Stable C7/9 cusp formation for the Novikov equation
Abstract
We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity C7/9. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved H1-type quantity excludes the C1/3 cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent 7/9, rather than the 3/5 exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the C7/9 cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.
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BibTeX
@misc{kim2026stable,
title = {{Stable $C^7/9$ cusp formation for the Novikov equation}},
author = {Yunjoo Kim and Dowan Koo and Bongsuk Kwon and Wanyong Shim},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29236},
eprint = {2608.29236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29236},
}
RIS
TY - UNPB
TI - Stable C^7/9 cusp formation for the Novikov equation
AU - Yunjoo Kim
AU - Dowan Koo
AU - Bongsuk Kwon
AU - Wanyong Shim
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29236
UR - https://arxiv.org/abs/2608.29236
AB - We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp Hölder regularity C^7/9. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved H^1-type quantity excludes the C^1/3 cusp associated with Burgers-type gradient blow-up, the precise Hölder exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent 7/9, rather than the 3/5 exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the C^7/9 cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.
ER -