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

- 52 Downloads
- 1 Citations

## 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 α_{ 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.

## Preview

Unable to display preview. Download preview PDF.

## REFERENCES

- 1.
- 2.B. Pelloni, “Well-posed boundary value problems for linear evolution equations on a finite interval,”
*Math. Proc. Camb. Phil. Soc.*(in press).Google Scholar - 3.
- 4.
- 5.A. S. Fokas and L. Y. Sung, “Initial boundary value problems for linear evolution equations on the half-line,” DAMTP preprint, Cambridge Univ., Cambridge (2002).Google Scholar
- 6.A. S. Fokas and A. R. Its, “The nonlinear Schrödinger equation on a finite interval,” DAMTP preprint, Cambridge Univ., Cambridge (2002).Google Scholar
- 7.D. Antonopoulos, V. A. Dougalis, A. S. Fokas, and B. Pelloni, “Boundary value problems for Boussinesq-type system,”
*J. Math. Anal. Appl.*(submitted 2002).Google Scholar