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arXiv · 2512.13305

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

Kenneth H. Karlsen·Yan Rybalko
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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.

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Kenneth H. Karlsen, Yan Rybalko. 2025-12-15. Generic regularity and Lipschitz metric for a two-component Novikov system. https://arxiv.org/abs/2512.13305

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