A direct method for solving the generalized sine-Gordon equation

TL;DR

This paper introduces a direct method for solving the generalized sine-Gordon equation, enabling the construction of multisoliton solutions and revealing novel soliton behaviors without relying on inverse scattering techniques.

Contribution

It develops a new direct approach to solve the generalized sine-Gordon equation and constructs explicit multisoliton solutions, including novel soliton structures.

Findings

Constructed multisoliton solutions including kinks, loop solitons, and breathers.

Discovered a new soliton type where smaller solitons travel faster.

Connected the generalized sG equation to the short pulse equation in a scaling limit.

Abstract

The generalized sine-Gordon (sG) equation was derived as an integrable generalization of the sG equation. In this paper, we develop a direct method for solving the generalized sG equation without recourse to the inverse scattering method. In particular, we construct multisoliton solutions in the form of parametric representation. We obtain a variety of solutions which include kinks, loop solitons and breathers. The properties of these olutions are investigated in detail. We find a novel type of solitons with a peculiar structure that the smaller soliton travels faster than the larger soliton. We also show that the short pulse equation describing the propagation of ultra-short pulses is reduced from the generalized sG equation in an appropriate scaling limit. Subsequently, the reduction to the sG equation is briefly discussed.