The construction of exact rational solutions with constant asymptotic values at infinity of two-dimensional NVN integrable nonlinear evolution equations via the $\overline{\partial}$-dressing method

Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

“…One of the problems for future interest consists in looking for the corresponding analogues of these solutions in the obtained generalizations. The same question concerns lumps and rogue wave solutions that were investigated in several integrable systems recently [3,20,21,22,23,24,62,81] It is also known that inverse scattering and spectral methods [1,7,41,57] were applied to generate solutions of equations with self-consistent sources [6,27,46]. An extension of these methods to the obtained hierarchies and comparison with results that can be provided by BDTs (e.g., following [67]) presents an interest for us.…”

New exact solutions of two-dimensional integrable equations using the $\bar \partial $ -dressing method

“…2. Such a use of the reality condition was successfully applied in several earlier works devoted to constructing classes of exact solutions of some (2+1)-dimensional integrable nonlinear evolution equations using thē ∂-dressing method [4], [23]- [25]. But it is possible to not use the weak-field limit and to impose the reality condition u = u directly on exact complex solutions satisfying only the potentiality condition [28].…”

“…The reality condition for u in (2.4) and the potentiality condition for the operator L 1 lead to some restrictions on the kernel R 0 of the∂-problem. For example, for the elliptic version (σ = i, κ 1 = κ 2 = κ), i.e., for the Veselov-Novikov (VN) equation with the complex space variables ξ = z and η = z, the reality condition for u in the weak-field limit (χ = 1 in (1.6)) leads to the restriction [23] …”

Mach-Type Soliton in the Novikov-Veselov Equation

“…To obtain the real potential U , the following reality conditions [9] for resonance have to be considered…”