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This paper has been submitted for publication in Comm. Pure Appl. Math. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-2108019 (Buckingham), DMS-2307142 (Jenkins), DMS-1513054, DMS-1812625, and DMS-2204896 (Miller). Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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Suleimanov-Talanov self-focusing and the hierarchy of the focusing nonlinear Schrödinger equation

Robert J. Buckingham, Robert M. Jenkins, and Peter D. Miller

RJB: Department of Mathematical Sciences, University of Cincinnati
RMJ: Department of Mathematics, University of Central Florida
PDM: Department of Mathematics, University of Michigan, Ann Arbor

Abstract:

We study the self-focusing of wave packets from the point of view of the semiclassical focusing nonlinear Schrödinger equation. A type of finite-time collapse/blowup of the solution of the associated dispersionless limit was investigated by Talanov in the 1960s, and recently Suleimanov identified a special solution of the dispersive problem that formally regularizes the blowup and is related to the hierarchy of the Painlevé-III equation. In this paper we approximate the Talanov solutions in the full dispersive equation using a semiclassical soliton ensemble, a sequence of exact reflectionless solutions for a corresponding sequence of values of the semiclassical parameter epsilon tending to zero, approximating the Talanov initial data more and more accurately in the limit as epsilon tends to zero. In this setting, we rigorously establish the validity of the dispersive saturation of the Talanov blowup obtained by Suleimanov. We extend the result to the full hierarchy of higher focusing nonlinear Schrödinger equations, exhibiting new generalizations of the Talanov initial data that produce such dispersively regularized extreme focusing in both mixed and pure flows. We also argue that generic perturbations of the Talanov initial data lead to a different singularity of the dispersionless limit, namely a gradient catastrophe for which the dispersive regularization is instead based on the tritronquée solution of the Painlevé-I equation and the Peregrine breather solution which appears near points in space time corresponding to the poles of the former transcendental function as shown by Bertola and Tovbis.

Left: the amplitude of the dispersionless Talanov solution. Right: its dispersive regularization for different \(\epsilon\).