Exponential asymptotics for line solitons in two-dimensional periodic potentials

TL;DR

This paper develops an exponential asymptotics theory for line solitons in two-dimensional periodic potentials, revealing bifurcation behaviors and constructing multi-line-soliton states with analytical and numerical validation.

Contribution

It introduces a direct exponential asymptotics method for 2D problems, overcoming recurrence relation limitations and analyzing soliton bifurcations from Bloch-band edges and interior points.

Findings

Line solitons bifurcate from every Bloch-band edge for rational slopes.

Two line-soliton families exist for each rational slope.

A countable set of multi-line-soliton bound states can be analytically constructed.

Abstract

As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multi-scale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one-dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence-relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch-band edge; and for each rational slope, two line-soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multi-line-soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.