Resonantly interacting lump chains in the Mel'nikov equation

Abstract

We constructed resonant lump chain solutions to the Mel'nikov equation by utilizing Hirota bilinear approach, mainly includes the resonance between lump chains, resonance between lump and lump chains. By calculating the asymptotic form of the solution, the resonant interaction of these chains results in infinite phase shifts, which is analogous to the line soliton interactions in the Kadomtsev-Petviashvili-II equation. The interactions are classified into two types, oblique and parallel cases, depending on the velocities of the individual lump chains. Our study highlights several cases of obliquely collided lump chains with a Y-shaped structure. Furthermore, the parallel resonance of lump chains can lead to the transmission, splitting, or absorption of another lump chain. The interactions between lump and lump chains are classified into two types, lump is semi-localized in time or completely localized in time.