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arXiv · 2508.00594

Well-posedness of the periodic nonlinear Schr\"odinger equation with concentrated nonlinearity

Jinyeop Lee·Andrew Rout
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Abstract

We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. We also prove local well-posedness of the concentrated nonlinear Schr\"odinger equation below the energy space but above the endpoint of Sobolev embedding theorem. Our methods are based on compactness results and exploiting the Volterra integral equation structure of the problem. To our knowledge, this is the first rigorous solution theory for the concentrated nonlinear Schr\"odinger equation on the circle.

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Jinyeop Lee, Andrew Rout. 2025-08-01. Well-posedness of the periodic nonlinear Schr\"odinger equation with concentrated nonlinearity. https://arxiv.org/abs/2508.00594

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