1,720,985 research outputs found
Soliton interactions in certain square matrix nonlinear Schrödinger systems
No full textThis work deals with a class of square matrix nonlinear Schrödinger (MNLS) systems whose reductions include two equations that model hyperfine spin F= 1 spinor Bose–Einstein condensates in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable. Our main goal is to discuss the bright soliton solutions and their interactions for the focusing MNLS and for the two mixed sign systems within the framework of the inverse scattering transform. The nature of the solitons and their interactions depend on whether the associated norming constants (polarization matrices) are rank-one matrices (giving rise to ferromagnetic solitons) or full rank (corresponding to polar solitons). By computing the long-time asymptotics of the 2-soliton solutions, we determine how the polarization matrix of each soliton changes because of the interaction. Explicit formulas for the soliton interactions are given for all possible types of interacting solitons, namely ferromagnetic–ferromagnetic, polar–polar, and polar–ferromagnetic soliton interactions, and for all three inequivalent reductions of the MNLS systems that admit regular bright soliton solutions. We also present bound states, representing 2 solitons travelling with the same velocity, for all three systems
Spectral theory of the nonstationary schrodinger equation with a two-dimensionally perturbed arbitrary one-dimensional potential
No full textWe consider the nonstationary Schrodinger equation with the potential being a perturbation of a generic one-dimensional potential by means of a decaying two-dimensional function in the framework of the extended resolvent approach. We give the corresponding modification of the Jost and advanced/retarded solutions and spectral data and present relations between them. © 2005 Springer Science+Business Media, Inc
Inverse scattering transform for the complex short-pulse equation by a Riemann–Hilbert approach
No full textIn this paper, we develop the inverse scattering transform (IST) for the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity. The work extends to the complex case the Riemann–Hilbert approach to the IST for the real short-pulse equation proposed by A. Boutet de Monvel, D. Shepelsky and L. Zielinski in 2017. As a byproduct of the IST, soliton solutions are also obtained. Unlike the real SPE, in the complex case discrete eigenvalues are not necessarily restricted to the imaginary axis, and, as consequence, smooth 1-soliton solutions exist for any choice of discrete eigenvalue k1∈ C with Im k1< | Re k1|. The 2-soliton solution is obtained for arbitrary eigenvalues k1, k2, providing also the breather solution of the real SPE in the special case k2=-k1∗
The inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonzero boundary conditions.
No full textA rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values q±≡q0eiθ± as x→±∞ is presented. The direct problem is shown to be well posed for potentials q such that qx02010;q±∈L1,2(R±), for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables. © 2012 by the Massachusetts Institute of Technolog
Multiscale expansions and vector solitons of a two-dimensional nonlocal nonlinear Schrödinger system
No full textOne- and two-dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a Kadomtsev-Petviashvili equation and its cylindrical counterpart, Johnson's equation. This way, we derive approximate soliton solutions of the nonlocal NLS system, which have the form of: (a) dark or antidark soliton stripes and (b) dark lumps in the Cartesian geometry, as well as (c) ring dark or antidark solitons in the cylindrical geometry. The type of the soliton, namely dark or antidark, is determined by the degree of nonlocality: dark (antidark) soliton states are formed for weaker (stronger) nonlocality. We perform numerical simulations and show that the derived soliton solutions do exist and propagate undistorted in the original nonlocal NLS system
On-demand generation of dark soliton trains in Bose-Einstein condensates
No full textMatter-wave interference mechanisms in one-dimensional Bose-Einstein condensates that allow for the controlled generation of dark soliton trains upon choosing suitable box-type initial configurations are described. First, the direct scattering problem for the defocusing nonlinear Schrödinger equation with nonzero boundary conditions and general box-type initial configurations is discussed, and expressions for the discrete spectrum corresponding to the dark soliton excitations generated by the dynamics are obtained. It is found that the size of the initial box directly affects the number, size and velocity of the solitons, while the initial phase determines the parity of the solutions. The analytical results obtained for the untrapped system are compared to those of numerical simulations of the Gross-Pitaevskii equation, both in the absence and in the presence of a harmonic trap. The numerical results bear out the analytical results with excellent agreement
The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions
No full textThe inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values ql/r(t)=Al/re2iAl/r2t+il/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl θ r. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that (q(x,t)-q_{l/r}(t) L1,1(R±) with respect to x for all t ≥ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λ _{l/r}=\sqrt{k2+A2 t/r where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations
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