Soliton Stability of the 2D Nonlinear Schrödinger Equation

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Soliton Stability of the 2D Nonlinear Schrödinger Equation

  1. 1. Soliton Stability in 2D NLS Natalie Sheils sheilsn@seattleu.edu April 10, 2010 Natalie Sheils Soliton Stability in 2D NLS
  2. 2. Outline 1. Introduction to NLS Natalie Sheils Soliton Stability in 2D NLS
  3. 3. Outline 1. Introduction to NLS Trivial-Phase Solutions of NLS Natalie Sheils Soliton Stability in 2D NLS
  4. 4. Outline 1. Introduction to NLS Trivial-Phase Solutions of NLS Soliton Solution of NLS Natalie Sheils Soliton Stability in 2D NLS
  5. 5. Outline 1. Introduction to NLS Trivial-Phase Solutions of NLS Soliton Solution of NLS 2. Linear Stability Natalie Sheils Soliton Stability in 2D NLS
  6. 6. Outline 1. Introduction to NLS Trivial-Phase Solutions of NLS Soliton Solution of NLS 2. Linear Stability 3. High-Frequency Limit Natalie Sheils Soliton Stability in 2D NLS
  7. 7. Outline 1. Introduction to NLS Trivial-Phase Solutions of NLS Soliton Solution of NLS 2. Linear Stability 3. High-Frequency Limit 4. Future Work Natalie Sheils Soliton Stability in 2D NLS
  8. 8. Introduction to NLS The two-dimensional cubic nonlinear Schr¨dinger equation (NLS), o iψt + ψxx − ψyy + 2|ψ|2 ψ = 0. Natalie Sheils Soliton Stability in 2D NLS
  9. 9. Introduction to NLS The two-dimensional cubic nonlinear Schr¨dinger equation (NLS), o iψt + ψxx − ψyy + 2|ψ|2 ψ = 0. Among many other physical phenomena, NLS arises as a model of Natalie Sheils Soliton Stability in 2D NLS
  10. 10. Introduction to NLS The two-dimensional cubic nonlinear Schr¨dinger equation (NLS), o iψt + ψxx − ψyy + 2|ψ|2 ψ = 0. Among many other physical phenomena, NLS arises as a model of pulse propagation along optical fibers. Natalie Sheils Soliton Stability in 2D NLS
  11. 11. Introduction to NLS The two-dimensional cubic nonlinear Schr¨dinger equation (NLS), o iψt + ψxx − ψyy + 2|ψ|2 ψ = 0. Among many other physical phenomena, NLS arises as a model of pulse propagation along optical fibers. surface waves on deep water. Natalie Sheils Soliton Stability in 2D NLS
  12. 12. Introduction to NLS Figure: Wave tank in the Pritchard Fluid Mechanics Labratory in the Mathematics Department at Penn State University. Natalie Sheils Soliton Stability in 2D NLS
  13. 13. Trivial-Phase Solutions of NLS NLS admits a class of 1-D trivial-phase solutions of the form ψ(x, t) = φ(x)e iλt where φ is a real-valued function and λ is a real constant. Natalie Sheils Soliton Stability in 2D NLS
  14. 14. Trivial-Phase Solutions of NLS A specific NLS solution ψ is called a soliton solution. ψ(x, t) = sech(x)e it Natalie Sheils Soliton Stability in 2D NLS
  15. 15. Trivial-Phase Solutions of NLS A specific NLS solution ψ is called a soliton solution. ψ(x, t) = sech(x)e it Ψ x, 0 1.0 0.8 0.6 0.4 0.2 x 20 10 10 20 Natalie Sheils Soliton Stability in 2D NLS
  16. 16. Linear Stability In order to examine the stability of trivial-phase solutions to NLS, ψ(x, y , t) = φ(x)e it we add two-dimensional perturbations ψ(x, y , t) = e it (φ + u + i v + O( 2 )) where is a small real constant and u = u(x, y , t) and v = v (x, y , t) are real-valued functions. Natalie Sheils Soliton Stability in 2D NLS
