Homoclinic degeneracy for perturbed nonlinear Schrödinger equation

In this Letter, we consider a nonlinear Schrödinger (NLS) equation iut+uxx=−(|u|2−1)uiut+uxx=−(|u|2−1)u and obtain the explicit expressions of two new series of homoclinic and heteroclinic orbit solutions by utilizing the bilinear forms. When this equation is perturbed, we find that the homoclinic orbits degenerate but the heteroclinic orbits still exist. At the same time, the explicit expression of heteroclinic orbit solution to perturbed NLS equation is given by applying variable transformation. Moreover, the relation between global attractor and heteroclinic orbit is investigated.