Interaction of sine-Gordon kinks and breathers with a $\mathcal{PT}$-symmetric defect

TL;DR

This paper numerically investigates how sine-Gordon kinks and breathers interact with a localized $ ext{PT}$-symmetric defect, revealing phase shifts, energy exchange, and conditions for transmission or trapping.

Contribution

It introduces a detailed numerical analysis of sine-Gordon soliton interactions with $ ext{PT}$-symmetric defects, including phase shifts, critical velocities, and breather energy dynamics, which were not previously characterized.

Findings

Kinks always restore initial energy and momentum after passing the defect.

Kinks from the gain side always pass; from the loss side, passing depends on initial velocity.

Breathers can gain energy or split into kink-antikink pairs depending on parameters.

Abstract

The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized $\mathcal{PT}$-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated with the use of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect strongly depends on the breather parameters (velocity, frequency and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions the energy loss to radiation of small-amplitude extended waves is negligible.