Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

Abstract

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties. We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.

Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions

“…In contrast to the original method for the integrable systems with NZBCs developed by Zakharov [35] using a two-sheeted Riemann surface, Ablowitz et al introduced a uniformization variable [36] to solve the inverse problem on a standard complex z-plane. This manner was also used to analyze the IST of the NLS equation with NZBCs by Ablowitz, et al [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. Recently, we developed the approach to present a systematical theory for the IST of both focusing and defocusing modified KdV equations with NZBCs at infinity [53].…”

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

“…If we replace 1∕Δ from (69a) into (68), the linear systems for 1 , 2 and 3 , 4 decouple. Also, (27), (30), and (71) yield: 2 (̄2) ≡ 1∕ 2 ( 1 ) ≡ ( 2 (̄1)) * , 2 ( 2 ) = 1∕( 2 (̄2)) * ≡ 1∕ 2 (̄1), Ω( 1 ) = Ω * (̄1), Ω( 2 ) = −Ω(̄1), Ω(̄2) = −Ω( 1 ) ≡ −Ω * (̄1), and introducinḡ2…”

“…which reduces to the AL equation (3) for = ℎ and = ∕ℎ 2 . The IST for Equations (4) with rapidly decaying , as → ±∞ was considered in [28], and for the focusing AL with NZBCs in [30][31][32]. The IST for the defocusing case with NZBG and 0 < < 1 was developed in.…”

The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with nonzero boundary conditions

“…And Riemann-Hilbert method can be used to investigate the soliton solutions [16] and the long-time asymptotic of integrable systems [7]. Especially, in recent years, it has become a hot topic to investigate integrable systems with nonzero boundary conditions [17][18][19][20][21][22][23][24][25][26][27][28].…”