax pair give scalar Lax pairs with spectral parameters. Certain reductions produce periodic coefficients, leading to cluster algebra structures. Reductions of the discrete KdV equation are also considered, giving bi-Hamiltonian structures.
The document discusses approximate regeneration schemes for Markov chains. It introduces the concept of regeneration blocks between visits to an atom set. For chains without an atom, the Nummelin splitting technique extends the chain to be atomic. An approximate regeneration scheme is proposed using an estimated transition density over a small set to split the chain. This allows treating blocks of data as approximately i.i.d.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
We explain a methodology to compute families of homo/heteroclinic connections between periodic orbits. We show the relation of some homoclinic connections with resonant transitions in the RTBP
This document provides an overview of doubled geometry and double field theory. It introduces Lie and Courant algebroids, which are structures that underlie doubled geometry. Doubled geometry is defined on a manifold with fibers that transform as modules under the O(D,D) group. This allows for the description of duality-covariant and non-geometric backgrounds. Generalized tensors, connections, torsion, and curvature are defined on the doubled space. Double field theory is a field theory formulated on this doubled space.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
This document summarizes paraproduct operators with general dilations. It defines paraproducts and provides classical examples. It then introduces a non-classical example of paraproducts with respect to general dilations defined by groups of dilations on Cartesian product spaces. The author and co-author establish Lp estimates for such paraproduct operators by applying martingale estimates to dyadic structures and using square functions. The estimates depend only on the dilation structure and hold for certain exponent ranges.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document discusses Bayesian reliability analysis of systems when component lifetimes follow a geometric distribution. It provides the probability mass function of the geometric distribution and defines the prior and posterior distributions of the distribution parameter θ. It then derives the Bayesian reliability estimators for k-out-of-n, series, parallel and cold standby systems. Simulation studies are presented to analyze the Bayes risk of the estimators for different values of the model parameters.
1) The document provides information on various physics concepts including kinematics equations, forces, energy, and more.
2) It includes 13 key equations related to topics like velocity, acceleration, Newton's laws of motion, Hooke's law, energy, and others.
3) The document also provides details on astrophysics concepts like astronomical units and the Hubble constant, as well as common physics prefixes and units.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
The document is a lecture on nonlinear transport phenomena and patterns. It discusses heat transport models described by balance equations. The key equations presented are the heat transport balance equation and the resulting partial differential equation in one spatial variable. The lecture aims to discuss patterns supported by various transport equations, defined as non-monotonic solutions that maintain shape or evolve self-similarly over time.
This document summarizes Andrew Hone's talk on reductions of the discrete Hirota (discrete KP) equation. Plane wave reductions of the discrete Hirota equation yield Somos-type recurrence relations. Reductions of the discrete Hirota Lax pair give scalar Lax pairs with spectral parameters. Certain reductions produce periodic coefficients, leading to cluster algebra structures. Reductions of the discrete KdV equation are also considered, giving bi-Hamiltonian structures.
The document discusses approximate regeneration schemes for Markov chains. It introduces the concept of regeneration blocks between visits to an atom set. For chains without an atom, the Nummelin splitting technique extends the chain to be atomic. An approximate regeneration scheme is proposed using an estimated transition density over a small set to split the chain. This allows treating blocks of data as approximately i.i.d.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
We explain a methodology to compute families of homo/heteroclinic connections between periodic orbits. We show the relation of some homoclinic connections with resonant transitions in the RTBP
This document provides an overview of doubled geometry and double field theory. It introduces Lie and Courant algebroids, which are structures that underlie doubled geometry. Doubled geometry is defined on a manifold with fibers that transform as modules under the O(D,D) group. This allows for the description of duality-covariant and non-geometric backgrounds. Generalized tensors, connections, torsion, and curvature are defined on the doubled space. Double field theory is a field theory formulated on this doubled space.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
Dr. Arpan Bhattacharyya (Indian Institute Of Science, Bangalore)Rene Kotze
1. The document discusses entanglement entropy functionals for higher derivative gravity theories. It proposes new area functionals for computing entanglement entropy in higher derivative theories containing polynomials of curvature tensors.
2. These functionals are derived using the Lewkowycz-Maldacena interpretation of generalized entropy. However, attempting to derive the extremal surface equations from these functionals using bulk equations of motion leads to inconsistencies and ambiguities in some higher derivative theories like Gauss-Bonnet gravity.
3. The document suggests that the source of ambiguity lies in the limiting procedure used to extract the divergences near the conical singularity. Different limiting paths can lead to different extremal surface equations, indicating no unique prescription
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
This document summarizes paraproduct operators with general dilations. It defines paraproducts and provides classical examples. It then introduces a non-classical example of paraproducts with respect to general dilations defined by groups of dilations on Cartesian product spaces. The author and co-author establish Lp estimates for such paraproduct operators by applying martingale estimates to dyadic structures and using square functions. The estimates depend only on the dilation structure and hold for certain exponent ranges.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document discusses Bayesian reliability analysis of systems when component lifetimes follow a geometric distribution. It provides the probability mass function of the geometric distribution and defines the prior and posterior distributions of the distribution parameter θ. It then derives the Bayesian reliability estimators for k-out-of-n, series, parallel and cold standby systems. Simulation studies are presented to analyze the Bayes risk of the estimators for different values of the model parameters.
1) The document provides information on various physics concepts including kinematics equations, forces, energy, and more.
2) It includes 13 key equations related to topics like velocity, acceleration, Newton's laws of motion, Hooke's law, energy, and others.
3) The document also provides details on astrophysics concepts like astronomical units and the Hubble constant, as well as common physics prefixes and units.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
The document is a lecture on nonlinear transport phenomena and patterns. It discusses heat transport models described by balance equations. The key equations presented are the heat transport balance equation and the resulting partial differential equation in one spatial variable. The lecture aims to discuss patterns supported by various transport equations, defined as non-monotonic solutions that maintain shape or evolve self-similarly over time.
