Global existence of solutions to coupled PT-symmetric nonlinear Schrödinger equations

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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) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the 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.