×

Multi-dimensional homoclinic jumping and the discretized NLS equation. (English) Zbl 0957.37067

Author’s abstract: The author studies the existence of orbits which are depart from, and return to some invariant sets in a differential system being close to integrable under some rather sophisticated restrictions. The motivation of these restrictions is an application to some discretized variant of the perturbed nonlinear Schrödinger equation. Roughly speaking, the situation under the study can be described as follows.
Suppose some domain in \(\mathbb{R}^{2(n+m+1)}\) with usual symplectic structure possesses an invariant \(2m\)-dimensional symplectic submanifold \(N\) such that the system restricted to \(N\) is integrable and filled with invariant \(m\)-dimensional tori, and one of them, \(C\), consists of equilibria. These equilibria in transverse direction to \(N\) have each one pair of nonzero real eigenvalues and \(n\) pairs of purely imaginary eigenvalues. Eigenvalues and related eigenvectors are supposed not to depend on the base point on \(C\). There are \(2(n+m)\)-dimensional local center manifold \(M_0\) being normally hyperbolic due to the presence of real eigenvalues, and codimension 1 stable and unstable manifolds. It is supposed these locally stable and unstable manifolds coincide, forming two homoclinic manifolds. Furthermore, it is assumed each of these two homoclinic manifolds contains a one-parameter family of heteroclinic orbits that connect points on the torus \(C\) and, in addition, if one takes a heteroclinic orbit in the first family, and \(x_-\), \(x_+\) are its limiting points on \(C\), then there exists a heteroclinic orbit in the second family with the same limiting points. One further assumption requires the persistence of \(N\) under the perturbation. The author proves the existence of multi-pulse (jumping) homoclinic or heteroclinic orbits that are doubly asymptotic to the perturbed center manifold \(M_{\epsilon}\). These orbits are contained in the unstable manifold of \(N\), and in some cases also in the stable manifold of \(N\).

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI