Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

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Abstract

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n?N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.
Original languageEnglish
Pages (from-to)1598-1606
Number of pages9
JournalTheoretical and Mathematical Physics
Volume133
Issue number2
DOIs
Publication statusPublished - Nov 2002

Keywords

  • boundary value problems - Riemann?Hilbert problem - spectral analysis

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