Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

In this paper Lou＇s direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.

Multi-soliton solutions for the coupled modified nonlinear Schrdinger equations via Riemann–Hilbert approach

The coupled modified nonlinear Schrodinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann-Hilbert problem for the coupled modified nonlinear Schrodinger equations is formulated. And then, through solving the obtained Riemann-Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.

Computational Methods for Three Coupled Nonlinear Schrödinger Equations

In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

Four-soliton solution and soliton interactions of the generalized coupled nonlinear Schrodinger equation

Based on the generalized coupled nonlinear Schr¨odinger equation,we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method.The interactions among four solitons are also studied in detail.The results show that the interaction among four solitons mainly depends on the values of solution parameters;k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons;the pulse velocity and width mainly depend on the imaginary part of ki(i=1,2,3,4),while the pulse amplitude mainly depends on the real part of ki(i=1,2,3,4).

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schr(o|¨)dinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.

Covariant Prolongation Structure of Coupled Inhomogeneous Nonlinear Schrodinger Equation

We investigate the coupled inhomogeneous nonlinear Schrodinger equation by the covariant prolongationstructure theory, and obtain its Lax's representation. Moreover, we present the corresponding Riccati equations, Backlundtransformation, and one-soliton solution.

Standing Waves of the Coupled Nonlinear Schrdinger Equations

In this paper, we study the existence of standIng waves of the coupled nonlinear Schrodinger equations. The proofs of which rely on the Lyapunov-Schmidt methods and contraction mapping principle are due to F. Weinstein in .

Explicit High-Order Method to Solve Coupled Nonlinear Schrödinger Equations

Models of the coupled nonlinear Schrödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schrödinger equations. Many methods used to solve coupled nonlinear Schrödinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.

Sixth Order Finite Difference Method for Three Coupled Nonlinear Schrödinger Equations

In this work, we will derive a numerical method of sixth order in space and second order in time for solving 3-coupled nonlinear Schrödinger equations. The numerical method is unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we study the interaction dynamics of two solitons. It is found that both elastic and inelastic collisions can take place under suitable parametric conditions.

Some Kinds of New Composite Solutions of a Kind of Coupled Schrodinger Equation

With the help of the method that combines the first kind of elliptic equation with the function transformation, some kinds of new composite solutions of a kind of coupled Schr?dinger equation are constructed. First, a kind of function transformation is presented, and then the problem of solving solutions of a kind of coupled Schr?dinger equation can be changed to the problem of solving solutions of the first kind of elliptic equation. Then, with the help of the conclusions of the B?cklund transformation and so on of the first kind of elliptic equation, the new infinite sequence composite solutions of a kind of coupled Schr?dinger equation are constructed. These solutions are consisting of two-soliton solutions and two-period solutions and so on.

Numerical Approximation of Solution for the Coupled Nonlinear Schrodinger Equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O（τ2 ＋ h4） are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.

A Compact Scheme for Coupled Stochastic Nonlinear Schrodinger Equations

In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.

Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear Schrodinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium

Twin-core optical fibers are applied in such fields as the optical sensing and optical communication,and propagation of the pulses,Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper are the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients,which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium.Based on the integrable conditions,bilinear forms are derived,and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction.Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis.With different values of the delayed nonlinear response effect b(z),where z represents direction of the propagation,the linear-and parabolic-shaped one dark-dark soltions can be derived.Interactions between the parabolic-and periodic-shaped two dark-dark solitons are presented with b(z)as the linear and periodic functions,respectively.Directions of velocities of the two dark-dark solitons vary with z and the amplitudes of the solitons remain unchanged can be observed.Interactions between the two dark-dark solitons of different types are displayed,and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with z.We find that those interactions are elastic.

Explicit Multisoliton Solution to the Coupled Nonlinear Schrodinger Equations

Based on the inverse scattering transform for the coupled nonlinear Schrodinger （NLS） equations with vanishing boundary condition （VBC）, the multisoliton solution has been derived by some determinant techniques of some special matrices and determinants, especially the Cauchy-Binet formula. The oneand two-soliton solutions have been given as the illustration of the general formula of the multisoliton solution. Moreover, new nonsymmetric solutions corresponding to different number of zeros of the scattering data on the upper and lower half plane are discussed.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrodinger equation

The fractional quadric-cubic coupled nonlinear Schrodinger equation is concerned,and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method.The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated.Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index,respectively.The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied.Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

New Numerical Methods for the Coupled Nonlinear Schrdinger Equations

In this paper, three numerical schemes with high accuracy for the coupled Schrodinger equations are studied. The conserwtive properties of the schemes are obtained and the plane wave solution is analysised. The split step Runge-Kutta scheme is conditionally stable by linearized analyzed. The split step compact scheme and the split step spectral method are unconditionally stable. The trunction error of the schemes are discussed. The fusion of two solitions colliding with different β is shown in the figures. The numerical experments demonstrate that our algorithms are effective and reliable.

Optical solitons and modulation instability analysis of the(1+1)-dimensional coupled nonlinear Schrodinger equation

This study successfully reveals the dark,singular solitons,periodic wave and singular periodic wave solutions of the(1+1)-dimensional coupled nonlinear Schr?dinger equation by using the extended rational sine-cosine and rational sinh-cosh methods.The modulation instability analysis of the governing model is presented.By using the suitable values of the parameters involved,the 2-,3-dimensional and the contour graphs of some of the reported solutions are plotted.

Instability of Two Interacting Quasi-Monochromatic Waves in Shallow Water

The nonlinear interactions of waves with a double-peaked power spectrum have been studied in shallow water. The starting point is the prototypical equation for nonlinear unidirectional waves in shallow water, i.e. the Korteweg de Vries equation. By means of a multiple-scale technique two defocusing coupled Nonlinear SchrCMinger equations are derived. It is found analytically that plane wave solutions of such a system are unstable for small perturbations, showing that the existence of a new energy exchange mechanism which can influence the behavior of ocean waves in shallow water.

Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation

In this paper,the dynamics of the higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation¨are investigated via generalized Darboux transformation.Given a pair of linearly independent solutions of the Lax pair,the oneto three-soliton solutions are obtained via algebraic iteration.Furthermore,two and three solitons are respectively displayed via numerical simulation.Moreover,the dynamics of solitons are illustrated with corresponding evolution plots,such as elastic collisions,inelastic collisions,and bound states.It is found that there are some novel phenomena of interactions among solitons,which may provide a theoretical basis for studying optical solitons in experiments.

The Effect of Polarization Mode Dispersion on the Transmission System Using Dispersion Compensation Fiber

In this paper, we simulate the effect of Polarization Mode Dispersion( PMD ) on 40 Gb/s dispersion compensation transmission system by resolving the coupled nonlinear Schrodinger equation, and calculate the performance of system with different duty cycle order and chirp of the super Gaussian pulse. The results show that the performance of the system changes with these parameters, and there is a group of optimum parameters for the optimum performance of the system.