Interfaces supporting surface gap soliton ground states in the 1D nonlinear Schrödinger equation

Abstract

We consider the problem of verifying the existence of H 1 ground states of the 1D nonlinear Schrödinger equation for an interface of two periodic structures:and p > 1. The article [T. Dohnal, M. Plum and W. Reichel, "Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation," Comm. Math. Phys. 308, 511-542 (2011)] provides in the 1D case an existence criterion in the form of an integral inequality involving the linear potentials V 1 , V 2 and the Bloch waves of the operators − d 2 dx 2 +V 1,2 −λ. We choose here the classes of piecewise constant and piecewise linear potentials V 1,2 and check this criterion for a set of parameter values. In the piecewise constant case the Bloch waves are calculated explicitly and in the piecewise linear case verified enclosures of the Bloch waves are computed numerically. The integrals in the criterion are evaluated via interval arithmetic so that rigorous existence statements are produced. Examples of interfaces supporting ground states are reported including such, for which ground state existence follows for all periodic Γ 1,2 with ess sup Γ 1,2 > 0.

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

“…We will without loss of generality assume V 1 > V 2 in the following. Examples for elliptic problems involving interfaces modelled by potentials of this kind can be found in [14,Theorem 1], [15,Theorem 2] or [28]. To explain the motivation behind our study, we recall the interesting phenomenon called "double scattering".…”

Rigorous numerics for NLS: Bound states, spectra, and controllability

“…• the existence of surface gap solitons of the 1-D NLS (by numerical verification of an integral inequality) [15];…”