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.

Keywords

• Hirota method
• Lump solution
• Line rogue wave
• Long wave limit