Derivation and simulation of the M-lump solutions to two (2+1)-dimensional nonlinear equations

The N -rational solutions to two (2+1)-dimensional nonlinear evolution equations are constructed by utilizing the long wave limit method. M -lump solutions to the two equations are derived by making some parameters conjugate to each other. We present and discuss the 1-, 2- and 3-lump solutions to the two equations. The amplitude and shape of the one lump wave remain unchanged during the propagation. The dynamic properties of the collisions among multiple lump waves are analyzed, which indicate that the fusion and fission of multiple lump waves might occur. The multiple lump waves might merge into one lump wave, then split into multiple lump waves. The lines which multiple lump waves follow are various if we choose different parameters. These results are helpful to describe some nonlinear phenomena in the areas of optics, fluid dynamics and plasma.