On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schr\"{o}dinger equation

TL;DR

This paper constructs and analyzes symmetry preserving and broken soliton solutions of a nonlocal nonlinear Schrödinger equation, revealing complex collision behaviors and stability conditions through Darboux transformation methods.

Contribution

It introduces explicit multi-soliton solutions for the nonlocal NLS equation, highlighting novel stability properties and collision scenarios involving symmetry breaking.

Findings

Bright solitons are stable only with combined field and conjugate field.

Dark/antidark solitons are stable for individual fields and exhibit contrasting behaviors.

Multiple collision scenarios with exponential combinations are classified and demonstrated.

Abstract

We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schr\"{o}dinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show that they exhibit stable structures only when we combine the field and parity transformed complex conjugate field. Further, we derive two dark/antidark soliton solution with the help of DT method. Unlike the bright soliton case, dark/antidark soliton solution exhibits stable structure for the field and the parity transformed conjugate field separately. In the dark/antidark soliton solution case we observe a contrasting behaviour between the envelope of the field and parity transformed complex conjugate envelope of the field. For a particular parametric choice, we get dark (antidark) soliton for the field while the parity transformed complex conjugate field exhibits antidark (dark) soliton. Due to this surprising result, both the field and PT transformed complex conjugate field exhibit sixteen different combinations of collision scenario. We classify the parametric regions of dark and antidark solitons in both the field and parity transformed complex conjugate field by carrying out relevant asymptotic analysis. Further we present $2N$-dark/antidark soliton solution formula and demonstrate that this solution may have $2^{2N}\times 2^{2N}$ combinations of collisions.