An inverse spectral problem related to the Geng-Xue two-component Peakon equation / Hans Lundmark, Jacek Szmigielski

Auteur: Lundmark, Hans (1970-) - AuteurCo-auteur: Szmigielski, Jacek (1954-) - AuteurType de document: MonographieCollection: Memoirs of the American Mathematical Society ; 1155Langue: anglaisPays: Etats UnisÉditeur: Providence (R.I.) : American Mathematical Society, 2016Description: 1 vol. (VII-87 p.) : ill. ; 26 cm ISBN: 9781470420260 ; br. ISSN: 0065-9266Résumé: Authors’ 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..Bibliographie: Bibliogr. p. [85]-87. Index. Sujets MSC: 35Q53 Partial differential equations -- Equations of mathematical physics and other areas of application -- KdV-like equations (Korteweg-de Vries)
34L25 Ordinary differential equations -- Ordinary differential operators -- Scattering theory, inverse scattering
35P05 Partial differential equations -- Spectral theory and eigenvalue problems -- General topics in linear spectral theory
35Q51 Partial differential equations -- Equations of mathematical physics and other areas of application -- Soliton-like equations
35R30 Partial differential equations -- Miscellaneous topics -- Inverse problems
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Bibliogr. p. [85]-87. Index

Authors’ 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.

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