Multiparametric Rational Solutions of Order N to the KPI Equation and the Explicit Case of Order 3
P. Gaillard *
Universite de Bourgogne Franche Comte, Institut de Math´ematiques de Bourgogne, 9 Avenue Alain Savary BP 47870 21078 Dijon Cedex, France.
*Author to whom correspondence should be addressed.
Abstract
We present multiparametric rational solutions to the Kadomtsev-Petviashvili equation (KPI). These solutions of order N depend on 2N − 2 real parameters. Explicit expressions of the solutions at order 3 are given. They can be expressed as a quotient of a polynomial of degree 2N(N +1)−2 in x, y and t by a polynomial of degree 2N(N +1) in x, y and t, depending on 2N − 2 real parameters. We study the patterns of their modulus in the (x,y) plane for different values of time t and parameters.
Keywords: Multiparametric rational solution, Kadomtsev-Petviashvili equation, spatial dimensions, Riemann-Hilbert problem