Global existence of solutions to coupled $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schr\"odinger equations

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 $\mathcal{P}\mathcal{T}$-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 $H^1$.