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arXiv · 2209.11261

Global conservative solutions of the nonlocal NLS equation beyond blow-up

Yan Rybalko·Dmitry Shepelsky
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Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2\sigma q^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

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Yan Rybalko, Dmitry Shepelsky. 2022-09-22. Global conservative solutions of the nonlocal NLS equation beyond blow-up. https://doi.org/10.3934/dcds.2022173

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