How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation


About this Title

Hans Lundmark and Jacek Szmigielski

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 244, Number 1155
ISBNs: 978-1-4704-2026-0 (print); 978-1-4704-3510-3 (online)
DOI: http://dx.doi.org/10.1090/memo/1155
Published electronically: June 17, 2016

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • Acknowledgements
  • Chapter 1. Introduction
  • Chapter 2. Forward Spectral Problem
  • Chapter 3. The Discrete Case
  • Chapter 4. The Inverse Spectral Problem
  • Chapter 5. Concluding Remarks
  • Appendix A. Cauchy Biorthogonal Polynomials
  • Appendix B. The Forward Spectral Problem on the Real Line
  • Appendix C. Guide to Notation

Abstract


We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis–Procesi equations. Like the spectral problems for those equations, this one is of a ‘discrete cubic string’ type – a nonselfadjoint generalization of a classical inhomogeneous string – but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher–Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein’s solution of the inverse problem for the Stieltjes string.

References [Enhancements On Off] (What's this?)

American Mathematical Society