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arXiv · 2606.06636

On a rod-Kadomtsev-Petviashvili shallow water equation in two dimensions

Tiantian Zhao·Kai Yan
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Abstract

In this paper, we derive a new two-dimensional rod-Kadomtsev-Petviashvili (rod-KP) equation from the incompressible and irrotational three-dimensional Euler equation under the shallow water scaling. We establish the local well-posedness of the Cauchy problem in a suitable Sobolev space and derive the blow-up criterion for strong solutions via the energy method. Then, by exploring two appropriate conservation laws of the rod-KP equation, we are able to construct its global strong solution when the physical dimensionless parameter $σ=0$, and on the other hand produce the finite-time blow up solutions under some certain conditions when $σ\neq 0$. Furthermore, we present a uniqueness continuation property for the solutions. Finally, we investigate the existence of traveling-wave solutions in order to highlight the influence of weak transverse effects on wave stability, and we also exhibit the symmetry of solitary waves in the propagation direction.

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Tiantian Zhao, Kai Yan. 2026-06-04. On a rod-Kadomtsev-Petviashvili shallow water equation in two dimensions. https://arxiv.org/abs/2606.06636

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