Global Existence of Solutions to Coupled � � $\mathcal {PT}$ -Symmetric Nonlinear Schrödinger Equations
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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 $H^1$.