This document provides a summary of a project report on bifurcation analysis and its applications. It discusses key concepts in nonlinear systems such as equilibrium points, stability, linearization, and bifurcations including saddle node, transcritical, pitchfork and Hopf bifurcations. Examples are given to illustrate each type of bifurcation. Population models involving competition and prey-predator interactions are also discussed. The document outlines the contents which cover preliminary remarks, local theory of nonlinear systems, different types of bifurcations, and applications to population models.
This document summarizes recent discoveries and challenges in chaos theory presented by Xiong Wang from the Centre for Chaos and Complex Networks at City University of Hong Kong. Some key points discussed include: discovering a chaotic system with one stable equilibrium point, using symmetry to create systems with multiple equilibria whose stability can be tuned, and open questions around reconciling local stability with global chaotic behavior. The document concludes by acknowledging that chaos is a global phenomenon not precluded by local stability near equilibria.
The document provides an overview of Session 1 of a COMSOL training series, which introduces COMSOL software. It discusses multiphysics simulation, which involves modeling multiple interacting physical phenomena, like fluid flow, heat transfer, and electrodynamics. It gives examples of why simulation is useful, such as design validation, optimization, and analysis. It then outlines the basic steps of simulation: defining physical phenomena with PDEs, discretizing the domain, solving the PDEs, and visualizing results. Finally, it previews the example simulation of a microfluidic mixer.
This slide deals with different aspects of Comsol Multiphysics and it's possibility in the future as multiple physics properties can be studied simultaneously with the help of different inbuilt or user-defined modules in this software.
Gradient Dynamical Systems, Bifurcation Theory, Numerical Methods and Applica...Boris Fackovec
This document is a presentation on gradient dynamical systems, bifurcation theory, numerical methods, and applications. It discusses dynamical systems, phase portraits, linear and nonlinear systems of ordinary differential equations, bifurcations, bifurcation diagrams, numerical methods for constructing bifurcation diagrams, and two examples - a chemical reactor system and a cluster of three charged atoms.
JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron modelshirokazutanaka
This document provides an overview of topics to be covered in a lecture on single neuron models. It will discuss:
1) The basic anatomy and physiology of neurons including their morphology and membrane properties.
2) Phenomenological models of subthreshold dynamics like the integrate-and-fire, quadratic-and-fire, and resonate-and-fire models.
3) Biophysical models of spiking mechanisms including the Hodgkin-Huxley model and its use of ion channels and master equations.
4) Analysis techniques like phase plots and bifurcation analysis applied to models like FitzHugh-Nagumo and Hindmarsh-Rose.
5) Modern single neuron models such
This document discusses different types of hydraulic pressure control valves. It describes pressure relief valves, pilot operated relief valves, sequence control valves, and other types. Pressure relief valves limit pressure by diverting fluid to the reservoir when pressure reaches a set point. Pilot operated relief valves use a piston or spool controlled by a pilot valve. Sequence valves provide flow to a second actuator after the first reaches a threshold pressure. The document also provides examples of applications for different valve types.
This document discusses vapor-liquid equilibrium (VLE) calculations for various binary and ternary systems using the software HYSYS. It provides examples of calculating bubble point pressures, dew point pressures, and compositions for systems such as methanol/methyl acetate at different temperatures and compositions. It also assigns homework problems calculating VLE properties for systems like ethyl ethanoate/n-heptane and methane/ethylene/ethane using assumptions like Raoult's law.
This document provides an overview of using HYSYS simulation software to model and analyze chemical processes. It discusses setting up a HYSYS case by adding components, selecting a fluid package, and entering the simulation environment. It also covers defining process units like separators and heat exchangers, specifying stream properties, performing flash calculations, and generating workbooks. The document is intended as an introduction for students to learn the basic functionality of HYSYS through examples of common unit operations.
Simulation-Led Design Using SolidWorks® and COMSOL Multiphysics®Design World
Multiphysics has earned the reputation as an excellent approach for simulation in engineering and science. Applying multiphysics simulation early in the product development process brings you reliable computer models to verify and optimize your designs
This webinar will demonstrate how the COMSOL LiveLink for SolidWorks bridges the gap between design and analysis, integrating real-world simulation right into the CAD design environment of SolidWorks.
Attend this webinar to learn:
The importance of multiphysics modeling for true simulation of real-world applications
How to integrate analysis into the design process
the workflow of modeling with COMSOL Multiphysics and SolidWorks
Pressure Relief Valve Sizing for Single Phase FlowVikram Sharma
This presentation file provides a quick refresher to pressure relief valve sizing for single phase flow. The calculation guideline is as per API Std 520.
The document discusses the transportation processes that occur along coastal areas, noting that the same four transportation processes that occur in rivers (sliding, rolling, saltating, and suspended load) also take place in coastal environments through the actions of waves. Fine sediments eroded from cliffs and beaches are carried along the shoreline via longshore drift, which involves the swash and backwash of waves pushing material sideways in a zigzag pattern down the coast. Transportation results in the eventual rounding and smoothing of eroded materials from their original shapes and sizes.
Chaos theory is a mathematical field of study which states that non-linear dynamical systems
that are seemingly random are actually deterministic from much simpler equations. The
phenomenon of Chaos theory was introduced to the modern world by Edward Lorenz in 1972
with conceptualization of ‘Butterfly Effect’. As chaos theory was developed by inputs of
various mathematicians and scientists, it found applications in a large number of scientific
fields.
The purpose of the project is the interpretation of chaos theory which is not as familiar as
other theories. Everything in the universe is in some way or the other under control of Chaos
or product of Chaos. Every motion, behavior or tendency can be explained by Chaos Theory.
The prime objective of it is the illustration of Chaos Theory and Chaotic behavior.
This project includes origin, history, fields of application, real life application and limitations
of Chaos Theory. It explores understanding complexity and dynamics of Chaos.
Thermodynamics deals with the effects of work, heat and energy on systems. It considers macroscopic and microscopic changes. The laws of thermodynamics are:
1) Zeroth law - If two systems are in thermal equilibrium with a third, they are in equilibrium with each other.
