General Multi-Breather, High-Order Lump and Semi-Rational Solutions of the (2 + 1)-Dimensional Mel’nikov Equation

In this work, we show several types of (semi)-rational solutions for the two dimensional Mel’nikov equation. We derive multi-breather wave solutions. They present the oblique breathers, homoclinic orbits and their mixtures under two parameter constraints. We also obtain the general high-order lump solutions by employing the Schur polynomials. When internal parameters a_2m+1 are large, the patterns display the polygons containing lower-order lump pattern in the central region together with possible multiple fundamental lumps in the outer region. When time t is large, two types of lump patterns are presented by considering the prediction solutions. The first type of lump patterns contain fundamental lumps with triangular shapes. The second type of lump patterns contain fundamental lumps with non-triangular shapes in the outer region, together with possible fundamental lumps with triangular shapes in the inner region. Furthermore, we construct the semi-rational solutions to exhibit the fascinating processes of fusion and fission between lumps and line solitons. These solutions should be of great interest in a variety of high-dimensional multi-component systems.