poster: On-Site and Off-Site Solitary Waves of the Discrete Nonlinear Schrödinger Equation in Multiple Dimensions
speaker: Michael Jenkinson (Dept. of Applied Physics and Applied Math, Columbia University)abstract:
We construct several types of symmetric localized standing waves (solitons) to the \(d\)-dimensional discrete nonlinear Schrödinger equation (DNLS) with cubic nonlinearity for \(d = 1,2,3\) : \(i \partial_t u_n = h^{-2} (\delta^2 \vec{u})_n -
u_n
^2 u_n, \), where \(\delta^2\) denotes the discrete Laplacian on \(\mathbb{Z}^d\), using bifurcation methods about the continuum limit. Such waves and their energy differences play a role in the propagation of localized states of DNLS across the lattice. The energy differences, which we prove to exponentially small in a natural parameter, are related to the ”Peierls-Nabarro Barrier” in discrete systems, first investigated by M. Peyrard and M.D. Kruskal (1984).
Joint work with Michael I. Weinstein
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