Next-nearest neighbor interaction and localized solutions of polymer chains

We study localization in polymer chains modeled by the nonlinear discrete Schr?dinger equation (DNLS) with next-nearest-neighbor (n-n-n) interaction extending beyond the usual nearest-neighbor exchange approximation. Modulational instability of plane carrier waves is discussed and it is shown that localization gets amplified under the influence of an enhanced interaction radius. Furthermore, we construct exact localized solitonlike solutions of the n-n-n interaction DNLS. To this end the stationary lattice system is cast into a nonlinear map. The homoclinic orbits of unstable equilibria of this map are attributed to standing solitonlike solutions of the lattice system. We note that in comparison with the standard next-neighbor interaction DNLS which bears only one type of static soliton-like states (either staggering or unstaggering) the one with n-n-n interaction radius can support unstaggering as well as staggering stationary localized states with frequencies lying above respectively below the linear band. Generally, the stronger the n-n-n interaction on the DNLS lattice the smaller are the maximal amplitudes of the standing solitonlike solutions and the less rapid are their exponential decays. Received 4 October 2000     

Soliton Solutions of the Discrete Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

Exact soliton solutions of the dark discrete nonlinear Schrtidinger (DNLS) equation with nonvanishing boundary conditions are found and especially it is shown that the dark DNLS equation can have both dark and bright soliton solutions. Some solitary wave solutions of the DNLS equation with nonvanishing boundary conditions are also derived.     

On Kaup and Newell＇s Method for Solving DNLS Equation

Kaup and Newell＇s revised inverse scattering transform for the derivative nonlinear Schrodinger （DNLS） equation is investigated. We compared it with a more reasonable approach proposed recently, which is rigorously proven by the Liouville theorem. It is conduded that Kanp and Newell＇s revision is only suitable for giving single-soliton solution and can not be generalized to multi-soliton case.     

Generation of short-lived large-amplitude magnetohydrodynamic pulses by dispersive focusing

Large-amplitude MHD waves are routinely observed in space plasmas. We suggest that dispersive focusing, previously proposed for the excitation of freak waves in the ocean, can be also responsible for the excitation of short-lived large-amplitude MHD waves in space plasmas. The DNLS equation describes MHD waves propagating in plasmas at moderate angles with respect to the equilibrium magnetic field. We obtained an analytical solution of the linearised DNLS equation governing the generation of large-amplitude MHD waves from small-amplitude wave trains due to the dispersive focusing. Our numerical solutions of the full DNLS equation confirm this result.     

Inverse Scattering Transform in Squared Spectral Parameter for DNLS Equation under Vanishing Boundary Conditions

After a transformation, the inverse scattering transform for the derivative nonlinear Schr6dinger （DNLS） equation is developed in terms of squared spectral parameter. Following this approach, we obtain the orthogonality and completeness relations of free Jost solutions, which is impossibly constructed with usual spectral parameter in the previous works. With the help these relations, the Zakharov-Shabat equations as well as Marchenko equations of IST are derived in the standard way.     

Freak waves in laboratory and space plasmas

Generation of large-amplitude short-lived wave groups from small-amplitude initial perturbations in plasmas is discussed. Two particular wave modes existing in plasmas are considered. The first one is the ion-sound wave. In a plasmas with negative ions it is described by the Gardner equation when the negative ion concentration is close to critical. The results of numerical solution of the Gardner equation with the modulationally unstable initial condition are presented. These results clearly show the possibility of generation of freak ion-acoustic waves due to the modulational instability. The second wave mode is the Alfvén wave. When this wave propagates at a small angle with respect to the equilibrium magnetic field, and its wave length is comparable with the ion inertia length, it is described by the DNLS equation. Studying the evolution of an initial perturbation using the linearized DNLS equation shows that the generation of freak Alfvén waves is possible due to linear dispersive focusing. The numerical solution of the DNLS equation reveals that the nonlinear dispersive focusing can also produce freak Alfvén waves.     

On the Riemann–Hilbert problem of a generalized derivative nonlinear Schrödinger equation

In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schr?dinger(DNLS)equation.By establishing a matrix Riemann-Hilbert problem and reconstructing potential function q(x,t)from eigenfunctions{Gj(x,t,η)}3/1 in the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line.     

Comment on Revision of Kaup-Newell＇s Works on IST for DNLS Equation

In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.     

Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation

A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.     