2) First law - The change in internal energy of a closed system equals the heat supplied minus the work done.
3) Second law - Heat cannot spontaneously flow from a cold body to a hot body.
4) Third law - The entropy of a system approaches a constant value as the temperature approaches absolute zero.
The document discusses different types of compressors used to increase air pressure. It describes reciprocating compressors which use pistons to compress air inside cylinders. Rotary compressors like screw, vane, and lobe compressors compress air using rotating elements. Centrifugal and axial compressors accelerate air to increase pressure, with centrifugal compressors using impellers and axial using rotating and stationary blades in stages. The document provides details on components and operating principles of these compressor types.
A 64-year-old man presented with sudden onset of pain and loss of sensation in his right leg. Examination found absent pulses, decreased sensation, and an inability to move his toes, indicating acute limb ischemia. The document discusses the etiology, pathophysiology, clinical evaluation, investigations including Doppler ultrasound and angiography, and treatment approaches for acute limb ischemia including thrombolytics, surgery, and amputation. The goal of therapy is to restore blood flow, preserve the limb if possible, and prevent recurrence through anticoagulation.
Mathematical formulation of inverse scattering and korteweg de vries equationAlexander Decker
This document summarizes the mathematical formulation of inverse scattering and the Korteweg-de Vries (KdV) equation. It begins by defining inverse scattering as determining solutions to differential equations based on known asymptotic solutions, specifically by solving the Marchenko equation. It then discusses how the KdV equation describes shallow water waves and solitons, and how the inverse scattering transform method can be used to determine soliton solutions from arbitrary initial conditions. The document outlines the procedure, including deriving the scattering data from an initial potential function and using its time evolution to reconstruct solutions to the KdV equation at later times. It provides examples using reflectionless potentials, specifically obtaining the single-soliton solution from an initial sech^2
This document summarizes numerical studies of line soliton solutions to the Kadomtsev-Petviashvili (KP) equation. The KP equation models nonlinear shallow water waves and admits exact solitary wave solutions called line solitons. The document presents pseudospectral schemes for numerically solving the KP equation with initial conditions formed from pieces of exact one-soliton solutions. It demonstrates convergence of the numerical solutions to three types of exact two-soliton solutions: a (3142)-soliton, a Y-shaped soliton, and an O-shaped soliton. Parameters in the exact solutions are optimized to minimize the error between numerical and exact solutions.
The document discusses wave optics and electromagnetic waves. It defines key concepts like wavefronts, which connect points of equal phase, and rays, which describe the direction of wave propagation perpendicular to wavefronts. It explains Huygens' principle, which states that each point on a wavefront acts as a secondary source of spherical wavelets to determine the new wavefront position. The principle of superposition states that multiple waves add linearly at each point in space to determine the resulting disturbance. Interference occurs when waves are out of phase and their amplitudes diminish or vanish.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
Wavelets and Other Adaptive Methods is a document about wavelet analysis. It introduces wavelets as mathematical tools that can analyze local behavior using translated and scaled versions of basic wavelet functions. It describes Haar wavelets, including the Haar father and mother wavelet functions. It explains how wavelets can be constructed to form an orthonormal basis and discusses how wavelet regression can be used to estimate functions. The document provides examples of applications of wavelet analysis including signal processing and constructing wavelet bases.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
The Laplace transform is a mathematical tool that is useful for solving differential equations. It was developed by Pierre-Simon Laplace in the late 18th century. The Laplace transform takes a function of time and transforms it into a function of complex quantities. This transformation allows differential equations to be converted into algebraic equations that are easier to solve. Some common applications of the Laplace transform include modeling problems in semiconductors, wireless networks, vibrations, and electromagnetic fields.
This document discusses magnetic monopoles and solitons in field theory. It summarizes that solitons are finite-energy, non-dissipative solutions to classical wave equations that arise in non-linear theories. Magnetic monopoles can be constructed from potentials that have Dirac string singularities, requiring the Dirac quantization condition where magnetic charge is quantized. Several models are described where magnetic monopoles arise, including the 't Hooft-Polyakov model in 3+1 dimensions, where the mass of monopoles is related to the gauge boson mass. To date, no magnetic monopoles have been observed experimentally.
This document discusses the basic principles of seismic waves. It introduces longitudinal (P) waves and shear (S) waves, and derives the one-dimensional wave equation. It discusses wave phenomena like reflection, transmission, and refraction based on Snell's law at boundaries between layers. It also discusses the different arrivals of direct, reflected, and refracted/head waves that can be measured at the surface for seismic exploration purposes.
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A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
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Similar to Nonlinear transport phenomena: models, method of solving and unusual features (3) (20)
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
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تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
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Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
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Nonlinear transport phenomena: models, method of solving and unusual features (3)
1. Nonlinear transport phenomena:
models, method of solving and unusual
features: Lecture 3
Vsevolod Vladimirov
AGH University of Science and technology, Faculty of Applied
Mathematics
´
Krakow, August 6, 2010
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 1 / 34
2. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
3. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
4. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
5. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
6. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
7. Traveling wave solutions supported by GBE
We are interested in the following generalization of the BE,
called convection-diffusion - reaction equation:
τ ut t + ut + u ux = ν [un ux ]x + f (u) (1)
Our aim is to study the wave patterns, i.e. physically
meaningful traveling wave (TW) solutions, having the form
u(t, x) = U (ξ), ξ = x − V t.
We are specially interest in the existence of
solitons,
compactons,
some other solutions related to them
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34
8. Solitons and compactons are solitary waves, moving with
constant velocity V without change of their shape. The main
difference between them is seen on the graphs shown below:
A
Figure: Graph of the KdV soliton U (ξ) = Cosh2 [B ξ]
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 3 / 34
9. Figure: Graph of the Rosenau-Hyman compacton
ACos2 [B ξ], if |ξ| < 2 π,
U (ξ) =
0 otherwise
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 4 / 34
10. So how can we distinguish the solitary wave solutions and
compactons within the set of TW solutions?
