Supercritical blowup in coupled parity-time-symmetric nonlinear Schr\"odinger equations

TL;DR

This paper demonstrates finite-time supercritical blowup in a coupled parity-time-symmetric nonlinear Schrödinger system with gain and dissipation, using virial techniques and numerical illustrations.

Contribution

It proves supercritical blowup in a PT-symmetric coupled NLS system with gain and dissipation, covering both focusing and defocusing nonlinearities, which is a novel analytical result.

Findings

Finite-time supercritical blowup established.

Virial technique used for proofs.

Numerical illustrations support theoretical results.

Abstract

We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schr\"odinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.