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Three-phase solutions of the Kadomtsev-Petviashvili equation. (English) Zbl 0893.35112

The authors study real valued quasiperiodic three phase solutions of the Kadomtsev Petviashvili equation \[ u_{xt} + (3u^2)_{xx}+u_{xxxx} + 3u_{yy}=0 \] which describes a slow evolution of gravity induced waves of moderate amplitude on shallow water. The purpose of the paper is to investigate a class of three phase nonlinear waves of the Kadomtsev Petviashvili equation in order to make them available to physical applications. A three phase wave is determined by nine real parameters of physical significance (six entries of the real symmetric positive definite period matrix and three wave vectors) and three more arbitrary phase constants.
The authors define a fundamental region for parameters of a real valued theta function of three variables and study conditions for which the period matrix is indecomposable. Under the last condition a method to calculate the wave vectors and frequencies is described in detail. As a result the authors obtain a procedure to generate all smooth, real valued, three phase Kadomtsev Petviashvili solutions that can be expressed in terms of Riemann theta functions of three variables. They consider different representations of theta functions which are computationally efficient and investigate conditions under which three phase solutions are one or two phase solutions and/or time independent in some uniformly translating coordinate system.
By computer simulation the authors find large families of almost stationary solutions which can be of practical interest. Along with the approach stated above the authors also discuss shortly an alternative approach which was proposed earlier by Bobenko and Bordag when the solutions are computed in terms of certain Poincaré series. In order to make these results as accessible as possible the authors provide a computer program that permits to specify the parameters of three phase solutions and to observe the evolution of solutions in time.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
14H42 Theta functions and curves; Schottky problem
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