  17. 17. Linear Stability We substitute into NLS and simplify. We know is small, so the terms with the lowest order of are dominant. The O( 0 ) terms cancel out so O( 1 ) is the leading order. −ut = vxx − vyy + (2φ2 (x) − 1)v (1) vt = uxx − uyy + (6φ2 (x) − 1)u Natalie Sheils Soliton Stability in 2D NLS
  18. 18. Linear Stability Since the coefficients of the perturbed system (1) do not explicitly depend on y or t, we may separate variables, and let u(x, y , t) = U(x)e iρy +Ωt + c.c. v (x, y , t) = V (x)e iρy +Ωt + c.c. Natalie Sheils Soliton Stability in 2D NLS
  19. 19. Linear Stability Since the coefficients of the perturbed system (1) do not explicitly depend on y or t, we may separate variables, and let u(x, y , t) = U(x)e iρy +Ωt + c.c. v (x, y , t) = V (x)e iρy +Ωt + c.c. Ω gives us the following conditions for spectral stability: Natalie Sheils Soliton Stability in 2D NLS
  20. 20. Linear Stability Since the coefficients of the perturbed system (1) do not explicitly depend on y or t, we may separate variables, and let u(x, y , t) = U(x)e iρy +Ωt + c.c. v (x, y , t) = V (x)e iρy +Ωt + c.c. Ω gives us the following conditions for spectral stability: If any Ω has positive real part, the solution is unstable. Natalie Sheils Soliton Stability in 2D NLS
  21. 21. Linear Stability Since the coefficients of the perturbed system (1) do not explicitly depend on y or t, we may separate variables, and let u(x, y , t) = U(x)e iρy +Ωt + c.c. v (x, y , t) = V (x)e iρy +Ωt + c.c. Ω gives us the following conditions for spectral stability: If any Ω has positive real part, the solution is unstable. If all Ω have negative real parts, the solution is stable. Natalie Sheils Soliton Stability in 2D NLS
  22. 22. Linear Stability Since the coefficients of the perturbed system (1) do not explicitly depend on y or t, we may separate variables, and let u(x, y , t) = U(x)e iρy +Ωt + c.c. v (x, y , t) = V (x)e iρy +Ωt + c.c. Ω gives us the following conditions for spectral stability: If any Ω has positive real part, the solution is unstable. If all Ω have negative real parts, the solution is stable. If all Ω are purely imaginary, the solution is stable. Natalie Sheils Soliton Stability in 2D NLS
  23. 23. Linear Stability Then, U and V satisfy the following differential equations. ΩU = V − ρ2 V − 2V φ2 − V (2) −ΩV = U − ρ2 U − 6Uφ2 − U Natalie Sheils Soliton Stability in 2D NLS
  24. 24. Linear Stability Natalie Sheils Soliton Stability in 2D NLS
  25. 25. High-Frequency Limit Natalie Sheils Soliton Stability in 2D NLS
  26. 26. High-Frequency Limit ! "#$%&'$()!*+,-.(/#! ! 01234567051849:!;70<51:5!;1! ! =8052<8706<<996!;7059684:!;1! Natalie Sheils Soliton Stability in 2D NLS
  27. 27. High-Frequency Limit In our linear stability problem (2) we assume ρ is large and U(x) ∼ u0 (µx) + ρ−1 u1 (µx) + ρ−2 u2 (µx) + . . . V (x) ∼ v0 (µx) + ρ−1 v1 (µx) + ρ−2 v2 (µx) + . . . µ ∼ ρ + µ0 + µ1 ρ−1 + µ2 ρ−2 + . . . Ω ∼ ω−2 ρ2 + ω3 ρ−3 . Pick z = µx. Natalie Sheils Soliton Stability in 2D NLS
  28. 28. High-Frequency Limit At leading order in ρ, equation (2) becomes: (4) 2 v0 + 2v0 + (1 + ω−2 )v0 = 0. Natalie Sheils Soliton Stability in 2D NLS
  29. 29. High-Frequency Limit At leading order in ρ, equation (2) becomes: (4) 2 v0 + 2v0 + (1 + ω−2 )v0 = 0. In solving this equation, we want v0 to be bounded. This implies that ω−2 is purely imaginary and −1 < iω−2 < 1. Pick w = iω−2 . Natalie Sheils Soliton Stability in 2D NLS
