Electron. J. Diff. Equ., Vol. 2014 (2014), No. 153, pp. 1-9.

Stability of solitary waves for a three-wave interaction model

Orlando Lopes

Abstract:
In this article we consider the normalized one-dimensional three-wave interaction model
$$\displaylines{ i\frac{\partial z_1}{\partial t}=- \frac{d^2z_1}{dx^2}- z_3{\bar z}_2\cr i\frac{\partial z_2}{\partial t}=- \frac{d^2z_2}{dx^2}- z_3{\bar z}_1\cr i\frac{\partial z_3}{\partial t}=- \frac{d^2z_3}{dx^2}- z_1z_2. }$$
Solitary waves for this model are solutions of the form
$$ z_1(t,x)=e^{i\omega_1 t} u_1(x)\quad z_2(t,x)=e^{i\omega_2 t} u_2(x)\quad z_3(t,x) =e^{i(\omega_1+\omega_2) t} u_3(x), $$
where $\omega_1$ and $\omega_2$ are positive frequencies, and $u_i(x)$, $i=1,2,3$ are real-valued functions that satisfy the ODE system
$$\displaylines{ - \frac{d^2u_1}{dx^2} - u_2u_3+\omega_1u_1=0 \cr - \frac{d^2u_2}{dx^2} - u_1u_3+\omega_2u_2=0 \cr - \frac{d^2u_3}{dx^2} - u_1u_2+(\omega_1+\omega_2)u_3=0. }$$
For the case $\omega_1=\omega_2=\omega$, we prove existence, uniqueness and stability of solitary waves corresponding to positive solutions $u_i(x)$ that tend to zero as x tends to infinity. The full model has more parameters, and the case we consider corresponds to the exact phase matching. However, as we will see, even in the simpler case, a formal proof of stability depends on a nontrivial spectral analysis of the linearized operator. This is so because the spectral analysis depends on some calculations on a full neighborhood of the parameter $(\omega,\omega)$ and the solution is not known explicitly.

Submitted January 10, 2013. Published June 30, 2014.
Math Subject Classifications: 34A34.
Key Words: Dispersive equations; variational methods; stability.

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Orlando Lopes
IMEUSP- Rua do Matao, 1010, Caixa postal 66281
CEP: 05315-970, Sao Paulo, SP, Brazil
email: olopes@ime.usp.br

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