Existence of Generalized Heteroclinic Solutions of the Coupled Schr¨odinger System under a Small Perturbation

The following coupled Schrodinger system with a small perturbation
is considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.

Existence of Generalized Heteroclinic Solutions of the Coupled Schrodinger System under a Small Perturbation

The following coupled Schrödinger system with a small perturbation $$\begin{array}{*{20}c} {u_{xx} + u - u^3 + \beta uv^2 + \varepsilon f(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R},} \\ {v_{xx} - v + v^3 + \beta u^2 v + \varepsilon g(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R}} \\ \end{array}$$ is considered, where β and ∈ are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution (called the generalized heteroclinic solution thereafter).