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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
1/2005

Oscillatory Solutions of Soliton Equations; Phase Transitions and Variational Problems

Spyridon Kamvissis

Abstract

Consider the Cauchy problem for the two following soliton equations: Korteweg-de-Vries with a small dispersion parameter and the nonlinear Schroedinger equation in 1+1 dimensions with cubic nonlinearity in the so-called semiclassical limit. It is known that at a certain "caustic" there is a "phase transition" in the solution of each of these equations. Oscillations of high frequency appear, so strong limits no longer exist. Soliton theory enables us to provide detailed information for the solutions as the small parameter goes to zero, prove the existence of weak limits, and describe completely the different oscillatory regions in terms of a variational problem of "electrostatic type". In the case of the focusing NLS this is a non-convex problem.

Received:
Jan 3, 2005
Published:
Jan 3, 2005
Keywords:
soliton equations, high frequency solutions, variational problems

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Preprint
2005 Repository Open Access
Spyridon Kamvissis

Oscillatory solutions of soliton equations : phase transitions and variational problems