$\bar\partial$-dressing method for a few ($2+1$)-dimensional integrable coupling systems

Several ($2+1$)-dimensional integrable coupling systems are derived from two sets of auxiliary linear problems, including the integrable coupling system of a ($2+1$)-dimensional generalization of the dispersive long-wave system, and a ($2+1$)-dimensional generalizations of the Burgers equation and the Davey–Stewartson system. We use the $\bar\partial$-dressing method to investigate these integrable coupling systems and obtain some exact solutions by solving the coupled $\bar\partial$-problem that we introduce. The methods and techniques presented in this paper may be good inspiration for dealing with similar problems and the corresponding integrable coupling systems.