Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions

Abstract

The inverse scattering transform (IST) with non-zero boundary conditions at infinity is presented for a matrix nonlinear Schrödinger-type equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and antiferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). Both the direct and the inverse problems are formulated in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the one-soliton solutions is analyzed in details in the self-focusing case, including some special cases not previously discussed in the literature.

Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations

“…In light of its potential applicative relevance, in this work, we develop the inverse scattering transform (IST) for the system (2) as a tool to solve the initial-value problem, as well as to obtain explicit soliton solutions. While the IST for "unreduced" matrix NLS systems, and for the "canonical" reductions corresponding to cases 1 and 2 (focusing and defocusing matrix NLS), is well established, both with zero and nonzero boundary conditions (see, for instance, [5,17,[31][32][33][34] and references therein), the IST and the soliton solutions corresponding to the reductions in cases 3 and 4 described above are novel, and present some interesting aspects and additional challenges with respect to the other two cases in that one needs to impose suitable constraints on the norming constants to guarantee that the soliton solutions are smooth for all x, t ∈ R. As a matter of fact, this work also provides several advances as far as the IST for general matrix NLS systems is concerned, including the well-known focusing and defocusing matrix NLS systems corresponding to cases 1 and 2. Specific focus of the work is to: (i) provide a rigorous definition of the norming constants that does not require any unjustified analytic extension of the scattering relations, and clarify the role of the rank of the norming constants in the spectral characterization of the corresponding solutions; (ii) properly account for all the symmetries in the potential matrix, and derive the corresponding symmetries in the scattering data (reflection coefficients and norming constants); (iii) formulate the inverse problem as a Riemann-Hilbert problem (RHP), instead of in terms of Marchenko equations; (iv) obtain novel soliton solutions for the reductions of Eq.…”

“…As mentioned before, the above equations are well known and well studied in the literature, and soliton solutions (both bright and dark, i.e., both with zero and nonzero boundary conditions) have been derived in the context of spinor BECs [10,11,[13][14][15][16][17][18]. Two other choices are possible, though.…”

Solitons and rogue waves in spinor Bose-Einstein condensates

“…Conversely, solutions in equivalence classes B, C and D describe ferromagnetic states, for which the total spin of the condensate is non-zero and the corresponding asymptotic state Φ − is not diagonal in general. Since the asymptotics Φ ± in a polar state are diagonal with only an overall phase difference [49], each φ j has the same asymptotic amplitudes as x → ±∞. Therefore, domain walls cannot form in a polar state.…”

“…[48]. All these soliton solutions are formulated in the context of the IST for this model, which was recently developed in [49]. We also discuss explicit spin polarization transformations of all these solutions that relate solitonic and rogue waves in spinor BECs and those in single component BECs.…”

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].…”