1. Soliton Stability in 2D NLS
Natalie Sheils
sheilsn@seattleu.edu
April 10, 2010
Natalie Sheils Soliton Stability in 2D NLS
2. Outline
1. Introduction to NLS
Natalie Sheils Soliton Stability in 2D NLS
3. Outline
1. Introduction to NLS
Trivial-Phase Solutions of NLS
Natalie Sheils Soliton Stability in 2D NLS
4. Outline
1. Introduction to NLS
Trivial-Phase Solutions of NLS
Soliton Solution of NLS
Natalie Sheils Soliton Stability in 2D NLS
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Linear Stability
Natalie Sheils Soliton Stability in 2D NLS
25. High-Frequency Limit
Natalie Sheils Soliton Stability in 2D NLS
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Future Work
Natalie Sheils Soliton Stability in 2D NLS
40. Future Work
Continue looking at orders of ρ.
Natalie Sheils Soliton Stability in 2D NLS
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. Acknowledgments
Dr. John Carter of Seattle University
Natalie Sheils Soliton Stability in 2D NLS
43. Acknowledgments
Dr. John Carter of Seattle University
Seattle University College of Science and Engineering
Natalie Sheils Soliton Stability in 2D NLS
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. Questions
Questions?
Natalie Sheils Soliton Stability in 2D NLS
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