An inverse spectral problem related to the Geng-Xue two-component peakon equation

TL;DR

This paper addresses a complex inverse spectral problem related to the Geng-Xue two-component peakon equation, introducing novel methods involving Cauchy biorthogonal polynomials and extending classical spectral theory to nonselfadjoint cases.

Contribution

It develops a new approach to solve the inverse spectral problem for a nonselfadjoint discrete cubic string, incorporating two Lax pairs and extending the application of Cauchy biorthogonal polynomials.

Findings

Spectrum is positive and simple due to oscillatory kernels.

Inverse problem solution generalizes Krein's method for nonselfadjoint cases.

Method applies to two-component peakon equations with novel spectral data.

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.