Breather and double-pole solutions of the derivative nonlinear Schrödinger equation from optical fibers

The derivative nonlinear Schrödinger (DNLS) equation, which governs the propagation of the femtosecond optical pulse in a monomodal optical fiber, is analytically studied in this Letter. Breather and double-pole solutions are derived from the two-soliton solution with the choice of parameters. It is found that the parameters in the DNLS equation cannot only control the phase and propagation direction of the breather and double pole, but also influence the interaction period of the breather. Elastic collisions between a breather and a dark/anti-dark soliton are studied by the qualitative analysis and graphical illustration. The stability of the breather and double-pole solutions is also analyzed.     

Comment on Revision of Kaup-Newell's Works on IST for DNLS Equation

In this note, it is shown that the revision of the Kaup-Newell's works on IST for DNLS equation is only available in the case of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.     

求解DNLS方程的反散射法的基本问题

对DNLS方程, 反散射法应选取k²=p为基本谱参数, 其中k为通常谱参数. 并引入一个新的谱参数及一个规范变换, 由此得到自由Jost解的规范正交系, 这就是反散射法的基本问题. 同时还得到了基于谱参数p的Marchenko方程和Zakharov-Shabat反散射方程. 关键词： DNLS方程 反散射法 自由Jost解 规范正交系     

Exact Soliton Solutions for DNLS Equation by the Method of Riemann Problem with Zeros

With the procedure to solve explicitly the equations of the Riemann matrix with poles, multisoliton solutions to the DNLS equation are found formally. A single soliton solution is given explicitly and the asymptotic behaviors of multisoliton solutions are discussed.     

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrödinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.     

On the chaotic aspects of three wave interaction in a magnetized plasma

Nonlinear processes in magnetized plasma are very much important for the proper understanding of many space and astrophysical events. One of the most important type of study has been done in the domain of Alfven waves. Here we show that a Galerkin type approximation of the DNLS (Derivative Nonlinear Schrödinger) equation describing such wave propagation leads to a new type of nonlinear dynamical systems, very much rich in chaotic properties. Starting with the detailed analysis of fixed points and stability zones we make an in depth study of the unstable periodic orbits, which span the whole attractor. Next the birth of a Hopf bifurcation is identified and normal form, limit cycle analyzed. In the course of our study the detailed structure of the attractor is analyzed. A possibility of internal crisis is also indicated. These results will help in the choice of the plasma parameters for the actual physical situation.     

Superfluidity versus disorder in the discrete nonlinear Schrödinger equation

We study the discrete nonlinear Schr?dinger equation (DNLS) in an annular geometry with on-site defects. The dynamics of a traveling plane-wave maps onto an effective nonrigid pendulum Hamiltonian. The different regimes include the complete reflection and refocusing of the initial wave, solitonic structures, and a superfluid state. In the superfluid regime, which occurs above a critical value of nonlinearity, a plane wave travels coherently through the randomly distributed defects. This superfluidity criterion for the DNLS is analogous to (yet very different from) the Landau superfluidity criteria in translationally invariant systems. Experimental implications for the physics of Bose-Einstein condensate gases trapped in optical potentials and of arrays of optical fibers are discussed.     

Conformal structure-preserving method for damped nonlinear Schrdinger equation

In this paper,we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger(DNLS) equation.Based on its damped multi-symplectic formulation,the DNLS system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF) method and implicit midpoint method are employed in spatial and temporal discretizations,respectively.For the dissipative part,we can solve it exactly.The proposed method conserves the conformal momentum conservation law in any local time-space region.With periodic boundary conditions,this method also preserves the total conformal momentum and the dissipation rate of momentum exactly.Numerical experiments are presented to demonstrate the conservative properties of the proposed method.     

Peregrine solitons and gradient catastrophes in discrete nonlinear Schrödinger systems

In the present work, we examine the potential robustness of extreme wave events associated with large amplitude fluctuations of the Peregrine soliton type, upon departure from the integrable analogue of the discrete nonlinear Schrödinger (DNLS) equation, namely the Ablowitz–Ladik (AL) model. Our model of choice will be the so-called Salerno model, which interpolates between the AL and the DNLS models. We find that rogue wave events are drastically distorted even for very slight perturbations of the homotopic parameter connecting the two models off of the integrable limit. Our results suggest that the Peregrine soliton structure is a rather sensitive feature of the integrable limit, which may not persist under “generic” perturbations of the limiting integrable case.     

Soliton Solutions of DNLS Equation Found by IST Anew and its Verification in Marchenko Formalism

A general procedure is proposed to derive the multi-soliton solutions of DNLS equation with vanishing boundary value, and the two-soliton solutions of it is given as an example. Furthermore, the verification of multi-soliton solutions is done through Marchenko formalism. PACS numbers: 05.45.Yv; 02.30.-f; 11.10.Ef