Classifying bilinear differential equations by linear superposition principle

Abstract

In this paper, we investigate the linear superposition principle of exponential traveling waves to construct a sub-class of [Formula: see text]-wave solutions of Hirota bilinear equations. A necessary and sufficient condition for Hirota bilinear equations possessing this specific sub-class of [Formula: see text]-wave solutions is presented. We apply this result to find [Formula: see text]-wave solutions to the (2+1)-dimensional KP equation, a (3+1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation, a (3+1)-dimensional generalized BKP equation and the (2+1)-dimensional BKP equation. The inverse question, i.e., constructing Hirota Bilinear equations possessing [Formula: see text]-wave solutions, is considered and a refined 3-step algorithm is proposed. As examples, we construct two very general kinds of Hirota bilinear equations of order 4 possessing [Formula: see text]-wave solutions among which one satisfies dispersion relation and another does not satisfy dispersion relation.