Abstract
This paper is devoted to study a coupled Schrödinger system with a small perturbation
where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schrödinger-KdV system.
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Deng, S., Guo, B. Generalized Homoclinic Solutions of a Coupled Schrödinger System Under a Small Perturbation. J Dyn Diff Equat 24, 761–776 (2012). https://doi.org/10.1007/s10884-012-9274-1
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DOI: https://doi.org/10.1007/s10884-012-9274-1