Modular and Landen transformations between finite solutions of Sine-Gordon Equation with phase N=1

TL;DR

This paper establishes Modular and Landen transformations connecting different finite solutions of the sine-Gordon equation with phase N=1, unifying breather and kink solutions through these transformations.

Contribution

It introduces Modular and Landen transformations that unify finite solutions of the sine-Gordon equation with phase N=1, linking heta representation and directly integrated solutions.

Findings

Unified finite solutions of sine-Gordon N=1 via transformations

Connected breather and kink solutions through transformations

Provided a framework for understanding solution relationships

Abstract

Among finite solutions of Sine-Gordon Equation with in phase N=1, we find Modular and Landen transformation between \theta representation and directly integrated solutions of breather and kink. Because of these transformations, we can say we unified the finite solutions of sine-Gordon equation with phase N=1.