Non-linear Smoothing for the Periodic Generalized Non-linear Schr\"odinger Equation

Ryan McConnell

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arXiv · 2203.00098

Non-linear Smoothing for the Periodic Generalized Non-linear Schr\"odinger Equation

Ryan McConnell

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Abstract

We consider the periodic non-linear Schr\"odinger equation with non-linearity given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the linear evolution is in a smoother space. We then study forced and damped defocusing non-linear Schr\"odinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smootheness for global attractors in the energy space.

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Ryan McConnell. 2022-02-28. Non-linear Smoothing for the Periodic Generalized Non-linear Schr\"odinger Equation. https://arxiv.org/abs/2203.00098

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