Maxwell Fronts in the Discrete Nonlinear Schr\"odinger Equations with Competing Nonlinearities

TL;DR

This paper explores Maxwell fronts in discrete nonlinear Schr"odinger equations with competing nonlinearities, analyzing their existence, stability, and behavior across different coupling regimes using advanced asymptotic methods.

Contribution

It characterizes Maxwell fronts in DNLS models with quadratic-cubic and cubic-quintic nonlinearities, providing new stability analysis and insights into multistability and front dynamics.

Findings

Maxwell fronts exist at the Maxwell point in models with competing nonlinearities.

Stability of Maxwell fronts varies across coupling regimes, analyzed via eigenvalue methods.

Novel asymptotic techniques reveal behavior of fronts in the continuum limit.

Abstract

In discrete nonlinear systems, the study of nonlinear waves has revealed intriguing phenomena in various fields such as nonlinear optics, biophysics, and condensed matter physics. Discrete nonlinear Schr\"odinger (DNLS) equations are often employed to model these dynamics, particularly in the context of Bose-Einstein condensates and optical waveguide arrays. While the classical DNLS with cubic nonlinearity admits well-known solitonic solutions, the introduction of competing nonlinearities, such as quadratic-cubic and cubic-quintic terms, gives rise to new behaviors, including multistability and front formation. One such emergent structure, the Maxwell front, is characterized by stationary interfaces between two energetically equivalent steady states, occurring at a critical parameter known as the Maxwell point. This paper investigates the existence and stability of Maxwell fronts in DNLS models with competing nonlinearities. Specifically, we examine the quadratic-cubic nonlinearity, as found in the discrete quantum droplets equation, and the cubic-quintic nonlinearity, both of which exhibit multistability. We explore the persistence of Maxwell fronts in both the anticontinuum limit (where the coupling between lattice sites is weak) and the continuum limit (where the coupling is strong). The stability of these fronts is analyzed through linear stability analysis, utilizing eigenvalue counting arguments and exponential asymptotic techniques. Our results provide new insights into multistability, front dynamics, and the role of competing nonlinearities in discrete wave systems. The main contributions of this work include the characterization of Maxwell fronts in DNLS equations with competing nonlinearities, the analysis of their stability across different coupling regimes, and the application of novel asymptotic methods to investigate their behavior in the continuum limit.