To answer this question, we restore to the geometric
interpretation of these solutions.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 5 / 34
11. So how can we distinguish the solitary wave solutions and
compactons within the set of TW solutions?
To answer this question, we restore to the geometric
interpretation of these solutions.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 5 / 34
12. Soliton on the phase plane of factorized system
Both solitons and compactons are the TW solutions, having the
form
u(t, x) = U (ξ), with ξ = x − V t.
We give the geometric interpretation of the soliton by
considering the Korteveg-de Vries (KdV) equation
ut + u ux + uxxx = 0
Inserting the TW ansatz into the KdV equation, we get:
d ¨ U 2 (ξ)
U (ξ) + − V U (ξ) = 0.
dξ 2
This equation is equivalent to the following dynamical system:
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 6 / 34
13. Soliton on the phase plane of factorized system
Both solitons and compactons are the TW solutions, having the
form
u(t, x) = U (ξ), with ξ = x − V t.
We give the geometric interpretation of the soliton by
considering the Korteveg-de Vries (KdV) equation
ut + u ux + uxxx = 0
Inserting the TW ansatz into the KdV equation, we get:
d ¨ U 2 (ξ)
U (ξ) + − V U (ξ) = 0.
dξ 2
This equation is equivalent to the following dynamical system:
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 6 / 34
14. Soliton on the phase plane of factorized system
Both solitons and compactons are the TW solutions, having the
form
u(t, x) = U (ξ), with ξ = x − V t.
We give the geometric interpretation of the soliton by
considering the Korteveg-de Vries (KdV) equation
ut + u ux + uxxx = 0
Inserting the TW ansatz into the KdV equation, we get:
d ¨ U 2 (ξ)
U (ξ) + − V U (ξ) = 0.
dξ 2
This equation is equivalent to the following dynamical system:
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 6 / 34
15. Soliton on the phase plane of factorized system
Both solitons and compactons are the TW solutions, having the
form
u(t, x) = U (ξ), with ξ = x − V t.
We give the geometric interpretation of the soliton by
considering the Korteveg-de Vries (KdV) equation
ut + u ux + uxxx = 0
Inserting the TW ansatz into the KdV equation, we get:
d ¨ U 2 (ξ)
U (ξ) + − V U (ξ) = 0.
dξ 2
This equation is equivalent to the following dynamical system:
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 6 / 34
16. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
17. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
18. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
19. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
20. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
21. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
22. ˙
U (ξ) = −W (ξ),
2 (ξ) (2)
W (ξ) = U 2 − V U (ξ)
˙
Lemma 1.The system (2) is a Hamiltonian system, with
W2 U3 U2
H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3)
2 6 2
Lemma 2.Hamiltonian function H(U, W ) remains constant
along a trajectory of the dynamical system (2).
Theorem 1. 1
All stationary points of the system (2) are placed at the
3 2
extremal points of the function Epot (U ) = U − V U ;
6 2
local maximum of Epot (U ) correspond to a saddle point,
while the local minimum corresponds to a center;
all possible coordinates of the phase trajectory corresponding
to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0.
1
Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34
23. U3 U2
Graph of the function Epot (U ) = 6 −V 2
Phase portrait of the system (2)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 8 / 34
24. U3 U2
Graph of the function Epot (U ) = 6 −V 2
Phase portrait of the system (2)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 8 / 34
25. U3 U2
Graph of the function Epot (U ) = 6 −V 2
Phase portrait of the system (2)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 8 / 34
26. Statement 1. The only trajectory that could correspond to the
soliton solution is the closed loop bi-asymptotic to the saddle
point.
Statement 2. The solitary wave solution U (ξ) is nonzero for
any ξ ∈ R.
˙ ˙
Proof. The linearized system U = −W, W = −V U is
¨
equivalent to the single equation U = V U . Solution to this
equation, that can approach zero, must have the form
U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero
for any nonzero ξ, and can attain zero when ξ → ± ∞.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34
27. Statement 1. The only trajectory that could correspond to the
soliton solution is the closed loop bi-asymptotic to the saddle
point.
Statement 2. The solitary wave solution U (ξ) is nonzero for
any ξ ∈ R.
˙ ˙
Proof. The linearized system U = −W, W = −V U is
¨
equivalent to the single equation U = V U . Solution to this
equation, that can approach zero, must have the form
U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero
for any nonzero ξ, and can attain zero when ξ → ± ∞.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34
28. Statement 1. The only trajectory that could correspond to the
soliton solution is the closed loop bi-asymptotic to the saddle
point.
Statement 2. The solitary wave solution U (ξ) is nonzero for
any ξ ∈ R.
˙ ˙
Proof. The linearized system U = −W, W = −V U is
¨
equivalent to the single equation U = V U . Solution to this
equation, that can approach zero, must have the form
U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero
for any nonzero ξ, and can attain zero when ξ → ± ∞.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34
29. Statement 1. The only trajectory that could correspond to the
soliton solution is the closed loop bi-asymptotic to the saddle
point.
Statement 2. The solitary wave solution U (ξ) is nonzero for
any ξ ∈ R.
˙ ˙
Proof. The linearized system U = −W, W = −V U is
¨
equivalent to the single equation U = V U . Solution to this
equation, that can approach zero, must have the form
U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero
for any nonzero ξ, and can attain zero when ξ → ± ∞.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34
30. Statement 1. The only trajectory that could correspond to the
soliton solution is the closed loop bi-asymptotic to the saddle
point.
Statement 2. The solitary wave solution U (ξ) is nonzero for
any ξ ∈ R.
