Global Existence of Solutions to Coupled 𝒫𝒯-Symmetric Nonlinear Schrödinger Equations

We study a system of two coupled nonlinear Schrödinger 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 (𝒫𝒯$$\mathcal {PT}$$) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the 𝒫𝒯$$\mathcal {PT}$$-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 H1, such that the H1-norm of the global solution may grow in time. In the Manakov case, we show analytically that the L2-norm of the global solution is bounded for all times and numerically that the H1-norm is also bounded. In the two-dimensional case, we obtain a constraint on the L2-norm of the initial data that ensures the existence of a global solution in the energy space H1.