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.

arXiv comment38 pages

References in corpus (2)

• Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector
• The bright $N$-soliton solution of a multi-component modified nonlinear Schrödinger equation

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• Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions