On elliptic cylindrical Kadomtsev-Petviashvili equation for surface waves

Abstract: There exist two classical versions of the Kadomtsev-Petviashvili (KP) equation [1], related to the Cartesian and cylindrical geometries of the waves (derivations for surface waves were given in [2] and [3], respectively). We derived and studied a version related to the elliptic-cylindrical geometry in [4] (joint work with Klein, Matveev and Smirnov). The derivation was given from the full set of Euler equations for surface gravity waves with the account of surface tension. The ecKP equation contains a parameter, and it reduces to the cKP equation both when this parameter tends to zero, and when the solutions are considered at distances much larger than that parameter. We showed that there exist transformations between all three versions of the KP equation associated with the physical problem formulation (KP, cKP and ecKP equations), and used them to obtain new classes of approximate solutions for the Euler equations. The solutions exist on the whole plane (at least formally). We hope that they could be useful in describing an intermediate asymptotics for the problems where sources, boundaries and obstacles have elliptic or nearly-elliptic geometry.

References:

[1] B.P. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970) 539-541.

[2] M.J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979) 691-715.

[3] R.S. Johnson, Water waves and Korteweg - de Vries equations, J. Fluid Mech., 97 (1980) 701-719.

[4] K.R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation, Chaos 23 (2013) 013126.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods