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.

Coupled Nonlinear Schrodinger Equation: Symmetries and Exact Solutions

The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger （CNLS） equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.

OPTIMAL POINT-WISE ERROR ESTIMATE OF A COMPACT FINITE DIFFERENCE SCHEME FOR THE COUPLED NONLINEAR SCHRODINGER EQUATIONS

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear SchrSdinger equation. The proposed scheme not only conserves the totM mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O（h4 ＋r2） in the discrete L∞ -norm with time step - and mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme.

Existence and Uniqueness of Positive Solutions to Three Coupled Nonlinear Schrdinger Equations

In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear SchrSdinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0≤ β=μ1 = μ2 =μ3 or β 〉 max{l,μ2,μ3}.

A Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrdinger Equations with Multiply Components

Considering the coupled nonlinear Schr¨odinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact finite difference method, Fourier pseudospectral method and wavelet collocation method for spatial discretizations, a series of high accurate conservative algorithms are presented. The proposed algorithms can preserve the corresponding discrete charge and energy conservation laws exactly, which would guarantee their numerical stabilities during long time computations.Furthermore, several analogous multi-symplectic algorithms are constructed as comparison. Numerical experiments for the unstable plane waves will show the advantages of the proposed algorithms over long time and verify the theoretical analysis.

Exact Soliton Solutions for the (2+1)-Dimensional Coupled Higher-Order Nonlinear Schr¨odinger Equations in Birefringent Optical-Fiber Communication

In birefringent optical fibers, the propagation of femtosecond soliton pulses is described by coupled higherorder nonlinear Schrodinger equations. In this paper, we will investigate the bright and dark soliton solutions of（2＋1）-dimensional coupled higher-order nonlinear Schrodinger equations, with the aid of symbolic computation and the Hirota method. On the basis of soliton solutions, we test and discuss the interactions graphically between the solitons in the x-z, x-t, and z-t planes.

Higher-Order Rogue Wave Pairs in the Coupled Cubic-Quintic Nonlinear Schr?dinger Equations

We study some novel patterns of rogue wave in the coupled cubic-quintic nonlinear Schr?dinger equations.Utilizing the generalized Darboux transformation, the higher-order rogue wave pairs of the coupled system are generated.Especially, the first-and second-order rogue wave pairs are discussed in detail. It demonstrates that two classical fundamental rogue waves can be emerged from the first-order case and four or six classical fundamental rogue waves from the second-order case. In the second-order rogue wave solution, the distribution structures can be in triangle,quadrilateral and ring shapes by fixing appropriate values of the free parameters. In contrast to single-component systems, there are always more abundant rogue wave structures in multi-component ones. It is shown that the two higher-order nonlinear coefficients ρ_1 and ρ_2 make some skews of the rogue waves.

Multi-Solitons and Combined Solitons with Self-Similar Behaviors in a Birefringent Fiber with Higher-Order Effects

A coupled variable-coefficient higher-order nonlinear Schrodinger equation in birefringent fiber is studied, and analytical multi-solton, combined bright and dark soliton, W-shaped and M-shaped soliton solutions are obtained. Nonlinear tunnelling of these combined solitons in dispersion barrier and dispersion well on an exponential background is discussed, and the decaying or increasing, even lossless tunnelling behaviors of combined solitons are decided by the decaying or increasing parameter.