˙ ˙
Proof. The linearized system U = −W, W = −V U is
¨
equivalent to the single equation U = V U . Solution to this
equation, that can approach zero, must have the form
U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero
for any nonzero ξ, and can attain zero when ξ → ± ∞.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34
31. Compactons on the phase plane of factorized system
Let us discuss the compacton TW solutions, basing on
Rosenau-Hyman equation
ut + u2 x
+ u2 xxx
= 0. (4)
Inserting the TW ansatz u(t, x) = U (ξ), ξ = x − V t into (4),
we get, after some manipulation, the dynamical system
dU
dT = −2 U 2 W,
dW , (5)
dT = U −V U + U 2 + 2 W 2
d d
where dT = 2 U2 dξ.
Lemma 3.The system (5) is a Hamiltonian system, with
1 4 V 3
H= U − U + U 2 W 2.
4 3
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 10 / 34
32. Compactons on the phase plane of factorized system
Let us discuss the compacton TW solutions, basing on
Rosenau-Hyman equation
ut + u2 x
+ u2 xxx
= 0. (4)
Inserting the TW ansatz u(t, x) = U (ξ), ξ = x − V t into (4),
we get, after some manipulation, the dynamical system
dU
dT = −2 U 2 W,
dW , (5)
dT = U −V U + U 2 + 2 W 2
d d
where dT = 2 U2 dξ.
Lemma 3.The system (5) is a Hamiltonian system, with
1 4 V 3
H= U − U + U 2 W 2.
4 3
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 10 / 34
33. Compactons on the phase plane of factorized system
Let us discuss the compacton TW solutions, basing on
Rosenau-Hyman equation
ut + u2 x
+ u2 xxx
= 0. (4)
Inserting the TW ansatz u(t, x) = U (ξ), ξ = x − V t into (4),
we get, after some manipulation, the dynamical system
dU
dT = −2 U 2 W,
dW , (5)
dT = U −V U + U 2 + 2 W 2
d d
where dT = 2 U2 dξ.
Lemma 3.The system (5) is a Hamiltonian system, with
1 4 V 3
H= U − U + U 2 W 2.
4 3
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 10 / 34
34. Graph of the function Epot (U )
Phase portrait of the system (5)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 11 / 34
35. Graph of the function Epot (U )
Phase portrait of the system (5)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 11 / 34
36. Graph of the function Epot (U )
Phase portrait of the system (5)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 11 / 34
37. Statement 3. The solution of to (5) corresponding to the
homoclinic loop has a compact support.
Sketch of the proof. It can be shown, that the saddle
separatrices enter (leave) the origin forming right angle with the
horizontal axis.Therefore, in proximity of the origin
W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic
equation −W = d U = C1 U α . It is evident, thus, that the
dξ
asymptotic solution takes the form
U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ)
1
approaches zero as ξ → ξ0 ± 0.
So we deal in fact with the glued (generalized) solution,
and its nonzero part corresponds to the homoclinic
loop.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34
38. Statement 3. The solution of to (5) corresponding to the
homoclinic loop has a compact support.
Sketch of the proof. It can be shown, that the saddle
separatrices enter (leave) the origin forming right angle with the
horizontal axis.Therefore, in proximity of the origin
W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic
equation −W = d U = C1 U α . It is evident, thus, that the
dξ
asymptotic solution takes the form
U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ)
1
approaches zero as ξ → ξ0 ± 0.
So we deal in fact with the glued (generalized) solution,
and its nonzero part corresponds to the homoclinic
loop.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34
39. Statement 3. The solution of to (5) corresponding to the
homoclinic loop has a compact support.
Sketch of the proof. It can be shown, that the saddle
separatrices enter (leave) the origin forming right angle with the
horizontal axis.Therefore, in proximity of the origin
W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic
equation −W = d U = C1 U α . It is evident, thus, that the
dξ
asymptotic solution takes the form
U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ)
1
approaches zero as ξ → ξ0 ± 0.
So we deal in fact with the glued (generalized) solution,
and its nonzero part corresponds to the homoclinic
loop.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34
40. Statement 3. The solution of to (5) corresponding to the
homoclinic loop has a compact support.
Sketch of the proof. It can be shown, that the saddle
separatrices enter (leave) the origin forming right angle with the
horizontal axis.Therefore, in proximity of the origin
W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic
equation −W = d U = C1 U α . It is evident, thus, that the
dξ
asymptotic solution takes the form
U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ)
1
approaches zero as ξ → ξ0 ± 0.
So we deal in fact with the glued (generalized) solution,
and its nonzero part corresponds to the homoclinic
loop.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34
41. Statement 3. The solution of to (5) corresponding to the
homoclinic loop has a compact support.
Sketch of the proof. It can be shown, that the saddle
separatrices enter (leave) the origin forming right angle with the
horizontal axis.Therefore, in proximity of the origin
W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic
equation −W = d U = C1 U α . It is evident, thus, that the
dξ
asymptotic solution takes the form
U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ)
1
approaches zero as ξ → ξ0 ± 0.