  30. 30. High-Frequency Limit Now we have √ √ √ √ −w −1 v0 = c1 e z + c2 e −z −w −1 + c3 e z w −1 + c4 e −z w −1 where ci ’s are complex constants. Natalie Sheils Soliton Stability in 2D NLS
  31. 31. High-Frequency Limit Now we have √ √ √ √ −w −1 v0 = c1 e z + c2 e −z −w −1 + c3 e z w −1 + c4 e −z w −1 where ci ’s are complex constants. If v0 is bounded, u0 is bounded and we find u0 to be √ √ √ √ −w −1 u0 = −ic1 e z − ic2 e −z −w −1 + ic3 e z w −1 + ic4 e −z w −1 . Natalie Sheils Soliton Stability in 2D NLS
  32. 32. High-Frequency Limit The next order of ρ is O(ρ): (4) (4) v1 + 2v1 + (1 − w 2 )v1 = −2iw µ0 u0 − 2µ0 v0 − 2µ0 v0 . Natalie Sheils Soliton Stability in 2D NLS
  33. 33. High-Frequency Limit The next order of ρ is O(ρ): (4) (4) v1 + 2v1 + (1 − w 2 )v1 = −2iw µ0 u0 − 2µ0 v0 − 2µ0 v0 . We want v1 to be bounded so we require the right-hand side of the equation to be orthogonal to the solution of the homogeneous equation. Natalie Sheils Soliton Stability in 2D NLS
  34. 34. High-Frequency Limit The next order of ρ is O(ρ): (4) (4) v1 + 2v1 + (1 − w 2 )v1 = −2iw µ0 u0 − 2µ0 v0 − 2µ0 v0 . We want v1 to be bounded so we require the right-hand side of the equation to be orthogonal to the solution of the homogeneous equation. In this case, we require our right-hand side to be zero. Then we have the following restriction: µ0 = 0. Natalie Sheils Soliton Stability in 2D NLS
  35. 35. High-Frequency Limit Now we have √ √ √ √ −w −1 v1 = c5 e z + c6 e −z −w −1 + c7 e z w −1 + c8 e −z w −1 and √ √ √ √ −w −1 u1 = −ic5 e z − ic6 e −z −w −1 + ic7 e z w −1 + ic8 e −z w −1 . Natalie Sheils Soliton Stability in 2D NLS
  36. 36. High-Frequency Limit For the next few orders of ρ the general solution of the homogeneous problem is the same as the previous orders. Natalie Sheils Soliton Stability in 2D NLS
  37. 37. High-Frequency Limit For the next few orders of ρ the general solution of the homogeneous problem is the same as the previous orders. We need to make sure the particular solution of the nonhomogeneous equation is bounded. Natalie Sheils Soliton Stability in 2D NLS
  38. 38. High-Frequency Limit For the next few orders of ρ the general solution of the homogeneous problem is the same as the previous orders. We need to make sure the particular solution of the nonhomogeneous equation is bounded. µ1 ∈ R µ2 =0. Natalie Sheils Soliton Stability in 2D NLS
  39. 39. Future Work Natalie Sheils Soliton Stability in 2D NLS
  40. 40. Future Work Continue looking at orders of ρ. Natalie Sheils Soliton Stability in 2D NLS
  41. 41. Future Work Continue looking at orders of ρ. We hope to find that ω3 is the first ωi with nonzero real part. Natalie Sheils Soliton Stability in 2D NLS
  42. 42. Acknowledgments Dr. John Carter of Seattle University Natalie Sheils Soliton Stability in 2D NLS
  43. 43. Acknowledgments Dr. John Carter of Seattle University Seattle University College of Science and Engineering Natalie Sheils Soliton Stability in 2D NLS
  44. 44. Acknowledgments Dr. John Carter of Seattle University Seattle University College of Science and Engineering Pacific Northwest Section of the Mathematical Association of America Natalie Sheils Soliton Stability in 2D NLS
  45. 45. Questions Questions? Natalie Sheils Soliton Stability in 2D NLS
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