Two spatial dimensional N-rogue waves and their dynamics in Mel’nikov equation

•General rogue waves of the Mel’nikov equation are presented in terms of determinants by means of the Hirota bilinear method.•The interesting and novel phenomena on rogue waves of the Mel’nikov equation are exhibited in a two spatial dimensional system.•This system admits bright- and dark-types rogue waves in a localized space.•The superpositions of such bright rogue waves are investigated graphically by different choices of the free parameters.

By means of the Hirota bilinear method, explicit representations of general rogue waves for the Mel’nikov equation are explored in terms of determinants. As applications, it is found that this system admits bright- and dark-types rogue waves localized in two dimensional space. Furthermore, the superposition of such bright rogue waves are investigated graphically by different choices of the free parameters.