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Semiclassical Soliton Ensembles

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, volume 154 of the Annals of Mathematics Studies series published by Princeton University Press, is a research monograph jointly authored with Spyridon Kamvissis and Ken McLaughlin. This book gives a complete analysis of the Riemann-Hilbert problem of inverse scattering associated with a general class of initial-value problems for the focusing nonlinear Schrödinger equation, in the "semiclassical" limit. In this limit, the class of solutions under consideration reduce to nonlinear superpositions of an enormous number of fundamental excitations called solitons. We call this type of solution a "semiclassical soliton ensemble". The Riemann-Hilbert based analysis avoids the pitfalls of a more direct approach based on the Lax-Levermore method. It predicts that the dynamics of a semiclassical soliton ensemble are qualitatively different in different parts of space-time, with sharp boundaries called "caustics" separating the different regions. A density plot of such a solution over the (x,t)-plane can be seen in the masthead of this page.

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