Lax pair, breather dynamics and symmetry analysis in nonlocal semi-discrete complex dispersionless system

Abstract

Dispersionless hierarchies are of much consideration because of their arising in many areas of physics and applied mathematics, like string theory, quantum field theories, etc. Among these hierarchies, we consider a generalized coupled dispersionless system which describes the profile of a current-fed string in the presence of an external magnetic field. In this work, we propose a novel parity-time symmetric, nonlocal semi-discrete complex coupled dispersionless system which is derived from the discretization of the corresponding Lax pair. Using the Darboux transformation and binary Darboux transformation, we construct multi-soliton and quasi-Grammian solutions expressed in terms of quasi-determinants instead of ratios of determinants. Explicit expressions for Grammian, breather, interaction of breather with kink, soliton and kink solutions are derived to analyze the system's behavior. Our study also explores both symmetry-preserving and symmetry-breaking solutions within the proposed model. To further substantiate these theoretical results, the solution dynamics are illustrated through surface and contour plots, providing a visual representation of their behavior.