Skip to main content
SpringerLink
Log in

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

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. S. Fokas and B. Pelloni, Math. Proc. Camb. Phil. Soc., 131, 521 (2001).

    Google Scholar 

  2. B. Pelloni, “Well-posed boundary value problems for linear evolution equations on a finite interval,” Math. Proc. Camb. Phil. Soc. (in press).

  3. A. S. Fokas, J. Math. Phys., 41, 4188 (2000).

    Google Scholar 

  4. A. S. Fokas, Proc. Roy. Soc. London A, 457, 371 (2001).

    Google Scholar 

  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).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Reading, Reading, UK

    B. Pelloni

Authors
  1. B. Pelloni
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pelloni, B. Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval. Theoretical and Mathematical Physics 133, 1598–1606 (2002). https://doi.org/10.1023/A:1021163230712

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021163230712

Search

Navigation