[Paper Review] New findings for the old problem: Exact solutions for domain walls in coupled real Ginzburg-Landau equations

Boris A. Malomed|arXiv (Cornell University)|Oct 27, 2021

Nonlinear Dynamics and Pattern Formation70 references58 citations

TL;DR

This paper presents new exact analytical solutions for domain wall (DW) states in coupled real Ginzburg-Landau (GL) equations, modeling systems like Rayleigh-Bénard convection, nonlinear optics, and Bose-Einstein condensates (BECs). It derives exact asymmetric DW solutions in an asymmetric diffusion system (vanishing diffusion in one component), symmetric DWs with linear coupling, composite states with bright solitons, and DW-like source/sink solutions for counterpropagating waves—each existing in continuous families with at least one free parameter, extending beyond prior isolated solutions.

ABSTRACT

This work reports new exact solutions for domain-wall (DW) states produced by a system of coupled real Ginzburg-Landau (GL) equations which model patterns in thermal convection, optics, and Bose-Einstein condensates (BECs). An exact solution for symmetric DW was known for a single value of the cross-interaction coefficient, G = 3 (defined so that its self-interaction counterpart is 1). Here an exact asymmetric DW is obtained for the system in which the diffusion term is absent in one component. It exists for all G > 1. Also produced is an exact solution for DW in the symmetric real-GL system which includes linear coupling. In addition, an effect of a trapping potential on the DW is considered, which is relevant to the case of BEC. In a system of three GL equations, an exact solution is obtained for a composite state including a two-component DW and a localized state in the third component. Bifurcations which create two lowest composite states are identified too. Lastly, exact solutions are found for the system of real GL equations for counterpropagating waves, which represent a sink or source of the waves, as well as for a system of three equations which includes a standing localized component.

Motivation & Objective

• To extend the known class of exact domain wall (DW) solutions in coupled real Ginzburg-Landau (GL) equations beyond the previously known isolated symmetric solution at G = 3.
• To address the lack of exact solutions for asymmetric systems, particularly when diffusion is absent in one component.
• To investigate the role of linear coupling and trapping potentials in generating new families of DW states.
• To construct composite solutions involving DWs in two components and localized states (e.g., bright solitons) in a third component.
• To provide exact solutions for systems modeling counterpropagating waves, including source and sink modes, which were previously only studied numerically.

Proposed method

• Derivation of exact analytical solutions using variational and ansatz-based methods for coupled real GL equations with specific parameter symmetries.
• Application of the Lyapunov functional formalism to ensure stability and guide solution construction in gradient systems.
• Use of a traveling-wave ansatz and symmetry reduction to simplify the system of PDEs into ODEs for stationary solutions.
• Incorporation of linear coupling terms between components to model inter-component interactions in BECs and optical systems.
• Introduction of a harmonic trapping potential to model experimental BEC setups, with exact solutions derived under this condition.
• Extension to three-component systems by introducing a localized third component coupled to a two-component DW, using matched asymptotic or direct solution techniques.

Experimental results

Research questions

• RQ1 Can exact asymmetric domain wall solutions be derived in a coupled real GL system where diffusion is absent in one component?
• RQ2 What are the conditions under which exact DW solutions exist in a symmetric system with linear coupling between components?
• RQ3 Can composite states consisting of a two-component domain wall and a localized state in a third component be constructed exactly?
• RQ4 Do exact solutions exist for systems modeling counterpropagating waves, such as source or sink modes in binary fluid convection?
• RQ5 How do bifurcations generate the lowest-energy composite states in three-component GL systems?

Key findings

• An exact asymmetric domain wall solution is derived for the system with zero diffusion in one component, valid for all cross-interaction coefficients G > 1.
• An exact symmetric domain wall solution is obtained in the presence of linear coupling between components, forming a continuous family of solutions.
• A composite state is constructed with a two-component domain wall and a bright soliton in the third component, representing a bound state of topological and localized excitations.
• Exact solutions for source and sink modes are found in a system of GL equations for counterpropagating waves, modeling traveling-wave convection in binary fluids.
• A new composite solution is derived for a system of three equations, featuring a kink or source formed by two counterpropagating components coupled to a standing localized mode in the third.
• All new solutions exist in continuous families with at least one free parameter, contrasting with the previously known isolated solution at G = 3.