Soliton Stability of the 2D Nonlinear Schrödinger Equation
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Soliton Stability in 2D NLS
Natalie Sheils
sheilsn@seattleu.edu
April 10, 2010
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLS
I Trivial-Phase Solutions of NLSI Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLSI Trivial-Phase Solutions of NLS
I Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLSI Trivial-Phase Solutions of NLSI Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLSI Trivial-Phase Solutions of NLSI Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLSI Trivial-Phase Solutions of NLSI Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Outline
1. Introduction to NLSI Trivial-Phase Solutions of NLSI Soliton Solution of NLS
2. Linear Stability
3. High-Frequency Limit
4. Future Work
Natalie Sheils Soliton Stability in 2D NLS
Introduction to NLS
The two-dimensional cubic nonlinear Schrodinger equation (NLS),
iψt + ψxx − ψyy + 2|ψ|2ψ = 0.
Among many other physical phenomena, NLS arises as a model of
I pulse propagation along optical fibers.
I surface waves on deep water.
Natalie Sheils Soliton Stability in 2D NLS
Introduction to NLS
The two-dimensional cubic nonlinear Schrodinger equation (NLS),
iψt + ψxx − ψyy + 2|ψ|2ψ = 0.
Among many other physical phenomena, NLS arises as a model of
I pulse propagation along optical fibers.
I surface waves on deep water.
Natalie Sheils Soliton Stability in 2D NLS
Introduction to NLS
The two-dimensional cubic nonlinear Schrodinger equation (NLS),
iψt + ψxx − ψyy + 2|ψ|2ψ = 0.
Among many other physical phenomena, NLS arises as a model of
I pulse propagation along optical fibers.
I surface waves on deep water.
Natalie Sheils Soliton Stability in 2D NLS
Introduction to NLS
The two-dimensional cubic nonlinear Schrodinger equation (NLS),
iψt + ψxx − ψyy + 2|ψ|2ψ = 0.
Among many other physical phenomena, NLS arises as a model of
I pulse propagation along optical fibers.
I surface waves on deep water.
Natalie Sheils Soliton Stability in 2D NLS
Introduction to NLS
Figure: Wave tank in the Pritchard Fluid Mechanics Labratory in theMathematics Department at Penn State University.
Natalie Sheils Soliton Stability in 2D NLS
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
Trivial-Phase Solutions of NLS
A specific NLS solution ψ is called a soliton solution.
ψ(x , t) = sech(x)e it
-20 -10 10 20x
0.2
0.4
0.6
0.8
1.0ΨHx, 0L
Natalie Sheils Soliton Stability in 2D NLS
Trivial-Phase Solutions of NLS
A specific NLS solution ψ is called a soliton solution.
ψ(x , t) = sech(x)e it
-20 -10 10 20x
0.2
0.4
0.6
0.8
1.0ΨHx, 0L
Natalie Sheils Soliton Stability in 2D NLS
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+ εu + iεv +O(ε2))
where ε is a small real constant and u = u(x , y , t) andv = v(x , y , t) are real-valued functions.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
We substitute into NLS and simplify. We know ε is small, so theterms with the lowest order of ε are dominant. The O(ε0) termscancel out so O(ε1) is the leading order.
−ut = vxx − vyy + (2φ2(x)− 1)v
vt = uxx − uyy + (6φ2(x)− 1)u(1)
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Since the coefficients of the perturbed system (1) do not explicitlydepend 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:
I If any Ω has positive real part, the solution is unstable.
I If all Ω have negative real parts, the solution is stable.
I If all Ω are purely imaginary, the solution is stable.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Since the coefficients of the perturbed system (1) do not explicitlydepend 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:
I If any Ω has positive real part, the solution is unstable.
I If all Ω have negative real parts, the solution is stable.
I If all Ω are purely imaginary, the solution is stable.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Since the coefficients of the perturbed system (1) do not explicitlydepend 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:
I If any Ω has positive real part, the solution is unstable.
I If all Ω have negative real parts, the solution is stable.
I If all Ω are purely imaginary, the solution is stable.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Since the coefficients of the perturbed system (1) do not explicitlydepend 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:
I If any Ω has positive real part, the solution is unstable.
I If all Ω have negative real parts, the solution is stable.
I If all Ω are purely imaginary, the solution is stable.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Since the coefficients of the perturbed system (1) do not explicitlydepend 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:
I If any Ω has positive real part, the solution is unstable.
I If all Ω have negative real parts, the solution is stable.
I If all Ω are purely imaginary, the solution is stable.
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Then, U and V satisfy the following differential equations.
ΩU = V − ρ2V − 2Vφ2 − V ′′
−ΩV = U − ρ2U − 6Uφ2 − U ′′(2)
Natalie Sheils Soliton Stability in 2D NLS
Linear Stability
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
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Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
In our linear stability problem (2) we assume ρ is large and
U(x) ∼ u0(µx) + ρ−1u1(µx) + ρ−2u2(µx) + . . .
V (x) ∼ v0(µx) + ρ−1v1(µx) + ρ−2v2(µx) + . . .
