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Global Existence of Solutions to Coupled \(\mathcal {PT}\)-Symmetric Nonlinear Schrödinger Equations

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Abstract

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 (\(\mathcal {PT}\)) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the \(\mathcal {PT}\)-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 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.

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Acknowledgments

The authors thank Vadim Vekslerchik for helpful discussion at an earlier stage of the project. The work of D.P. is supported by the Ministry of Education and Science of Russian Federation (the base part of the state task No. 2014/133). DAZ and VVK acknowledge support of FCT (Portugal) under the grants PEst-OE/FIS/UI0618/2014 and PTDC/FIS-OPT/1918/2012.

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Correspondence to Vladimir V. Konotop.

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Pelinovsky, D.E., Zezyulin, D.A. & Konotop, V.V. Global Existence of Solutions to Coupled \(\mathcal {PT}\)-Symmetric Nonlinear Schrödinger Equations. Int J Theor Phys 54, 3920–3931 (2015). https://doi.org/10.1007/s10773-014-2422-0

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  • DOI: https://doi.org/10.1007/s10773-014-2422-0

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