Envelope Periodic Solutions to Coupled Nonlinear Equations

The envelope periodic solutions to some nonlinear coupled equations are obtained by means of the Jacobielliptic function expansion method. And these envelope periodic solutions obtained by this method can degenerate tothe envelope shock wave solutions and/or the envelope solitary wave solutions.

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.

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.

Exact periodic solution in coupled nonlinear Schodinger equations

The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.

A class of coupled nonlinear Schrdinger equations:Painlev'e property,exact solutions,and application to atmospheric gravity waves

The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.

A Compact Finite Difference Schemes for Solving the Coupled Nonlinear Schrodinger-Boussinesq Equations

In this paper we are going to derive two numerical methods for solving the coupled nonlinear Schrodinger-Boussinesq equation. The first method is a nonlinear implicit scheme of second order accuracy in both directions time and space;the scheme is unconditionally stable. The second scheme is a nonlinear implicit scheme of second order accuracy in time and fourth order accuracy in space direction. A generalized method is also derived where the previous schemes can be obtained by some special values of l. The proposed methods will produced a coupled nonlinear tridiagonal system which can be solved by fixed point method. The exact solutions and the conserved quantities for two different tests are used to display the robustness of the proposed schemes.

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.

A Bilinear Bcklund Transformation and N-Soliton-Like Solution of Three Coupled Higher-Order Nonlinear Schrdinger Equations with Symbolic Computation

A bilinear Baecklund transformation is presented for the three coupled higher-order nonlinear Schroedinger equations with the inclusion of the group velocity dispersion, third-order dispersion and Kerr-law nonlinearity, which can describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses in wavelength-division multiplexed system. Starting from the Baecklund transformation, the analytical soliton solution is obtained from a trivial solution. Simultaneously, the N-soliton-like solution in double Wronskian form is constructed, and the corresponding proof is also given via the Wronskian technique. The results obtained from this paper might be valuable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communication systems.

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic-quintic nonlinear Schrdinger （CCQNLS） equations through the generalized Darboux transformation （DT）. A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higher-order localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed：（i） the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions; （ii） hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; （iii） hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α. These results further uncover some striking dynamic structures in the CCQNLS system.

New Exact Solutions for a Class of Nonlinear Coupled Differential Equations

More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.

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 .

STUDY OF EXPLICIT ANALYTIC SOLUTIONS FOR THE NONLINEAR COUPLED SCALAR FIELD EQUATIONS

By using two different transformations, several types of exact analytic solutions for a class of nonlinear coupled scalar field equation are obtained, which contain soliton solutions, singular solitary wave solutions and triangle function solutions. These results can be applied to other nonlinear equations. In addition, parts of conclusions in some references are corrected.

Solving coupled nonlinear Schrodinger equations via a direct discontinuous Galerkin method

In this work, we present the direct discontinuous Galerkin （DDG） method for the one-dimensional coupled nonlinear Schr5dinger （CNLS） equation. We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system. The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge Kutta method. Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.

Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schrodinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

Novel exact solutions of coupled nonlinear Schro¨dinger equations with time–space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.

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.