Lump waves from the degeneration of lump chains in the (2+1)-dimensional Boussinesq-type equation

We construct lump waves of the Boussinesq-type equation through two distinct approaches and present the corresponding figures. In the first method, lump chains are reduced to lump waves in two steps. Initially, appropriate parameters are selected to control the periods and amplitudes of the lump chains. When the periods and amplitudes of all lump chains tend to be identical, the lump chains exhibit coupling and form coupled lump chains. Then, as the periods of the coupled lump chains approach infinity, these coupled lump chains are finally reduced to lump waves. In the second approach, using the improved long-wave limit method, we directly reduce lump chains to lump waves.