Scattering of kinks in the $B\varphi^{4}$ model

TL;DR

This paper introduces the $B\varphi^4$ model, exploring how the parameter $B$ influences kink-antikink collision dynamics, revealing fractal structures, chaos, and nearly soliton behavior through detailed numerical analysis.

Contribution

It presents a new $B\varphi^4$ model and investigates how the parameter $B$ affects collision outcomes, including fractal structures and chaos, which was not previously studied.

Findings

Fractal escape windows for $B\leq 1$

Chaotic behavior emerges as $B$ approaches 3.3

Nearly soliton behavior observed at specific $B$ values

Abstract

In this study, based on the $\varphi^4$ model, a new model (called the $B\varphi^4$ model) is introduced in which the potential form for the values of the field whose magnitudes are greater than $1$ is multiplied by the positive number $B$. All features related to a single kink (antikink) solution remain unchanged and are independent of parameter $B$. However, when a kink interacts with an antikink in a collision, the results will significantly depend on parameter $B$. Hence, for kink-antikink collisions, many features such as the critical speed, output velocities for a fixed initial speed, two-bounce escape windows, extreme values, and fractal structure in terms of parameter $B$ are considered in detail numerically. The role of parameter $B$ in the emergence of a nearly soliton behavior in kink-antikink collisions at some initial speed intervals is clearly confirmed. The fractal structure in the diagrams of escape windows is seen for the regime $B\leq 1$. However, for the regime $B >1$, this behavior gradually becomes fuzzing and chaotic as it approaches $B = 3.3$. The case $B = 3.3$ is obtained again as the minimum of the critical speed curve as a function of $B$. For the regime $3.3< B \leq 10$, the chaotic behavior gradually decreases. However, a fractal structure is never observed. Nevertheless, it is shown that despite the fuzzing and shuffling of the escape windows, they follow the rules of the resonant energy exchange theory.