µ ∼ ρ+ µ0 + µ1ρ−1 + µ2ρ
−2 + . . .
Ω ∼ ω−2ρ2 + ω3ρ
−3.
Pick z = µx .
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
At leading order in ρ, equation (2) becomes:
v(4)0 + 2v ′′0 + (1 + ω2
−2)v0 = 0.
In solving this equation, we want v0 to be bounded. This impliesthat ω−2 is purely imaginary and −1 < iω−2 < 1. Pick w = iω−2 .
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
At leading order in ρ, equation (2) becomes:
v(4)0 + 2v ′′0 + (1 + ω2
−2)v0 = 0.
In solving this equation, we want v0 to be bounded. This impliesthat ω−2 is purely imaginary and −1 < iω−2 < 1. Pick w = iω−2 .
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
Now we have
v0 = c1ez√−w−1 + c2e
−z√−w−1 + c3e
z√
w−1 + c4e−z√
w−1
where ci ’s are complex constants.
If v0 is bounded, u0 is bounded and we find u0 to be
u0 = −ic1ez√−w−1 − ic2e
−z√−w−1 + ic3e
z√
w−1 + ic4e−z√
w−1.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
Now we have
v0 = c1ez√−w−1 + c2e
−z√−w−1 + c3e
z√
w−1 + c4e−z√
w−1
where ci ’s are complex constants.
If v0 is bounded, u0 is bounded and we find u0 to be
u0 = −ic1ez√−w−1 − ic2e
−z√−w−1 + ic3e
z√
w−1 + ic4e−z√
w−1.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
The next order of ρ is O(ρ):
v(4)1 + 2v ′′1 + (1− w2)v1 = −2iwµ0u
′′0 − 2µ0v
′′0 − 2µ0v
(4)0 .
We want v1 to be bounded so we require the right-hand side of theequation to be orthogonal to the solution of the homogeneousequation.
In this case, we require our right-hand side to be zero. Then wehave the following restriction:
µ0 = 0.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
The next order of ρ is O(ρ):
v(4)1 + 2v ′′1 + (1− w2)v1 = −2iwµ0u
′′0 − 2µ0v
′′0 − 2µ0v
(4)0 .
We want v1 to be bounded so we require the right-hand side of theequation to be orthogonal to the solution of the homogeneousequation.
In this case, we require our right-hand side to be zero. Then wehave the following restriction:
µ0 = 0.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
The next order of ρ is O(ρ):
v(4)1 + 2v ′′1 + (1− w2)v1 = −2iwµ0u
′′0 − 2µ0v
′′0 − 2µ0v
(4)0 .
We want v1 to be bounded so we require the right-hand side of theequation to be orthogonal to the solution of the homogeneousequation.
In this case, we require our right-hand side to be zero. Then wehave the following restriction:
µ0 = 0.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
Now we have
v1 = c5ez√−w−1 + c6e
−z√−w−1 + c7e
z√
w−1 + c8e−z√
w−1
and
u1 = −ic5ez√−w−1 − ic6e
−z√−w−1 + ic7e
z√
w−1 + ic8e−z√
w−1.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
I For the next few orders of ρ the general solution of thehomogeneous problem is the same as the previous orders.
I We need to make sure the particular solution of thenonhomogeneous equation is bounded.
I µ1 ∈ RI µ2=0.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
I For the next few orders of ρ the general solution of thehomogeneous problem is the same as the previous orders.
I We need to make sure the particular solution of thenonhomogeneous equation is bounded.
I µ1 ∈ RI µ2=0.
Natalie Sheils Soliton Stability in 2D NLS
High-Frequency Limit
I For the next few orders of ρ the general solution of thehomogeneous problem is the same as the previous orders.
I We need to make sure the particular solution of thenonhomogeneous equation is bounded.
I µ1 ∈ RI µ2=0.
Natalie Sheils Soliton Stability in 2D NLS
Future Work
I Continue looking at orders of ρ.
I We hope to find that ω3 is the first ωi with nonzero real part.
Natalie Sheils Soliton Stability in 2D NLS
Future Work
I Continue looking at orders of ρ.
I We hope to find that ω3 is the first ωi with nonzero real part.
Natalie Sheils Soliton Stability in 2D NLS
Future Work
I Continue looking at orders of ρ.
I We hope to find that ω3 is the first ωi with nonzero real part.
Natalie Sheils Soliton Stability in 2D NLS
Acknowledgments
I Dr. John Carter of Seattle University
I Seattle University College of Science and Engineering
I Pacific Northwest Section of the Mathematical Association ofAmerica
Natalie Sheils Soliton Stability in 2D NLS
Acknowledgments
I Dr. John Carter of Seattle University
I Seattle University College of Science and Engineering
I Pacific Northwest Section of the Mathematical Association ofAmerica
Natalie Sheils Soliton Stability in 2D NLS
Acknowledgments
I Dr. John Carter of Seattle University
I Seattle University College of Science and Engineering
I Pacific Northwest Section of the Mathematical Association ofAmerica
Natalie Sheils Soliton Stability in 2D NLS
Questions
Questions?
Natalie Sheils Soliton Stability in 2D NLS
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