##
D.E. Pelinovsky, D.A. Zezyulin, and V.V. Konotop

#
Global existence of solutions to coupled PT-symmetric nonlinear Schrodiger equations,

###
International Journal of Theoretical Physics 54 (2015), 3920-3931

**Abstract:**

We study a system of two coupled nonlinear Schrodinger 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 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}.

**Keywords:**

PT-symmetry, coupled nonlinear Schrodinger equation, Manakov system,
global existence, blow-up in finite time, apriori energy estimates.