^{1,a)}, C. Klein

^{2}, V. B. Matveev

^{2}and A. O. Smirnov

^{3}

### Abstract

There exist two versions of the Kadomtsev-Petviashvili (KP) 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.

The “elliptic cylindrical Kadomtsev-Petviashvili (ecKP) equation,” where

*a*is a parameter and , is derived for surfacegravity waves with nearly elliptic front, generalising the cylindrical KP equation for nearly concentric waves and describing the intermediate asymptotics. We find transformations between the derived ecKP equation and two existing versions of the KP equation for water wave problems, for nearly plane and nearly concentric waves, as well as the Lax pair for the ecKP equation. The transformations are used to construct important classes of exact solutions of the derived ecKP equation and corresponding new asymptotic solutions for the Euler equations from the known solutions of the KP equation. The ecKP equation is a universal integrable model applicable to generic weakly nonlinear weakly dispersive waves with nearly elliptic wave fronts.

We thank G. A. El, E. V. Ferapontov, and R. H. J. Grimshaw for useful discussions, and referees for constructive comments and helpful references. K.R.K. and A.O.S. acknowledge support and hospitality of the Institut de Mathématiques de Bourgogne, where they held visiting positions in the spring-summer of 2012, which has made this collaboration possible. C.K. and V.B.M. thank for financial support by the ANR via the program ANR-09-BLAN-0117-01.

I. INTRODUCTION

II. DERIVATION OF THE ELLIPTIC CYLINDRICAL KP EQUATION

III. TRANSFORMATIONS BETWEEN KP, CKP, AND ECKP EQUATIONS

IV. SPECIAL SOLUTIONS OF ecKP-I AND ecKP-II EQUATIONS

V. APPROXIMATE SOLUTIONS FOR SURFACE WAVES

VI. CONCLUDING REMARKS

### Key Topics

- Fluid equations
- 27.0
- Surface waves
- 12.0
- Numerical solutions
- 11.0
- Free surface
- 10.0
- Numerical modeling
- 10.0

## Figures

Solution to the ecKP-I equation obtained as the image of the lump (31) with under the action of the map (30) for *a* = 0.01 and several values of τ.

Solution to the ecKP-I equation obtained as the image of the lump (31) with under the action of the map (30) for *a* = 1 and several values of τ.

2-soliton solution of the ecKP-II equation for *a* = 0.01 with for several values of τ.

2-soliton solution of the ecKP-II equation for *a* = 0.01 with for several values of τ.

2-soliton solution of the ecKP-II equation for *a* = 1 with for several values of τ.

2-soliton solution of the ecKP-II equation for *a* = 1 with for several values of τ.

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = 0 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 1 (bottom right).

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = 0 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 1 (bottom right).

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = 0.1 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 1 (bottom right).

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = 0.1 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 1 (bottom right).

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = –0.5 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 2 (bottom right).

Surface wave corresponding to the one-soliton solution (34) of the ecKP-II equation with *K* = 1 and *L* = –0.5 for *t* = 0 (top left), *t* = 0.25 (top right), *t* = 0.5 (bottom left), and *t* = 2 (bottom right).

Surface waves corresponding to the one-soliton solution of the ecKP-II Eq. (34) with *K* = 1.5 and *L* = 0 for *t* = 0 (left) and *t* = 2 (right).

Surface waves corresponding to the one-soliton solution of the ecKP-II Eq. (34) with *K* = 1.5 and *L* = 0 for *t* = 0 (left) and *t* = 2 (right).

Surface waves corresponding to the one-soliton solution of the ecKP-II Eq. (34) with *K* = 1.6 and *L* = 0.1 for *t* = 0 (left) and *t* = 2 (right).

Surface waves corresponding to the one-soliton solution of the ecKP-II Eq. (34) with *K* = 1.6 and *L* = 0.1 for *t* = 0 (left) and *t* = 2 (right).

Surface waves corresponding to the exceptional one-soliton solution of the ecKP-II Eq. (34) with *K* = 1.5 and *L* defined by Eq. (40) ( ) for *t* = 0 (left) and *t* = 1 (right).

Surface wave corresponding to the solution of the ecKP-II equation obtained from the canonical KP-soliton with for *t* = 0.5 (left) and *t* = 1 (right).

Surface wave corresponding to the solution of the ecKP-II equation obtained from the canonical KP-soliton with for *t* = 0.5 (left) and *t* = 1 (right).

Surface waves corresponding to the two-soliton solution of the ecKP-II equation with for *t* = 1 (top left), *t* = 2 (top right), *t* = 3 (bottom left), and *t* = 4 (bottom right).

Surface waves corresponding to the two-soliton solution of the ecKP-II equation with for *t* = 1 (top left), *t* = 2 (top right), *t* = 3 (bottom left), and *t* = 4 (bottom right).

Surface waves corresponding to the two-soliton solution of the ecKP-II equation with for *t* = 0 (top left), *t* = 0.5 (top right), *t* = 1 (bottom left), and *t* = 2 (bottom right).

Surface waves corresponding to the two-soliton solution of the ecKP-II equation with for *t* = 0 (top left), *t* = 0.5 (top right), *t* = 1 (bottom left), and *t* = 2 (bottom right).

Surface waves corresponding to the ecKP-I lump solution with , for *t* = 0 (top left), *t* = 0.5 (top right), *t* = 1 (bottom left), and *t* = 2 (bottom right).

Surface waves corresponding to the ecKP-I lump solution with , for *t* = 0 (top left), *t* = 0.5 (top right), *t* = 1 (bottom left), and *t* = 2 (bottom right).

Genus 2 solution (42) to the ecKP-II equation for *a* = 0.01 generated by the curve for several values of τ.

Genus 2 solution (42) to the ecKP-II equation for *a* = 0.01 generated by the curve for several values of τ.

Genus 2 solution (42) to the ecKP-II equation for *a* = 1 generated by the curve for several values of τ.

Genus 2 solution (42) to the ecKP-II equation for *a* = 1 generated by the curve for several values of τ.

Genus 3 solution (42) to the ecKP-II equation for *a* = 0.01 (left)and *a* = 1 (right), generated by the curve at .

Genus 3 solution (42) to the ecKP-II equation for *a* = 0.01 (left)and *a* = 1 (right), generated by the curve at .

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