1,721,049 research outputs found
Solvability of the direct and inverse problems for the nonlinear Schrodinger equation
No full textIn this paper we study rigorous spectral theory and solvability for both the direct and inverse problems of the Dirac operator associated with the nonlinear Schrödinger equation. We review known results and techniques, as well as incorporating new ones, in a comprehensive, unified framework. We identify functional spaces in which both direct and inverse problems are well posed, have a unique solution and the corresponding direct and inverse maps are one to one
Inverse Scattering Transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening
No full textIn this paper we study the propagation of optical pulses in an optical medium with coherent three level atomic transitions. The interaction between the pulses and the medium is described by the coupled
Maxwell–Bloch equations, which we investigate by applying the method of inverse scattering transform. The details of the inverse scattering method and the non-trivial evolution of the associated scattering data are discussed. The one- and two-soliton solutions, polarization shifts due to two-soliton interactions, and the explicit form of the transmission matrix associated with pure soliton solutions are also derived
On a "Quasi" Integrable Discrete Eckhaus Equation
No full textn this paper, a discrete version of the Eckhaus equation is introduced. The discretiza- tion is obtained by considering a discrete analog of the transformation taking the con- tinuous Eckhaus equation to the continuous linear, free Schr ̈odinger equation. The resulting discrete Eckhaus equation is a nonlinear system of two coupled second-order difference evolution equations. This nonlinear (1+1)-dimensional system is reduced to solving a first-order, ordinary, nonlinear, difference equation. In the real domain, this nonlinear difference equation is effective in reducing the complexity of the discrete Eckhaus equation. But, in the complex domain it is found that the nonlinear difference equation has a nontrivial Julia set and can actually produce chaotic dynamics. Hence, this discrete Eckhaus equation is considered to be “quasi” integrable. The chaotic behavior is numerically demonstrated in the complex plane and it is shown that the discrete Eckhaus equation retains many of the qualitative features of its continuous counterpart
Discrete Vector Solitons: Composite Solitons, Yang–Baxter Maps and Computation
No full textCollisions of solitons for an integrable discretization of the coupled nonlinear Schrodinger equation are investigated. By a generalization of Manakov’s well-known formulas for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is equivalent to a sequence pairwise interactions and, moreover, the net result of the interaction is independent of the order in which such collisions occur. Further, the order-invariance is shown to be related to the fact that the map that determines the interaction of two such solitons satisfies the Yang–Baxter relation. The associated matrix factorization problem is discussed in detail and the notion of
fundamental and composite solitons is elucidated. Moreover, it is shown that, in analogy with the continuous case, collisions of fundamental solitons can be described by explicit fractional linear transformations of a complex-valued scalar polarization state. Because the parameters controlling the energy switching
between the two components exhibit nontrivial information transformation, they can, in principle, be used to implement logic operations
Soliton interactions in the vector NLS equation
No full textCollisions of solitons for two coupled and N-coupled NLS equation are investigated from various viewpoints. By suitably employing Manakov's well-known formulae for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is pairwise and the net result of the interaction is independent of the order in which such collisions occur. Further, this is shown to be related to the fact that the map determining the interaction of two solitons with nontrivial internal degrees of freedom (e.g. vector solitons) satisfies the Yang–Baxter relation. The associated matrix factorization problem is discussed in detail. Soliton interactions are also described in terms of linear fractional transformations, and the problem of existence of a solution for a basic three-collision gate, which has recently been introduced, is analysed
Integrable Nonlinear Schrodinger Systems and their SolitonDynamics
No full textNonlinear Schrodinger (NLS) systems are important examples of
physically-significant nonlinear evolution equations that can be solved by the inverse scattering transform (IST) method. In fact, the IST for discrete and continuous, as well as scalar and vector, NLS systems all fit into the same framework, which is reviewed here. The parallel presentation of the IST for each of these systems not only clarifies the common structure of the IST,
but also highlights the key variations. Importantly, these variations manifest themselves in the dynamics of the solutions. With the IST approach, one can explicitly construct the soliton solutions of each of these systems, as well as formulas from which one can determine the dynamics of soliton interaction. In
particular, vector solitons, both continuous and discrete, are partially characterized by a polarization vector, which is shifted by soliton interaction. Here, we give a complete account of the nature of this polarization shift. The polarization vector can be used to encode the value of a binary digit (“bit”) and the soliton interaction arranged so as to effect logical computations
Inverse scattering transform for the vector nonlinear Schrödinger equation with non-vanishing boundary conditions
No full textThe inverse scattering transform for the vector defocusing nonlinear Schrödinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system
Discrete and continuous nonlinear Schrödinger systems
Full text linkThis book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector
- …
