On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation

TL;DR

This paper introduces a new integrable elliptic cylindrical KP equation, expanding the mathematical models for wave phenomena in different geometries and providing new solutions for water wave problems.

Contribution

It derives and analyzes a novel elliptic cylindrical KP equation, establishing transformations between different KP versions and generating new approximate water wave solutions.

Findings

Derived a new integrable elliptic cylindrical KP equation

Established transformations between Cartesian, cylindrical, and elliptic cylindrical KP equations

Produced new classes of approximate solutions for water waves

Abstract

There exist two versions of the Kadomtsev-Petviashvili equation, related to the Cartesian and cylindrical geometries of the waves. In this paper we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly-nonlinear weakly-dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.