A direct method for studying soliton equations
type: Item
instance of: scholarly article
title: A direct method for studying soliton equations
zbMATH Open document ID: 0970.35121
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 2001-10-21T00:00:00Z
review text: The article provides a comprehensive survey of functional-analytic methods in the soliton theory. Also, it cannot be regarded as an introduction to the subject since many crucial concepts are assumed to be known. To outline the main idea for the particular case of the KdV equation, let \(E\) be a Banach space, \(A\in{\mathcal L}(E)\) a continuous operator, \(B\in{\mathcal A}(E)\) an operator in a Banach quasi-ideal. Denoting \(L(x,t)= \exp(Ax+ A^3t)B\) and assuming the existence of the inverse \((1+ L)^{-1}\). Then the family of operators \(U= ((1+ L)^{-1}(AL+ LA))_x\), \(A(E)\) is a solution of the operator KdV equation \(U_t= U_{xxx}+ 3(UU_x+ U_xU)\). To restitute the scalar case, let \(\tau\) be a continuous trace on \(A(E)\), \(P\in{\mathcal L}(E)\) a projection of rank 1. Then, assuming \(UP= U\), \(U_xP= U_x\), the function \(u= \tau(U)\) resolves the common KdV equation. The authors report on the traces on operator ideals, Marchenko and Hirota methods, the finite-dimensional cases (especially the negaton asymptotics), the diagonal operators (leading to infinite soliton superpositions), generalizations for the unbounded operators (by using the \(C_0\)-semigroups), and the Cauchy initial problem. Several graphs of soliton superpositions and extensive literature conclude this instructive exposition.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 47L20
zbMATH DE Number: 1556956
zbMATH Keywords: soliton
zbMATH Keywords: negaton
zbMATH Keywords: operator equations
zbMATH Keywords: Banach quasi-ideal
zbMATH Keywords: operator KdV equation
zbMATH Keywords: traces on operator ideals
author: Bernd Carl
author: C. Schiebold
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1556956