So we deal in fact with the glued (generalized) solution,
and its nonzero part corresponds to the homoclinic
loop.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34
42. Very important conclusion
The localized wave patterns, such as solitary waves and
compactons are represented in the phase plane of the
factorized system by the HOMOCLINIC LOOP, i.e. the
phase trajectory bi-asymptotic to a saddle point.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 13 / 34
43. Factorization of the modeling system
Let’s return to the generalized C-R-D equation:
α utt + ut + u ux − κ (un ux )x = f (u). (6)
Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this
system,
we get:
˙
∆(U )U = ∆(U ) W, (7)
˙
∆(U )W = − f (U ) + κn U n−1 2
W + (V − U ) W ,
where ∆(U ) = κ U n − α V 2 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34
44. Factorization of the modeling system
Let’s return to the generalized C-R-D equation:
α utt + ut + u ux − κ (un ux )x = f (u). (6)
Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this
system,
we get:
˙
∆(U )U = ∆(U ) W, (7)
˙
∆(U )W = − f (U ) + κn U n−1 2
W + (V − U ) W ,
where ∆(U ) = κ U n − α V 2 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34
45. Factorization of the modeling system
Let’s return to the generalized C-R-D equation:
α utt + ut + u ux − κ (un ux )x = f (u). (6)
Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this
system,
we get:
˙
∆(U )U = ∆(U ) W, (7)
˙
∆(U )W = − f (U ) + κn U n−1 2
W + (V − U ) W ,
where ∆(U ) = κ U n − α V 2 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34
46. Factorization of the modeling system
Let’s return to the generalized C-R-D equation:
α utt + ut + u ux − κ (un ux )x = f (u). (6)
Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this
system,
we get:
˙
∆(U )U = ∆(U ) W, (7)
˙
∆(U )W = − f (U ) + κn U n−1 2
W + (V − U ) W ,
where ∆(U ) = κ U n − α V 2 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34
47. Factorization of the modeling system
Let’s return to the generalized C-R-D equation:
α utt + ut + u ux − κ (un ux )x = f (u). (6)
Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this
system,
we get:
˙
∆(U )U = ∆(U ) W, (7)
˙
∆(U )W = − f (U ) + κn U n−1 2
W + (V − U ) W ,
where ∆(U ) = κ U n − α V 2 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34
48. The time-delayed C − R − D equation is of dissipative
type. Therefore:
the factorized system is not Hamiltonian;
the homoclinic trajectory will appear at the specific
values of the parameters!!!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 15 / 34
49. The time-delayed C − R − D equation is of dissipative
type. Therefore:
the factorized system is not Hamiltonian;
the homoclinic trajectory will appear at the specific
values of the parameters!!!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 15 / 34
50. The time-delayed C − R − D equation is of dissipative
type. Therefore:
the factorized system is not Hamiltonian;
the homoclinic trajectory will appear at the specific
values of the parameters!!!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 15 / 34
51. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
52. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
53. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
54. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
55. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
56. Our further strategy:
we choose specific function f (u) such that the
factorized system will have at least two stationary
points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ;
we state the condition for which the stationary point
(U1 , 0) becomes a center;
we state, the conditions which guarantee that the other
point (U0 , 0) simultaneously is a saddle;
we analyze the appearance of the limit cycle with zero
radius at the point (U1 , 0);
we check numerically the possibility of homoclinic
bifurcation appearance.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34
57. Further assumptions: We assume that
f (U ) = (U − U0 )m (U − U1 ) ψ(U ), U1 > U0 ≥ 0,
where ψ(U )|<U0 , U1 > = 0.
Under these assumption our system has two stationary points
(U0 , 0) and (U1 , 0) lying on the horizontal axis of the phase
space (U, W ), and no any other stationary point inside the
segment < U0 , U1 >.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 17 / 34
58. Creation of a stable limit cycle
Theorem 1.
1. If the following conditions hold
∆(U1 ) · ψ(U1 ) ≡ κ U1 − α V 2 ψ(U1 ) > 0,
n
(8)
and
n−1
|∆(U1 )| ϕ(U1 ) + κ n U1 |ϕ(U1 )| > 0,
˙ (9)
where
ϕ(U ) = (U − U0 )m ψ(U ), ∆(U ) = κ U n − α V 2 then in
vicinity of the stationary point (U1 , 0) a stable limit
cycle with zero radius is created, when the wave pack
velocity V approaches the bifurcation value Vcr1 = U1 .
2. Under these conditions, the other stationary point
(U0 , 0) is a topological saddle, or, at least, contains a
saddle sector in the half-plane U > U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 18 / 34
59. Creation of a stable limit cycle
Theorem 1.
1. If the following conditions hold
∆(U1 ) · ψ(U1 ) ≡ κ U1 − α V 2 ψ(U1 ) > 0,
n
(8)
and
n−1
|∆(U1 )| ϕ(U1 ) + κ n U1 |ϕ(U1 )| > 0,
˙ (9)
where
ϕ(U ) = (U − U0 )m ψ(U ), ∆(U ) = κ U n − α V 2 then in
vicinity of the stationary point (U1 , 0) a stable limit
cycle with zero radius is created, when the wave pack
velocity V approaches the bifurcation value Vcr1 = U1 .
2. Under these conditions, the other stationary point
(U0 , 0) is a topological saddle, or, at least, contains a
saddle sector in the half-plane U > U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 18 / 34
60. Creation of a stable limit cycle
Theorem 1.
1. If the following conditions hold
∆(U1 ) · ψ(U1 ) ≡ κ U1 − α V 2 ψ(U1 ) > 0,
n
(8)
and
n−1
|∆(U1 )| ϕ(U1 ) + κ n U1 |ϕ(U1 )| > 0,
˙ (9)
where
ϕ(U ) = (U − U0 )m ψ(U ), ∆(U ) = κ U n − α V 2 then in
vicinity of the stationary point (U1 , 0) a stable limit
cycle with zero radius is created, when the wave pack
velocity V approaches the bifurcation value Vcr1 = U1 .
2. Under these conditions, the other stationary point
(U0 , 0) is a topological saddle, or, at least, contains a
saddle sector in the half-plane U > U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 18 / 34
61. A peculiarity of our factorized system
˙
∆(U )U = ∆(U ) W, (10)
˙
∆(U )W = − f (U ) + κn U n−1 W 2 + (V − U ) W ,
where f (U ) = (U − U1 ) (U − U0 )m ψ(U ), is the presence of the
line of singularities ∆(U ) = κ U n − α V 2 = 0, moving along
the phase plane, as the bifurcation parameter V is
changed.
This creates an extra mechanism of the limit cycle destruction,
different from the homoclinic bifurcation.
But it is just the singular line, which makes possible the
presence of such TW solutions, as compactons, shock fronts
and the peakons.
A necessary condition of their appearance reads as follows:
the singular line must contain the topological saddle at the
moment of the homoclinic bifurcation!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 19 / 34
62. A peculiarity of our factorized system
˙
∆(U )U = ∆(U ) W, (10)
˙
∆(U )W = − f (U ) + κn U n−1 W 2 + (V − U ) W ,
where f (U ) = (U − U1 ) (U − U0 )m ψ(U ), is the presence of the
line of singularities ∆(U ) = κ U n − α V 2 = 0, moving along
the phase plane, as the bifurcation parameter V is
changed.
