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Maxwell Fronts in the Discrete Nonlinear Schrödinger Equations with Competing Nonlinearities

Farrell Theodore Adriano, Hadi Susanto

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 Schrödinger (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.

Maxwell Fronts in the Discrete Nonlinear Schrödinger Equations with Competing Nonlinearities

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 Schrödinger (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.
Paper Structure (12 sections, 1 theorem, 82 equations, 11 figures)

This paper contains 12 sections, 1 theorem, 82 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Quadratic-Cubic DNLS
  4. Anticontinuum Limit
  5. C = 0
  6. C small
  7. Continuum Limit
  8. h = 0
  9. h small
  10. Numerical Results
  11. Cubic-Quintic Nonlinearity
  12. Conclusion

Key Result

theorem 1

Assume that the stationary solution $\bm{\varphi}$ is a front solution satisfying $\varphi_{n} \to \varphi^{\mathrm{low}}$ as $n \to -\infty$ and $\varphi_{n} \to \varphi^{\mathrm{up}}$ as $n \to \infty$, and that $\bm{\varphi}$ does not change signs on $\mathbb{Z}$. Suppose further that $L_{\pm}$ h

Figures (11)

  • Figure 1: Phase portrait of the quadratic-cubic stationary equation \ref{['eqn:quad_cubic_dnls_stationary']} in the continuum limit $C \to \infty$ i.e. Eq. \ref{['eqn:2-3 continuum eqn']} for values of $\mu$ around the Maxwell point $\mu_{M} = -2/9$. The left panel shows the case of $\mu = -0.2244 < \mu_{M}$, the middle panel shows the case $\mu = \mu_{M}$, and the right panel shows the case $\mu = -0.22 > \mu_{M}$. The black curves in each panel represent the homoclinic/heteroclinic orbits connecting the uniform states.
  • Figure 2: Real part of the eigenvalue of $\mathcal{L}$ for the onsite quadratic-cubic Maxwell front for $C \in [0,0.1]$. The black solid line shows the numerically computed eigenvalue, the orange dashed line shows the asymptotic approximation \ref{['eqn:ac_lambda_approx']}, and the green dashed line shows the one-site approximation \ref{['eqn:ac lambda approx one site approx']}.
  • Figure 3: Bifurcation diagrams of quadratic-cubic Maxwell fronts with codes of length $0,1,2$, and $3$. Blue lines represent stable fronts and red lines represent unstable fronts.
  • Figure 4: Profiles (left) and spectrum (right) of quadratic-cubic Maxwell fronts of length $2$ and $3$ for $C = 0.02$.
  • Figure 5: Eigenvalue of the largest real part of onsite and intersite quadratic-cubic Maxwell fronts for $C \in [0,1.1]$. The inset on the left panel shows a zoomed in view of the onsite eigenvalue for $C \in [0.4,0.56]$, the vertical axis of the inset is on a logarithmic scale.
  • ...and 6 more figures

Theorems & Definitions (2)

  • theorem 1
  • proof