Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Authors : Pierre Gaillard

Pages : 154-163

Doi:10.32323/ujma.978875

View : 28 | Download : 10

Publication Date : 2021-12-30

Article Type : Research

Abstract :Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions depend on $2N-1$ real parameters.  A third representation in terms of a quotient of two real polynomials depending on $2N-2$ real parameters is given; the numerator is a polynomial of degree $2N(N+1)-2$ in $x$, $y$ and $t$ and the denominator is a polynomial of degree $2N(N+1)$ in $x$, $y$ and $t$. The maximum absolute value is equal to $2(2N+1)^{2}-2$.  We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane $(x,y)$ and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular.
Keywords : Kadomtsev Petviasvili eqaution, Fredholm determinants, Wronskians, Rational solutions