[Paper Review] Dark-bright solitons in coupled NLS equations with unequal dispersion coefficients

E. G. Charalampidis, P. G. Kevrekidis+2 more|arXiv (Cornell University)|Jul 4, 2014

Advanced Mathematical Physics Problems3 citations

TL;DR

This paper investigates dark-bright soliton solutions in a two-component nonlinear Schrödinger system with unequal dispersion coefficients, treating the dark soliton as a frozen background to study bright soliton formation. It identifies bifurcation points in the linear limit, analyzes stability across ground and excited states via numerical simulations, and reveals regimes of stability for multi-peak bright solitons with zero crossings.

ABSTRACT

We study a two-component nonlinear Schr\odinger system with repulsive nonlinear interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in the one-component. Treating it as a frozen one, we explore the possibility of the formation of bright solitonic bound states in the other component. We identify bifurcation points of such states in the linear limit for the bright component, and explore their continuation in the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability not only of the single-peak ground state (the dark-bright solitary wave), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability manifestation.

Motivation & Objective

• To understand the formation of bright solitonic bound states in a two-component nonlinear Schrödinger system with repulsive interactions and unequal dispersion coefficients.
• To explore the conditions under which stable bright solitons can coexist with a dark solitary wave in one component.
• To identify bifurcation points in the linear limit that give rise to bright soliton solutions.
• To investigate the stability of both ground states and excited states of the bright component, including those with zero crossings.
• To numerically determine regimes of stability and simulate the dynamical outcomes when instability occurs.

Proposed method

• Formulates a two-component coupled nonlinear Schrödinger system with distinct dispersion coefficients for each component and repulsive nonlinearities.
• Treats the dark soliton in one component as a frozen background, reducing the problem to a nonlinear eigenvalue problem for the bright component.
• Applies linear stability analysis in the limit of vanishing nonlinearity to locate bifurcation points for bright soliton solutions.
• Explores an analytically tractable limit where the dispersion coefficient of the bright component vanishes.
• Uses direct numerical simulations to assess the stability of identified states and to study the dynamics of instability.
• Employs continuation techniques to trace the evolution of soliton states from the linear to the nonlinear regime.

Experimental results

Research questions

• RQ1 Under what conditions can a bright soliton form as a bound state on a dark soliton background in a two-component NLS system with unequal dispersion?
• RQ2 How do bifurcation points in the linear limit influence the emergence of stable bright soliton solutions in the nonlinear regime?
• RQ3 What are the stability characteristics of excited bright soliton states with one or more zero crossings in the bright component?
• RQ4 In what parameter regimes do dark-bright solitons remain dynamically stable under perturbations?
• RQ5 What are the dynamical outcomes when dark-bright soliton states become unstable, as revealed by direct simulations?

Key findings

• Bifurcation points for bright soliton solutions are identified in the linear limit, marking the onset of nonlinear bound states.
• A regime of potential stability is found for the single-peak ground state (dark-bright soliton) across a range of dispersion parameter ratios.
• Excited states of the bright component, including those with one or more zero crossings, are numerically identified as potentially stable in specific parameter regimes.
• The limit of vanishing dispersion in the bright component provides an analytically tractable framework for understanding soliton formation.
• When instability occurs, direct numerical simulations show that the system evolves into delocalized or fragmented structures, depending on the initial conditions and parameter values.
• The study reveals that stability is sensitive to the ratio of dispersion coefficients and the presence of zero crossings in the bright component's wave function.