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

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 ($\mathcal{P}\mathcal{T}$) 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 $H^1$.

arXiv comment11 pages, 1 figure; accepted for International Journal of Theoretical Physics http://www.springer.com/physics/journal/10773

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• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Stabilization of the coupled pendula chain under parametric PT-symmetric driving force
• Long-time stability of breathers in Hamiltonian $\mathcal{P}\mathcal{T}$-symmetric lattices
• Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation
• $L^2$-Sobolev space bijectivity of the inverse scattering of a $3\times 3$ AKNS system
• Classical Hamiltonian Systems with Balanced loss and gain