LDR 03199 2200373 4500 010 _a9781470420260 _bbr. 011 _a0065-9266 090 _a17010 101 _aeng 102 _aUS 100 _a20150209 frey50 200 _aAn inverse spectral problem related to the Geng-Xue two-component Peakon equation _bM _fHans Lundmark, Jacek Szmigielski 210 _aProvidence (R.I.) _cAmerican Mathematical Society _d2016 215 _a1 vol. (VII-87 p.) _d26 _cill. 225 _9170211 _aMemoirs of the American Mathematical Society _v1155 _x0065-9266 320 _aBibliogr. p. [85]-87. Index 330 _aAuthors’ 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. 410 _9170209 _aAMS _tMemoirs of the American Mathematical Society _x0065-9266 676 _a2010 686 _20 _9163755 _a35Q53 _bPartial differential equations -- Equations of mathematical physics and other areas of application _xKdV-like equations (Korteweg-de Vries) 686 _20 _9163471 _a34L25 _bOrdinary differential equations -- Ordinary differential operators _xScattering theory, inverse scattering 686 _20 _9163738 _a35P05 _bPartial differential equations -- Spectral theory and eigenvalue problems _xGeneral topics in linear spectral theory 686 _20 _9163754 _a35Q51 _bPartial differential equations -- Equations of mathematical physics and other areas of application _xSoliton-like equations 686 _20 _9163792 _a35R30 _bPartial differential equations -- Miscellaneous topics _xInverse problems 700 _4070 _aLundmark _bHans _f1970- _9183282 701 _4070 _aSzmigielski _bJacek _f1954- _9183283 856 _uhttps://zbmath.org/?q=an:1375.34030 _zzbMath 856 _uhttps://mathscinet.ams.org/mathscinet-getitem?mr=3545110 _zMSN 856 _uhttps://arxiv.org/pdf/1304.0854.pdf _zArXiv 856 _uhttp://www.ams.org/books/memo/1155/ _zAMS-résumé 905 _aaw _b2018 906 _aaw _b2018-11-27 001 17010 995 _f12535-01 _xachat _918695 _cCMI _20 _kSéries AMS _o0 _eCouloir _bCMI