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An inverse spectral problem related to the Geng–Xue two-component peakon equation
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-4137-8272
University of Saskatchewan.
2016 (English)In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 244, no 1155Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2016. Vol. 244, no 1155
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-139695DOI: 10.1090/memo/1155ISI: 000383777800001Scopus ID: 2-s2.0-84989871909OAI: oai:DiVA.org:liu-139695DiVA: diva2:1130585
Available from: 2017-08-10 Created: 2017-08-10 Last updated: 2017-08-30Bibliographically approved

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