Stability of solitary waves for a three-wave interaction model

Lopes, Orlando

Electronic journal of differential equations, Tome 2014 (2014)

Résumé

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

$\displaylines{ - \frac{d^2u_1}{dx^2} - u_2u_3+\omega_1u_1=0 \cr - \frac{d^2u_2}{dx^2} - u_1u_3+\omega_2u_2=0 \cr - \frac{d^2u_3}{dx^2} - u_1u_2+(\omega_1+\omega_2)u_3=0. }$

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.

Détail

Classification : 34A34
Keywords: dispersive equations, variational methods, stability