Electronic Journal of Differential Equations (Jun 2014)
Stability of solitary waves for a three-wave interaction model
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.
