Riemann-Hilbert approach to the modified nonlinear Schröinger equation with non-vanishing asymptotic boundary conditions

Abstract

The modified nonlinear Schrödinger (NLS) equation was proposed to describe the nonlinear propagation of the Alfven waves and the femtosecond optical pulses in a nonlinear single-mode optical fiber. In this paper, the inverse scattering transform for the modified NLS equation with non-vanishing asymptotic boundary at infinity is presented. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. The asymptotic behaviors, analyticity and the symmetries of the Jost solutions of Lax pair for the modified NLS equation, as well as the spectral matrix are analyzed in details. Then a matrix Riemann-Hilbert (RH) problem associated with the problem of nonzero asymptotic boundary conditions is established, from which N -soliton solutions is obtained via the corresponding reconstruction formulae. As an illustrate examples of N -soliton formula, two kinds of one-soliton solutions and three kinds of two-soliton solutions are explicitly presented according to different distribution of the spectrum. The dynamical feature of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions. Finally, we analyze the differences between our results and those on zero boundary case.

Inverse scattering transformation for the Fokas-Lenells equation with nonzero boundary conditions

“…The inverse scattering transform is an important method to study physically important nonlinear wave equations with Lax pair such as the NLS equation, the modified KdV equation, Sine-Gordan equation [16,17]. As an improved version of inverse scattering transform, the Riemann-Hilbert method has been widely adopted to solve nonlinear integrable models [18][19][20][21][22][23] The paper is organized as follows. In Section 2, starting from a Lax pair, a transformation is introduced to invert the boundary conditions into constants, and then Jost solutions are obtained.…”

Long-time asymptotic behavior of the modified Schrödinger equation via Dbar-steepest descent method

“…The inverse transformation and dressing method were used to construct N-soliton solutions of the modified NLS equation (1.1) with zero boundary conditions were considered [14][15][16]. Recently, we presented inverse transformation for the modified NLS equation (1.1) with nonzero boundary conditions by using Riemann-Hilbert method [19]. From the determinant expressions of N-soliton solutions of the modified NLS equation (1.1), the asymptotic behaviors of the N-soliton solutions in the case of t → ∞ was directly derived [17].…”

Riemann-Hilbert approach for the NLSLab equation with nonzero boundary conditions

“…The study of IST for the focusing case with NZBCs is reference [38], but only part of the research results were included and no more solutions were given. Partial results were also obtained in reference [39,40,41,42,43], which was used to study the stability of the Peregrine solitons under perturbations. Regarding the MI difficulties, we can get an overview of the the subject and a historical perspective from Zakharov and Ostrovsky's [44] article.…”