Non-existence and existence of localized solitary waves for the two-dimensional long-wave–short-wave interaction equations

In this study, we establish the non-existence and existence results for the localized solitary waves of the two-dimensional long-wave–short-wave interaction equations. Both the non-existence and existence results are based on Pohozaev-type identities. We prove the existence of solitary waves by showing that the solitary waves are the minimizers of an associated variational problem.