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arXiv · 2106.06367

Modified scattering for the one-dimensional Schr\"odinger equation with a subcritical dissipative nonlinearity

Xuan Liu·Ting Zhang
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Abstract

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger equation with a subcritical dissipative nonlinearity $\lambda |u|^\alpha u$, where $0<\alpha<2$, and $\lambda $ is a complex constant satisfying $\text{Im} \lambda >\frac{\alpha |\text{Re} \lambda |}{2\sqrt{ \alpha +1}}$. For arbitrary large initial data, we present the uniform time decay estimates when $4/3<\alpha <2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<\alpha <2$. The proof is based on the vector fields method and a semiclassical analysis method.

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Xuan Liu, Ting Zhang. 2021-06-11. Modified scattering for the one-dimensional Schr\"odinger equation with a subcritical dissipative nonlinearity. https://arxiv.org/abs/2106.06367

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