Global existence of solutions to coupled ${\cal PT}$-symmetric nonlinear Schr\"odinger equations

TL;DR

This paper investigates the existence and boundedness of solutions to coupled ${ m PT}$-symmetric nonlinear Schr"odinger equations, proving global existence in energy space and analyzing conditions for bounded norms in one and two dimensions.

Contribution

It provides the first rigorous proof of global solutions for ${ m PT}$-symmetric coupled nonlinear Schr"odinger equations, including the Manakov case, in both one and two dimensions.

Findings

Global solutions exist in 1D with possible growth in $H^1$-norm.

In the Manakov case, $L^2$-norm is bounded and $H^1$-norm is numerically bounded.

In 2D, initial data constraints ensure global existence.

Abstract

We study a system of two coupled nonlinear Schr\"{o}dinger equations, where one equation includes gain and the other one includes losses. Strengths of the gain and the loss are equal, i.e., the resulting system is parity-time (${\cal PT}$) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the ${\cal PT}$-generalization of the Manakov model. In the one-dimensional case, we prove the existence of a global solution to the Cauchy problem in energy space $H^1$, such that the $H^1$-norm of the global solution may grow in time. In the Manakov case, we show analytically that the $L^2$-norm of the global solution is bounded for all times and numerically that the $H^1$-norm is also bounded. In the two-dimensional case, we obtain a constraint on the $L^2$-norm of the initial data that ensures the existence of a global solution in the energy space $H^1$.