Rational and semi-rational solutions of the modified Kadomtsev–Petviashvili equation and the $$\varvec{(2 + 1)}$$-dimensional Konopelchenko–Dubrovsky equation

Abstract

General rational and semi-rational solutions of the modified Kadomtsev–Petviashvili (mKP) equation and the Konopelchenko–Dubrovsky equation are obtained based on the bilinear method and the KP hierarchy reduction technique. These solutions are expressed in terms of $$N \times N$$ determinants. The dynamics of the solutions, which exhibit various patterns, are thoroughly analyzed. It is shown that the rational solutions may describe the elastic interaction of a single-peak wave with either a double-peak (M-shape) wave or another single-peak wave for $$N=1$$ . Depending on the choice of parameters, the semi-rational solutions are found to depict the inelastic interaction between two (Y-shape) or three waves for $$N=1$$ . The second-order ( $$N=2$$ ) rational solutions exhibit the elastic interaction of three single-peak waves with either one double-peak wave or another single-peak wave. Inelastic interaction is displayed by proper choices of the parameters for semi-rational solutions. When $$N > 2$$ , similar local dynamical behaviors of the rational and semi-rational solutions have been observed.