Generic regularity and Lipschitz metric for a two-component Novikov system

TL;DR

This paper studies the regularity and stability of solutions to a two-component Novikov system, introducing a new Lipschitz metric based on geometric analysis to better understand solution behavior and interactions.

Contribution

It develops a geometric framework and a new Lipschitz metric for analyzing the regularity and stability of solutions to a complex integrable PDE system.

Findings

Solutions are $C^k$ regular away from finite characteristic curves.

A Lipschitz metric is constructed using a Finsler norm.

The metric measures minimal energy transportation between solutions.

Abstract

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions between peakon-like structures. Our study has two main objectives: first, to analyze the generic regularity of global conservative solutions; and second, to construct a new metric that guarantees the Lipschitz continuity of the flow. Building on the geometric framework developed by Bressan and Chen for quasilinear second-order wave equations, we prove that the solution retains $C^k$ regularity away from a finite number of piecewise $C^{k-1}$ characteristic curves. Furthermore, we provide a description of the solution behavior in the vicinity of these curves. By introducing a Finsler norm on tangent vectors in the space of solutions, expressed in the transformed Bressan-Constantin variables, we introduce a Lipschitz metric representing the minimal energy transportation cost between two solutions.