Existence of generalized heteroclinic solutions of the coupled Schrödinger system under a small perturbation
Shengfu Deng , Boling Guo , Tingchun Wang
Chinese Annals of Mathematics, Series B ›› 2014, Vol. 35 ›› Issue (6) : 857 -872.
Existence of generalized heteroclinic solutions of the coupled Schrödinger 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).
Coupled Schrödinger system / Heteroclinic solutions / Reversibility
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