On the multipeakon system of a two-component Novikov equation

TL;DR

This paper analyzes a two-component Novikov equation with peakon solutions, focusing on spectral problems, long-term behavior, and inverse spectral methods, extending classical techniques to a novel non-self-adjoint system.

Contribution

It introduces a spectral analysis framework for a non-self-adjoint two-component Novikov equation and proves global existence of peakon flows using Moser's deformation method.

Findings

Eigenvalues computed via isospectral deformation

Global existence of peakon solutions established

Inverse problem solved using Weyl functions and Krein's method

Abstract

We are exploring variations of the Novikov equation that have weak solutions called peakons. Our focus is on a two-component Novikov equation with a non-self-adjoint $4\times 4$ Lax operator for which we examine the related forward and inverse spectral maps for the peakon sector. To tackle the forward spectral problem, we convert it into a matrix eigenvalue problem, for which the original boundary value problem is a two-fold cover. We then use an isospectral deformation to the long-time regime to calculate the relevant eigenvalues. To support the long-term deformation, we prove the global existence of the peakon flows using ideas based on Moser's deformation method, which was used in his study of the finite Toda lattice. We subsequently solve the inverse problem by studying a trio of Weyl functions that can be approximated simultaneously by rational functions that involve tensor products and an additional symmetry condition. This part of our paper is rooted in Krein's solution to the inverse problem for the Stieltjes string.