Two-dimensional new coherent structures of lump-soliton solutions for the Mel’nikov equation

Abstract

Under investigation in this paper are new novel coherent structures of two-dimensional lump-soliton for the Mel’nikov equation. The Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction method are applied to construct a particular family of determinant semi-rational solutions exhibiting various coherent waves to the Mel’nikov equation. We first investigate some novel coherent waves, [Formula: see text]th-order lumps first appear from the [Formula: see text] dark line solitons and finally disappear into those [Formula: see text] dark line solitons after living on the constant background for a very short period. In contrast to the usual lump, those lumps in the coherent structures of lump-soliton are not only localized in two-dimensional space and but also localized in time.