Application of the $\bar\partial$-dressing method to a $(2+1)$-dimensional equation

A remarkable method for investigating solutions of nonlinear soliton equation is the $\bar\partial$-dressing method. Although there are other methods that can also be used for that aim, the $\bar\partial$-dressing method is the most transparent and leads directly to the final results. The $(2+1)$-dimensional Sawada–Kotera equation is studied by analyzing the eigenfunction and the Green's function of its Lax representation as well as by the inverse spectral transformation, yielding a new $\bar\partial$ problem. The solution is constructed based on solving the $\bar\partial$-problem by choosing a proper spectral transformation. Furthermore, once the time evolution of the spectral data is determined, we are able to completely obtain a formal solution of the Sawada–Kotera equation.