Dbar dressing method to nonlinear Schrödinger equation with nonzero boundary conditions

Abstract

The Dbar dressing method is extended to study the focusing/defocusing nonlinear Schrödinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered. The function is meromorphic outside an annulus with center 0 and satisfies a local Dbar problem inside the annulus. The theory of such function is extended to construct the Lax pair of the NLS equation with nonzero boundary condition. In this procedure, the relation between the NLS potential and the solution of the Dbar problem is established. A certain distribution for the Dbar problem is introduced to obtain the focusing/defocusing NLS equation and the conservation laws. The explicit solutions of the focusing/defocusing NLS equation with nonzero boundary condition are given from a special distribution.

Dbar-dressing methods to nonlinear defocusing Hirota equation with nonzero boundary conditions

“…Substituting the expansion of the Laurent series for 14) and (19), by comparing the coefficient of z, we obtained…”

“…θ(z; x, t)=λ[x +(4βk 2 +2αk +2βq 2 0 )t]. (10) In recent years, the solution of the integrable equations with NZBCs has been studied extensively and the research methods are abundant [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Among these works, the inverse scattering transform and the Riemann-Hilbert problem play an important role.…”

“…The ∂-dressing method is a generalization of the Riemann-Hilbert problem but can obtain the solution more easier. Thus many scholars apply ∂-dressing method to study nonlinear integrable equations [18,19,[23][24][25][26][27][28][29][30][31][32][33][34][35][36], which include nonlinear integrable equations with NZBCs [18,19]. The ∂-dressing method for Gerdjikov-Ivanov(GI) equation with NZBCs was studied in [18].…”

“…The ∂-dressing method for Gerdjikov-Ivanov(GI) equation with NZBCs was studied in [18]. The ∂-dressing method for the nonlinear Schrödinger equation with NZBCs was studied in [19], and the explicit solutions of the focusing and defocusing nonlinear Schrödinger equation are calculated from a special distribution.…”

Dbar-approach to coupled nonlocal NLS equation and general nonlocal Reduction

“…∂ (Dbar)-problem is an effective tool to study nonlinear evolution equations and to give their explicit solutions [7-9, 15, 16, 36, 37, 39]. Recently, we extended the Dbar approach to study the NLS equation with nonzero boundary condition [38]. The existing research methods to nonlocal integrable equations are mainly the inverse scattering method (the Riemann-Hilbert problem) and the Darboux transformation.…”