Singularity formation and blowup of complex-valued solutions of the modified KdV equation

Jerry L. Bona; Stéphane Vento; Fred B. Weissler

doi:10.3934/dcds.2013.33.4811

Article Contents

2013, Volume 33, Issue 11&12: 4811-4840. Doi: 10.3934/dcds.2013.33.4811

This issue Previous Article Floquet representations and asymptotic behavior of periodic evolution families Next Article Periodic traveling--wave solutions of nonlinear dispersive evolution equations

Singularity formation and blowup of complex-valued solutions of the modified KdV equation

1.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607
2.
Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539 Laboratoire Analyse, Géométrie et Applications, 99 avenue J.B. Clément - 93430 Villetaneuse

Received: November 2011

Revised: August 2012

Published: October 2013

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Abstract

The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $ u_t + 6u^2u_x + u_{xxx} = 0 $ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $ - oo \lt x \lt oo $, exponentially decreasing to zero as $|x| \to oo$, that blow up in finite time.

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Mathematics Subject Classification: 35Q53, 35Q51, 37K40, 35A21, 35B44.

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