Three‐Phase Solutions of the Kadomtsev–Petviashvili Equation

The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N, based on a Riemann theta-function of N variables. For N=1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions.

[1]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[2]  Igor Krichever,et al.  METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS , 1977 .

[3]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water , 1989, Journal of Fluid Mechanics.

[4]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[5]  N. Scheffner,et al.  A note on the generation and narrowness of periodic rip currents , 1991 .

[6]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[7]  Akira Nakamura,et al.  A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evolution Equations. I. Exact Two-Periodic Wave Solution , 1979 .

[8]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water. Part 2. Asymmetric waves , 1995, Journal of Fluid Mechanics.

[9]  Ljudmila A. Bordag,et al.  Periodic multiphase solutions of the Kadomtsev-Petviashvili equation , 1989 .

[10]  Akira Nakamura,et al.  A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evolution Equations. : II. Exact One- and Two-Periodic Wave Solution of the Coupled Bilinear Equations , 1980 .

[11]  H. Segur,et al.  An Analytical Model of Periodic Waves in Shallow Water , 1985 .

[12]  W. K. Hayman,et al.  TOPICS IN COMPLEX FUNCTION THEORY VOL. II , 1971 .

[13]  B. Dubrovin,et al.  REAL THETA-FUNCTION SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI EQUATION , 1989 .

[14]  Takahiro Shiota,et al.  Characterization of Jacobian varieties in terms of soliton equations , 1986 .

[15]  David W. Lewis,et al.  Matrix theory , 1991 .

[16]  H. Minkowski,et al.  Diskontinuitätsbereich für arithmetische Äquivalenz , 1989 .