A direct method for solving the generalized sine-Gordon equation II

TL;DR

This paper analytically solves the generalized sine-Gordon equation with a specific parameter, revealing new solution structures like kinks and breathers, and connects it to known equations through limits and conservation laws.

Contribution

It provides the first analytical solutions for the generalized sine-Gordon equation with , highlighting new solution types and a novel method for deriving conservation laws.

Findings

Identifies kink and breather solutions for  case.

Shows reduction to short pulse and sine-Gordon equations in limits.

Introduces a Backlund transformation for conservation laws.

Abstract

The generalized sine-Gordon (sG) equation $u_{tx}=(1+\nu\partial_x^2)\sin\,u$ was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with $\nu=-1$. Here, we address the equation with $\nu=1$. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. Last, we provide a recipe for deriving an infinite number of conservation laws by using a novel B\"acklund transformation connecting solutions of the sG and generalized sG equations.