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

Lijun Miao; Jingwei Yu; Linlin Qiu; Lijun Miao; Jingwei Yu; Linlin Qiu

doi:10.3934/era.2025159

Electronic Research Archive

2025, Volume 33, Issue 6: 3571-3583. doi: 10.3934/era.2025159

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Research article

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

School of Mathematics, Liaoning Normal University, Dalian 116029, China

Received: 28 September 2024 Revised: 27 February 2025 Accepted: 04 June 2025 Published: 11 June 2025

This paper was focused on global existence for the stochastic nonlinear Schrödinger equation with time-dependent loss/gain, which read $ \mathrm{i}du+(\Delta u+\lambda|u|^\alpha u+ia(t)u)dt = dW $. We proved the global existence and uniqueness of the solution in $ H^1(\mathbb{R}^N) $ 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.
- stochastic nonlinear Schrödinger equation,
- global existence,
- time-dependent loss/gain,
- additive noise,
- energy space
Citation: Lijun Miao, Jingwei Yu, Linlin Qiu. On global existence for the stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain[J]. Electronic Research Archive, 2025, 33(6): 3571-3583. doi: 10.3934/era.2025159

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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 $ \mathrm{i}du+(\Delta u+\lambda|u|^\alpha u+ia(t)u)dt = dW $. We proved the global existence and uniqueness of the solution in $ H^1(\mathbb{R}^N) $ 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.

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