Other Families of Rational Solutions to the KPI Equation

Abstract

Aims / Objectives: We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written 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 for each positive integer N. Place and Duration of Study: Institut de math´ematiques de Bourgogne, Universit´e de Bourgogne Franche-Cont´e between January 2020 and January 2021. Conclusion: We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x; y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of two peaks in the case of order 2.