Generalized Nonlinear Schrodinger Equation Research Articles

Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation

This paper studies finite difference schemes for solving the generalized nonlinear Schrödinger (GNLS) equationiut−uxx+q(|u|2)u=f(x,t)u. A new linearlized Crank–Nicolson-type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the GNLS equation. Several schemes, including Crank–Nicolson-type schemes, Hopscotch-type schemes, split step Fourier scheme, and pseudospectral scheme, are adopted for solving three model problems of GNLS equation which arise from many physical problems. withq(s)=s2,q(s)=ln(1+s), andq(s)=−4s/(1+s), respectively. The numerical results demonstrate that the linearized Crank–Nicolson scheme is efficient and robust.

On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain

The integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher-order generalized nonlinear Schrödinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower-order continuum isotropic spin chain, is identified by carrying out a Painlevé singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.

On the dynamics of the radially symmetric Heisenberg ferromagnetic spin system

By considering the geometrical equivalence of the radially symmetric Heisenberg ferromagnetic spin system in n-arbitrary spatial dimensions and the generalized nonlinear Schrödinger equation (GNLSE) with radial symmetry, it is shown that they possess the Painlevé property only for the (n=2) circularly (planar radially) symmetric case. For the circularly symmetric case, suitable (2×2) matrix eigenvalue equations are constructed, involving nonisospectral flows and their gauge equivalence is shown. The connection with inhomogeneous systems and, in particular, the linearly x-dependent system is pointed out. Appropriate Bäcklund transformations (BT) and explicit soliton solutions for both the spin systems and the GNLSEs are also derived.

Time-dependent quantum-fluid density-functional study of high-energy proton-helium collisions.

A quantum-fluid density-functional theory (QF DFT) is proposed, yielding a time-dependent (TD) generalized nonlinear Schr\"odinger equation (GNLSE) as the equation of motion (EOM) for dealing with proton--helium-atom collisions from ``start'' to ``finish.'' The EOM contains the Weizs\"acker term as the kinetic-energy functional, apart from local exchange and correlation functionals. The GNLSE is numerically solved in cylindrical polar coordinates by a leapfrog-type finite-difference algorithm. Various TD quantities such as difference density (DD), induced-dipole moment (IDM), dipole polarizability tensor component, reaction probability, etc., have been studied to obtain physical insights into the mechanism of the TD collision process. In particular, the DD and the oscillating IDM permit a natural partitioning of the p-He collision process into approach, encounter, and departure regimes. The TD DD profiles reveal that, as a result of the interaction, p\ensuremath{\sigma} densities mix substantially into the 1s density of the He atom. Critical comments are made on the usefulness of the QF DFT approach for understanding TD processes.

Optical switching between damped bistable soliton states using periodic amplifiers

The possibility of cyclic switching between bistable soliton states of the undamped generalized nonlinear Schrodinger equation (GNLSE) has been previously studied by the authors. It is shown that it is possible to maintain bistable solitary wave solutions of the GNLSE in the presence of damping by using localized periodic amplifiers and switch states from low to high and vice versa while maintaining stability. Provided that materials with the appropriate nonlinear mechanisms can be found or fabricated, this simulation points in the direction of two-state optical transmission systems that permit signal processing.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>