[Paper Review] Mixed solitons in (2+1) dimensional multicomponent long-wave--short-wave system

T. Kanna, M. Vijayajayanthi+1 more|arXiv (Cornell University)|Dec 1, 2012

Nonlinear Waves and Solitons3 citations

TL;DR

This paper derives a (2+1)-dimensional multicomponent long-wave–short-wave resonance interaction (LSRI) system for weak nonlinear dispersive waves in a Kerr-type medium and applies Hirota’s bilinear method to construct mixed bright-dark soliton solutions. The key finding is that in systems with three or more short-wave components, bright solitons exhibit energy-exchanging collisions due to higher-dimensional coupling, while dark solitons undergo standard elastic collisions, revealing rich dynamics governed by soliton parameters and ω-j terms that control bound states and collision behavior.

ABSTRACT

We derive a (2+1)-dimensional multicomponent long-wave$-$short-wave resonance interaction (LSRI) system as the evolution equation for propagation of $N$-dispersive waves in weak Kerr type nonlinear medium in the small amplitude limit. The mixed (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher dimensional LSRI system, are obtained by applying the Hirota's bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for $N$-component LSRI system with $N>3$, if the bright solitons appear in atleast two components, interesting collision behaviour takes place resulting in energy exchange among the bright solitons. However the dark solitons undergo standard elastic collision accompanied by a position-shift and a phase-shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.

Motivation & Objective

• To derive a general (2+1)-dimensional multicomponent LSRI system for N-dispersive waves in a weak Kerr nonlinear medium using multiple scale perturbation.
• To identify integrable cases of the multicomponent LSRI system by selecting specific parameter choices.
• To construct mixed (bright-dark) one- and two-soliton solutions using Hirota’s direct method for the integrable system.
• To analyze the collision dynamics of mixed solitons, particularly the emergence of energy-sharing collisions in systems with more than two short-wave components.
• To investigate the role of ωj-parameters and α-parameters in controlling soliton bound states and collision behavior in higher-dimensional settings.

Proposed method

• Derives the (2+1)-dimensional multicomponent LSRI system via the multiple scale perturbation method in the small amplitude limit for N-dispersive waves in a nonlinear medium.
• Reduces the general system to an integrable form by selecting specific parameter values, enabling exact soliton solutions.
• Applies Hirota’s bilinearization method to the integrable system to derive exact one- and two-soliton solutions, including mixed bright-dark solitons.
• Analyzes soliton dynamics by varying the ωj-parameters, which arise from the higher-dimensional nature of the system and influence soliton binding and collision.
• Uses the α-parameters to control beating effects in bound states and to tune the interaction behavior of solitons.
• Performs numerical simulations to visualize soliton bound states and collision processes in the x–y plane for different parameter regimes.

Experimental results

Research questions

• RQ1 How does the inclusion of multiple short-wave components in a (2+1)-dimensional LSRI system affect the collision dynamics of mixed bright-dark solitons?
• RQ2 What role do the ωj-parameters—arising from the higher-dimensional nature of the system—play in transitioning solitons from bound states to colliding states?
• RQ3 Can energy-exchanging collisions between bright solitons occur in a multicomponent LSRI system with three or more short-wave components, and what conditions enable this?
• RQ4 How do the α-parameters influence the stability and beating effects in two-soliton bound states, particularly when the one-soliton dynamics remain unaffected?
• RQ5 What is the nature of the interaction between bright and dark parts of mixed solitons in systems with two short-wave components versus those with three or more?

Key findings

• In the two short-wave components case, the bright and dark parts of mixed solitons behave like scalar solitons, exhibiting standard elastic collisions without energy exchange.
• For N > 3 components, particularly in the three short-wave component case with two bright and one dark part, the bright solitons undergo energy-exchanging collisions characterized by intensity redistribution and amplitude-dependent position shifts.
• The dark soliton parts in all configurations undergo only standard elastic collisions with phase and position shifts, independent of the bright component dynamics.
• The ωj-parameters, which emerge due to the (2+1)-dimensional structure, control the transition from bound soliton states to colliding solitons: when ωj = 0, bound states form; when ωj ≠ 0, solitons collide elastically.
• The α-parameters, while having negligible effect on single-soliton propagation, significantly influence bound states and can be tuned to suppress beating effects in two-soliton bound states.
• In the three-component case with non-zero ωj, the system supports energy-exchanging collisions of bright parts in the x–y plane, a phenomenon absent when ωj = 0, which leads to only stationary bound states.