New Exact Solutions of Some Two-Dimensional Integrable Nonlinear Equations via ∂-Dressing Method

In the last two decades the Inverse Spectral Transform (IST) method has been generalized and successfully applied to various (2 + 1)-dimensional nonlinear evolution equations such as Kadomtsev–Petviashvili, Davey–Stewardson, Nizhnik–Veselov–Novikov, Zakharov–Manakov system, Ishimory, two dimensional integrable sine-Gordon and others (see books [1, 2, 3, 4] and references therein). The nonlocal Riemann–Hilbert [5], ∂-problem [6] and more general ∂-dressing method of Zakharov and Manakov [7, 8] are now basic tools for solving (2 + 1)dimensional integrable nonlinear equations (see also the reviews [10, 11, 12] and books [1, 2, 3, 4]). In the present short paper new exact solutions calculated via ∂-dressing method of some two-dimensional integrable nonlinear equations such as Nizhnik–Veselov–Novikov (NVN) [13, 14], generalized Kaup–Kuperschmidt (2DKK) [16, 17] and generalized Savada–Kotera (2DSK) [16, 17] equations are reviewed. It is well known that ∂-dressing method is very powerful method for the solution of integrable nonlinear evolution equations. This method has been discovered by Zakharov and Manakov [7, 8] (see also the books [3, 4]) and applies now successfully as to (1 + 1)-dimensional and also to (2 + 1)-dimensional integrable nonlinear evolution equations. The ∂-dressing method allows to construct Lax pairs (auxiliary linear problems); to solve initial and boundary value problems, to calculate the broad classes of exact solutions of integrable nonlinear equations. By the use of ∂-dressing method one can construct simultaneously broad classes of exactly solvable potentials (variable coefficients of linear PDE’s) and corresponding wave functions of auxiliary linear problems. Let us remind following to [7, 8] basic ingredients of ∂-dressing method for (2+1)-dimensional case. At first one postulates nonlocal ∂-problem: