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

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 $\sigma =0$, and on the other hand produce the finite-time blow up solutions under some certain conditions when $\sigma \ne 0$. 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.

36 pages