Exponential asymptotics for solitons in PT-symmetric periodic potentials

TL;DR

This paper develops exponential asymptotics to analyze PT-symmetric solitons in periodic potentials, revealing bifurcation, stability properties, and multi-soliton bound states, with analytical results validated by numerical comparisons.

Contribution

It introduces a novel exponential asymptotics approach for complex PT-symmetric potentials, uncovering soliton bifurcations, stability criteria, and multi-soliton states.

Findings

Two soliton families bifurcate from band edges.

One family is always unstable, the other can be stable.

Analytical predictions match numerical results with minor differences.

Abstract

Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT-symmetric multi-soliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.