Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation

Abstract

We consider the nonlinear Schrödinger equation $(-\Delta +V(x))u=\Gamma (x)|u{|}^{p-1}u$, $x\in {\mathbb{R}}^n$ with $V(x)=V_1(x){\chi }_{\{x_1>0\}}(x)+V_2(x){\chi }_{\{x_1<0\}}(x)$ and $\Gamma (x)={\Gamma }_1(x){\chi }_{\{x_1>0\}}(x)+{\Gamma }_2(x){\chi }_{\{x_1<0\}}(x)$ and with $V_1,V_2,{\Gamma }_1,{\Gamma }_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min \sigma (-\Delta +V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1,\Gamma \equiv {\Gamma }_1$ and $V\equiv V_2,\Gamma \equiv {\Gamma }_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\frac{d^2}{d{x}^2}+V_1(x)$ and $-\frac{d^2}{d{x}^2}+V_2(x)$.

definition of ground and bound states added, assumption (H2) weakened (sign changing nonlinearity is now allowed); 33 pages, 4 figures

References in corpus (6)

• Bose-Einstein Condensates in Optical Lattices: Band-Gap Structure and Solitons
• Observation of surface gap solitons in semi-infinite waveguide arrays
• Observation of two-dimensional surface solitons in anisotropic waveguide arrays
• Observation of two-dimensional lattice interface solitons
• Surface solitons at interfaces of arrays with spatially-modulated nonlinearity
• Surface gap solitons at a nonlinearity interface

Cited by in corpus (2)

• Bound States for Nano-Tubes with a Dislocation
• Mean--field stability for the junction of quasi 1D systems with Coulomb interactions