Rational localized wave patterns in the form of Schur polynomials for the (2 + 1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili-I equation in fluid dynamics

Abstract

In this paper, we derive a family of rational localized wave solutions with any order in the Bogoyavlenskii–Kadomtsev–Petviashvili-I equation in terms of the Kadomtsev–Petviashvili reduction method. These rational localized waves are expressed by the Grammian determinants, and the entries of the determinant are presented by means of the Schur polynomials, which provide convenience in discussing the dynamics of localized wave solutions. According to the parity of the element indexes in the higher-order determinants, we mainly discuss two different types of higher-order rational localized wave solutions. Tuning the free parameters of the higher-order rational localized waves, lump-type localized waves of various polygon patterns, such as triangle, quadrangle, and pentagon, are obtained. It is shown that when one of these free parameters in the higher-order rational localized waves becomes sufficiently large, the localized wave solutions given by the odd indexes element are made up of the first-order fundamental rational localized wave. However, the higher-order rational localized wave solutions given by the even indexes element can exhibit not only the polygon wave patterns given by the first-order fundamental rational localized wave but also other novel hybrid wave patterns. These hybrid wave patterns consist of the first-order fundamental rational localized wave and other higher-order fundamental rational localized wave. These results will help us to better understand the wave patterns and control of nonlinear localized waves in fluid dynamics.