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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)
Published electronically: June 17, 2016

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Table of Contents


  • 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


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.

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