Riemann–Hilbert approach to the modified nonlinear Schrödinger 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, we present the inverse scattering transform for the modified NLS equation iut+uxx+2|u|2u+i1α(|u|2u)x=0,α>0,with nonvanishing boundary values at infinity u(x,t)∼u±e−4iα2t+2iαx,x→±∞.An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. The direct scattering problem is shown to be well posed for potentials u such that u−u±∈L1,2(R±), for which existence and analyticity properties of eigenfunctions and scattering data are established. Their asymptotic behaviors and the symmetries are analyzed in details based on the Lax pair for the modified NLS equation. Then the inverse scattering problem is formulated as a Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions. The existence and uniqueness of solution for the mixed RH problem for t>0 are strictly proved by decomposing it into a pure RH problem and a scattering RH problem. The N-soliton solutions for the modified NLS equation are obtained via a reconstruction formulae between solution of the modified NLS equation and the solution of above mixed RH problem. As an illustrate example of N-soliton formula, for N=1 and N=2, two kinds of one-soliton solutions and three kinds of two-soliton solutions are explicitly presented, respectively 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.