Interaction Properties of the Periodic and Step-like Solutions of the Double-Sine-Gordon Equation

TL;DR

This paper investigates the interaction properties of periodic and step-like solutions of the double-Sine-Gordon equation, revealing phase transitions and differing behaviors as the potential parameter varies, with implications for 1+1 dimensional many-body systems.

Contribution

It provides a detailed analysis of how solutions of the double-Sine-Gordon equation behave under different parameters, identifying two distinct phases and their phase transition characteristics.

Findings

Step-like solutions behave similarly to Sine-Gordon solutions.

Periodic solutions exhibit two phases with different behaviors.

Response functions differ across phases, indicating a phase transition.

Abstract

The periodic and step-like solutions of the double-Sine-Gordon equation are investigated, with different initial conditions and for various values of the potential parameter $\epsilon$. We plot energy and force diagrams, as functions of the inter-soliton distance for such solutions. This allows us to consider our system as an interacting many-body system in 1+1 dimension. We therefore plot state diagrams (pressure vs. average density) for step-like as well as periodic solutions. Step-like solutions are shown to behave similarly to their counterparts in the Sine-Gordon system. However, periodic solutions show a fundamentally different behavior as the parameter $\epsilon$ is increased. We show that two distinct phases of periodic solutions exist which exhibit manifestly different behavior. Response functions for these phases are shown to behave differently, joining at an apparent phase transition point.