On global existence for the stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain

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Abstract

This paper was focused on global existence for the stochastic nonlinear Schrödinger equation with time-dependent loss/gain, which read idu + (∆u + λ|u|^α u + ia(t)u)dt = dW. We proved the global existence and uniqueness of the solution in $H^{1}\left(\mathbb{R}^{N}\right)$ through the uniform boundedness of the momentum and energy functionals. The global existence result of the solution for this type of equation depended on the ranges of time-dependent loss/gain coefficient.

Keywords

stochastic nonlinear Schrödinger equation / global existence / time-dependent loss/gain / additive noise / energy space

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