Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor Equations

TL;DR

This paper introduces a new relativistic integrable nonlinear spinor model in 1+1 dimensions, constructs exact dissipative soliton solutions using Hirota's method, and analyzes their particle-like properties and interactions.

Contribution

It presents a novel relativistic spinor model, applies Hirota's bilinear method to find exact dissipative soliton solutions, and explores their fusion, fission, and resonant interactions.

Findings

Exact one and two dissipative soliton solutions obtained

Dissipaton solutions exhibit particle-like nonlinear excitations

Resonant interactions confirmed through asymptotic analysis

Abstract

A new relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions, gauge equivalent to Papanicolau spin model, defined on the one sheet hyperboloid is introduced. By using the double numbers, the model is represented as hyperbolic complex valued relativistic massive Thirring type model. By Hirota's bilinear method, an exact one and two dissipative soliton solutions of this model are constructed. Calculation of first three integrals of motion for one dissipaton solution shows that the last one represents a particle-like nonlinear excitation, with relativistic dispersion and highly nonlinear mass. A nontrivial solution of the system of algebraic equations, showing fusion and fission of relativistic dissipatons is found. Asymptotic analysis of exact two dissipaton solution confirms resonant character of our dissipaton interactions.