Interfaces Supporting Surface Gap Soliton Ground States in the 1D Nonlinear Schroedinger Equation

TL;DR

This paper verifies the existence of ground states for the 1D nonlinear Schrödinger equation with interface conditions using explicit and numerical methods, providing rigorous existence results for various potential configurations.

Contribution

It applies an integral inequality criterion to piecewise constant and linear potentials, combining explicit calculations and interval arithmetic to rigorously confirm ground state existence.

Findings

Explicit Bloch wave calculations for piecewise constant potentials.

Numerical enclosures of Bloch waves for piecewise linear potentials.

Existence of ground states for all periodic nonlinearities with positive essential supremum.

Abstract

We consider the problem of verifying the existence of $H^1$ ground states of the 1D nonlinear Schr\"odinger equation for an interface of two periodic structures: $$-u" +V(x)u -\lambda u = \Gamma(x) |u|^{p-1}u \ {on} \R$$ with $V(x) = V_1(x), \Gamma(x)=\Gamma_1(x)$ for $x\geq 0$ and $V(x) = V_2(x), \Gamma(x)=\Gamma_2(x)$ for $x<0$. Here $V_1,V_2,\Gamma_1,\Gamma_2$ are periodic, $\lambda <\min\sigma(-\tfrac{d^2}{dx^2}+V)$, and $p>1$. The article [T. Dohnal, M. Plum and W. Reichel, "Surface Gap Soliton Ground States for the Nonlinear Schr\"odinger Equation," \textit{Comm. Math. Phys.} \textbf{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 $-\tfrac{d^2}{dx^2}+V_{1,2}-\lambda$. 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 $\Gamma_ {1,2}$ with $\esssup \Gamma_{1,2}>0$.