Rational and semi-rational solutions to the Mel’nikov equation

Abstract

We investigate rational and semi-rational solutions to the ( 2 + 1 ) -dimension Mel’nikov equation based on the Hirota bilinear method and the long wave limit approach. It is shown that the derived rational solutions display lump solitons in ( x , y ) -plane and line rogue waves in ( x , t ) -plane. As to the semi-rational solutions consisting of the hybrid of the soliton and rational solution, they exhibit the interaction between solitons and lumps in ( x , y ) -plane. While in ( x , t ) -plane the hybrid solutions demonstrate the interaction between line rogue waves and solitons. Dynamics of solutions are analysed with plots.