Analytical methods of asymmetry double sine-Gordon equation in infinite one-dimensional system

TL;DR

This paper develops an analytical method using Mobius transformation to solve the asymmetry double sine-Gordon equation, reducing it to finding roots of a quartic polynomial, thus providing a formal solution to a traditionally nonintegrable equation.

Contribution

The paper introduces a novel analytical approach employing Mobius transformation to solve the asymmetry double sine-Gordon equation, which was previously considered nonintegrable.

Findings

Reduced the problem to solving a quartic polynomial.

Provided a formal solution method for a nonintegrable equation.

Demonstrated the applicability of Mobius transformation in nonlinear equations.

Abstract

Traditionally, Double Sine-Gordon Equation (DSGE) is seen as a nonintegrable equation. That means we cannot find general solutions in asymmetry DSGE. In this paper, we develop analytical method to solve this equation by Mobius transformation. And finally, this can reduce the problem to find roots of polynomial of four degree in one element. We have known this can be solved by square formally because its degree less than five. Although complexity as a solution, but in this sense, we can say we formally solve this nonintegrable equation.