This creates an extra mechanism of the limit cycle destruction,
different from the homoclinic bifurcation.
But it is just the singular line, which makes possible the
presence of such TW solutions, as compactons, shock fronts
and the peakons.
A necessary condition of their appearance reads as follows:
the singular line must contain the topological saddle at the
moment of the homoclinic bifurcation!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 19 / 34
63. A peculiarity of our factorized system
˙
∆(U )U = ∆(U ) W, (10)
˙
∆(U )W = − f (U ) + κn U n−1 W 2 + (V − U ) W ,
where f (U ) = (U − U1 ) (U − U0 )m ψ(U ), is the presence of the
line of singularities ∆(U ) = κ U n − α V 2 = 0, moving along
the phase plane, as the bifurcation parameter V is
changed.
This creates an extra mechanism of the limit cycle destruction,
different from the homoclinic bifurcation.
But it is just the singular line, which makes possible the
presence of such TW solutions, as compactons, shock fronts
and the peakons.
A necessary condition of their appearance reads as follows:
the singular line must contain the topological saddle at the
moment of the homoclinic bifurcation!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 19 / 34
64. A peculiarity of our factorized system
˙
∆(U )U = ∆(U ) W, (10)
˙
∆(U )W = − f (U ) + κn U n−1 W 2 + (V − U ) W ,
where f (U ) = (U − U1 ) (U − U0 )m ψ(U ), is the presence of the
line of singularities ∆(U ) = κ U n − α V 2 = 0, moving along
the phase plane, as the bifurcation parameter V is
changed.
This creates an extra mechanism of the limit cycle destruction,
different from the homoclinic bifurcation.
But it is just the singular line, which makes possible the
presence of such TW solutions, as compactons, shock fronts
and the peakons.
A necessary condition of their appearance reads as follows:
the singular line must contain the topological saddle at the
moment of the homoclinic bifurcation!
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 19 / 34
65. To what kind of localized solution corresponds the
homoclinic loop?
Asymptotic study of the dynamical system, corresponding to
source equation
α utt + ut + u ux − κ (un ux )x = (u − U0 )m (u − U1 ) ψ(u) (11)
enable to state that:
homoclinic loop corresponds to the compactly-supported
TW if 0 < m < 1, and n ∈ N, and U∗ = U0 , where
U∗ ⇔ ∆(U ) = 0 ;
homoclinic loop corresponds to the soliton-like TW if
m = 1, n ∈ N and U∗ < U0 ;
homoclinic loop corresponds to a semi-compacton (or
shock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 20 / 34
66. To what kind of localized solution corresponds the
homoclinic loop?
Asymptotic study of the dynamical system, corresponding to
source equation
α utt + ut + u ux − κ (un ux )x = (u − U0 )m (u − U1 ) ψ(u) (11)
enable to state that:
homoclinic loop corresponds to the compactly-supported
TW if 0 < m < 1, and n ∈ N, and U∗ = U0 , where
U∗ ⇔ ∆(U ) = 0 ;
homoclinic loop corresponds to the soliton-like TW if
m = 1, n ∈ N and U∗ < U0 ;
homoclinic loop corresponds to a semi-compacton (or
shock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 20 / 34
67. To what kind of localized solution corresponds the
homoclinic loop?
Asymptotic study of the dynamical system, corresponding to
source equation
α utt + ut + u ux − κ (un ux )x = (u − U0 )m (u − U1 ) ψ(u) (11)
enable to state that:
homoclinic loop corresponds to the compactly-supported
TW if 0 < m < 1, and n ∈ N, and U∗ = U0 , where
U∗ ⇔ ∆(U ) = 0 ;
homoclinic loop corresponds to the soliton-like TW if
m = 1, n ∈ N and U∗ < U0 ;
homoclinic loop corresponds to a semi-compacton (or
shock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 20 / 34
68. To what kind of localized solution corresponds the
homoclinic loop?
Asymptotic study of the dynamical system, corresponding to
source equation
α utt + ut + u ux − κ (un ux )x = (u − U0 )m (u − U1 ) ψ(u) (11)
enable to state that:
homoclinic loop corresponds to the compactly-supported
TW if 0 < m < 1, and n ∈ N, and U∗ = U0 , where
U∗ ⇔ ∆(U ) = 0 ;
homoclinic loop corresponds to the soliton-like TW if
m = 1, n ∈ N and U∗ < U0 ;
homoclinic loop corresponds to a semi-compacton (or
shock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0 .
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 20 / 34
69. Figure: Vicinity of the origin for various combinations of the
parameters m, n
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 21 / 34
70. Numerical investigation of factorized system
Numerical simulations of the system (10) were carried
out with κ = 1, U1 = 3, U0 = 1. The remaining
parameters varied from one case to another.
We discuss the results concerning the details of the
phase portraits in terms of the reference frame
(X, W ) = (U − U0 , W ).
The localized wave patterns are presented in ”physical”
coordinates (ξ, U ), where ξ = x − V t is the TW
coordinate
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 22 / 34
71. Numerical investigation of factorized system
Numerical simulations of the system (10) were carried
out with κ = 1, U1 = 3, U0 = 1. The remaining
parameters varied from one case to another.
We discuss the results concerning the details of the
phase portraits in terms of the reference frame
(X, W ) = (U − U0 , W ).
The localized wave patterns are presented in ”physical”
coordinates (ξ, U ), where ξ = x − V t is the TW
coordinate
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 22 / 34
72. Numerical investigation of factorized system
Numerical simulations of the system (10) were carried
out with κ = 1, U1 = 3, U0 = 1. The remaining
parameters varied from one case to another.
We discuss the results concerning the details of the
phase portraits in terms of the reference frame
(X, W ) = (U − U0 , W ).
