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

Abstract

We review new classes of exact solutions with functional parameters with constant asymptotic values at infinity of the Nizhnik-Veselov-Novikov equation and new classes of exact solutions with functional parameters of two-dimensional generalizations of the Kaup-Kupershmidt and Sawada-Kotera equations, constructed using the Zakharov-Manakov∂-dressing method. We present subclasses of multisoliton and periodic solutions of these equations and give examples of linear superpositions of exact solutions of the Nizhnik-Veselov-Novikov equation.

About linear superpositions of special exact solutions of Nizhnik-Veselov-Novikov equation

“…with complex constant coefficients A n and complex discrete spectral parameters M n = Λ n leads to simple determinant formula 22,24…”

“…It was shown in the papers 22,24 that to such real solutions u leads the following choice of parameters a n = −a n := ia n0 , µ n = − ǫ λ n , n = 1, . .…”

“…Several classes of exact solutions of VN equation (1) have been constructed in last three decades (1980 -2010) via different methods 10,11,[13][14][15][16][17][18][19][20][21][22][23][24] , see also the books 3,4 . These solutions include finite-zone type solutions 11 , rationally localized solutions [13][14][15][16][17]20,21 or lumps, solutions with functional parameters 10,18,23,24 , multi-line soliton solutions 22,24 and so on. Underline that the first auxiliary linear problem (3) is nothing but the 2D stationary Schrödinger equation so exact solutions of VN equation constructed via all IST approaches are also transparent potentials of this Schrödinger equation.…”

“…are also exact solutions of VN equation (1). For convenience here some useful formulas of ∂-dressing method for VN equation (1) 3,4,[20][21][22][23][24] are presented. Central object of this method is the scalar wave function χ(λ; z,z, t)…”

“…In the present note we do not use the limit of weak fields and impose the reality condition u = u directly to calculated complex solutions satisfying only to potentiality condition. This approach makes it possible to receive besides multi-line soliton solutions also plane wave type singular periodic solutions (this was shown at first in 22,24 ) and their's superpositions.…”