Intrinsic localized modes for DNLS equation with competing nonlinearities: bifurcations

TL;DR

This paper investigates intrinsic localized modes in DNLS equations with competing nonlinearities, analyzing bifurcations and solution branches as parameters vary, revealing new nonsymmetric ILMs and bifurcation scenarios.

Contribution

It introduces a numerical continuation method for ILMs in DNLS with competing nonlinearities, identifying bifurcation structures and solutions without anti-continuum counterparts.

Findings

All ILM branches originate at anti-continuum limit except a finite number.

Existence of nonsymmetric ILMs with no ACL counterparts.

Two unlimited $ ext{infinity}$-branches exist for each $ ext{gamma}$, undergoing bifurcations.

Abstract

We study nonlinear excitations described by DNLS-type equations with so-called competing nonlinearities. These are the nonlinearities that consist of two power terms with coefficients of different sign. A key feature of these models is the presence of two governing parameters: $\alpha$, which characterizes the coupling between lattice sites, and $\gamma$, which quantifies the balance between competing nonlinearities. Our study focuses on intrinsic localized modes (ILMs) -- solutions that exhibit spatial localization over a few lattice sites. The basic example for our study is the cubic-quartic equation that recently has been used to describe 3D BEC cloud in the mean field approximation with Lee-Huang-Yang corrections. We employ numerical continuation from the anti-continuum limit (ACL) where the coupling between the lattice sites is neglected (the case $\alpha=0$). We analyze $\alpha$-dependent branches of the basic ILMs and their bifurcations when $\gamma$ varies. Our study shows that all branches of ILMs originated at anti-continuum limit, except a finite number, bifurcate and do not exist for large values of $\alpha$. We present tables of bifurcations for the ILMs that involve not more than 3 excited lattice sites. It is shown that the model supports nonsymmetric ILMs that have no counterparts in ACL. Also we study the branches of ILMs that can be continued unlimitedly when $\alpha\to \infty$ (called here $\infty$-branches). It was found that for any $\gamma$ there are exactly two (up to symmetries) $\infty$-branches. When $\gamma$ grows these branches undergo a sequence of bifurcations. Finally, we compare our results with the results for the quadratic-cubic equation and the cubic-quintic equation and found no qualitative difference in (a) tables of bifurcations, (b) presence of solutions without ACL counterpart and (c) scenario of switching of $\infty$-branches.