The localized wave patterns are presented in ”physical”
coordinates (ξ, U ), where ξ = x − V t is the TW
coordinate
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 22 / 34
73. Figure: Homoclinic solution of the system (10) with
ϕ(U ) = (U − U0 )1/2 (m = 1/2)(left) and the corresponding compactly
supported TW solution to Eq. (11) (right), obtained for n = 1,
α = 0.12, Vcr2 ∼ 2.68687 and U∗ − U0 = −0.133684
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 23 / 34
74. Figure: Homoclinic solution of the system (10) with
ϕ(U ) = (U − U0 )1/2 (m = 1/2) (left) and the corresponding TW
solution to Eq. (11) (right), obtained for n = 1, α = 0.13827,
Vcr2 ∼ 2.68892 and U∗ − U0 = 0.99973
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 24 / 34
75. Figure: Homoclinic solution of the system (10) with
ϕ(U ) = (U − U0 )1 (m = 1) (left), the corresponding tandem of
well-localized soliton-like solutions to Eq. (11) (center), and the
soliton-like solution (right), obtained for n = 1, α = 0.06,
Vcr2 ∼ 2.65795 and U∗ − U0 = −0.576119
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 25 / 34
76. Figure: Homoclinic solution of the system (10) with ϕ(U ) = (U − U0 )1
(m = 1) (left), the corresponding tandem od solitary wave solutions to
Eq. (11) (center) and a single solitary wave solution (right), obtained
for n = 1, α = 0.142, Vcr2 ∼ 2.65489 and U∗ − U0 = 0.000878617
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 26 / 34
77. Figure: Tandems od shock-like solutions to Eq. (11), corresponding to
ϕ(U ) = (U − U0 )1 (m = 1), U∗ ≈ U0 , n = 2 (left), n = 3 (center), and
n = 4 (right)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 27 / 34
78. Figure: Shock-like solutions to Eq. (11), corresponding to
ϕ(U ) = (U − U0 )3 (m = 3), U∗ ≈ U0 , n = 1 (left), n = 3 (center), and
n = 4 (right)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 28 / 34
79. Figure: Shock-like solutions to Eq. (11), corresponding to
ϕ(U ) = (U − U0 )m , U∗ ≈ U0 , n = 4, m = 1 (left), m = 2 (center), and
m = 3 (right)
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 29 / 34
80. Figure: Periodic solution of the system (10) with ϕ(U ) = −(U − U0 )1
(m = 1) (left) and the corresponding tandem of generalized peak-like
solutions to Eq. (11) (right), obtained for n = 1, α = 0.552,
Vcr2 ∼ 3.00593 and U∗ − U1 = 1.98765
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 30 / 34
81. Figure: Homoclinic solution of the system (10) with
ϕ(U ) = −(U − U0 )1/2 (m = 1/2) (left) and the corresponding tandem
of generalized peak-like solutions to Eq. (11) (right), obtained for
n = 1, α = 0.562, Vcr2 ∼ 3.14497 and U∗ − U1 = 2.55863
=
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 31 / 34
82. Summary
1. The time-delayed C − R − D system possesses a large
variety of the localized wave patterns.
2. The type of the pattern is strongly depend on the
values of the parameters
3. An open question is the analytical description of the
localized wave patterns within the given model.
4. Another open question is the stability and the
attractive features of the localized wave patterns.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 32 / 34
83. Summary
1. The time-delayed C − R − D system possesses a large
variety of the localized wave patterns.
2. The type of the pattern is strongly depend on the
values of the parameters
3. An open question is the analytical description of the
localized wave patterns within the given model.
4. Another open question is the stability and the
attractive features of the localized wave patterns.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 32 / 34
84. Summary
1. The time-delayed C − R − D system possesses a large
variety of the localized wave patterns.
2. The type of the pattern is strongly depend on the
values of the parameters
3. An open question is the analytical description of the
localized wave patterns within the given model.
4. Another open question is the stability and the
attractive features of the localized wave patterns.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 32 / 34
85. Summary
1. The time-delayed C − R − D system possesses a large
variety of the localized wave patterns.
2. The type of the pattern is strongly depend on the
values of the parameters
3. An open question is the analytical description of the
localized wave patterns within the given model.
4. Another open question is the stability and the
attractive features of the localized wave patterns.
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 32 / 34
86. THANKS FOR YOUR ATTENTION
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 33 / 34
87. Delayed C-R-D equation can be formally obtained if we
consider the integro-differential equation
t
1 t−t
ut = exp[− ] [κ (un ux )x − u ux + f (u)] (t , x)
α −∞ α
instead of
ut = κ (un ux )x − u ux + f (u).
Integrating the integro-differential equation w.r.t. temporal
variable, w obtain the target equation
α ut t + ut = κ (un ux )x − u ux + f (u).
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 34 / 34
88. Delayed C-R-D equation can be formally obtained if we
consider the integro-differential equation
t
1 t−t
ut = exp[− ] [κ (un ux )x − u ux + f (u)] (t , x)
α −∞ α
instead of
ut = κ (un ux )x − u ux + f (u).
Integrating the integro-differential equation w.r.t. temporal
variable, w obtain the target equation
α ut t + ut = κ (un ux )x − u ux + f (u).
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 34 / 34
89. Delayed C-R-D equation can be formally obtained if we
consider the integro-differential equation
t
1 t−t
ut = exp[− ] [κ (un ux )x − u ux + f (u)] (t , x)
α −∞ α
instead of
ut = κ (un ux )x − u ux + f (u).
Integrating the integro-differential equation w.r.t. temporal
variable, w obtain the target equation
α ut t + ut = κ (un ux )x − u ux + f (u).
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 34 / 34
90. Delayed C-R-D equation can be formally obtained if we
consider the integro-differential equation
t
1 t−t
ut = exp[− ] [κ (un ux )x − u ux + f (u)] (t , x)
α −∞ α
instead of
ut = κ (un ux )x − u ux + f (u).
Integrating the integro-differential equation w.r.t. temporal
variable, w obtain the target equation
α ut t + ut = κ (un ux )x − u ux + f (u).
KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 34 / 34