Exact Solutions for a Higher-Order Nonlinear Schrodinger Equation in Atmospheric Dynamics

By giving prior assumptions on the form of the solutions, we succeed to find several exact solutions for a higher-order nonlinear Schroetinger equation derived from one important model in the study of atmospheric and ocean dynamical systems. Our analytical solutions include bright and dark solitary waves, and periodical solutions, which can be used to explain atmospheric phenomena.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.

Sub-ODE's New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F12 = AF2 ＋ BF2＋p ＋ CF2＋2p （where F1= dF/dε, p 〉 0） are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 （2007） 115] and IS. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 （2008） 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematics.

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.

Nonautonomous solitons in the continuous wave background of the variable-coefficient higher-order nonlinear Schrodinger equation

We reduce the variable-coefficient higher-order nonlinear Schrodinger equation （VCHNLSE） into the constantcoefficient （CC） one. Based on the reduction transformation and solutions of CCHNLSE, we obtain analytical soliton solutions embedded in the continuous wave background for the VCHNLSE. Then the excitation in advancement and sustainment of soliton arrays, and postponed disappearance and sustainment of the bright soliton embedded in the background are discussed in an exponential system.

The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.

Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations

This paper mainly deals with the higher-order coupled Kirchhoff-type equations with nonlinear strong damped and source terms in a bounded domain. We obtain some results that are estimation of the upper bounds of Hausdorff dimension and Fractal dimension of the global attractor.

The pth-order periodic solutions for a family of N-coupled nonlinear SchrSdinger equations

By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrodinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.

Periodic Waves of a Discrete Higher Order Nonlinear SchrSdinger Equation

The Hirota equation is a higher order extension of the nonlinear Schr6dinger equation by incorporating third order dispersion and one form of self steepening effect, New periodic waves for the discrete Hirota equation are given in terms of elliptic functions. The continuum limit converges to the analogous result for the continuous Hirota equation, while the long wave limit of these discrete periodic patterns reproduces the known resulr of the integrable Ablowitz-Ladik system.

PERMANENCE AND ASYMPTOTIC PROPERTIES OF NONLINEAR DELAY DIFFERENCE EQUATIONS

The asymptotic behavior of a class of nonlinear delay difference equation wax studied . Some sufficient conditions are obtained for permanence and global attractivity . The results can be applied to a claxs of nonlinear delay difference equations and to the delay discrete Logistic model and some known results are included.

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrodinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schr?dinger equation,since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices.Based on the Lax pair,the binary Darboux transformation is constructed under certain constraints,thus the multi-dark soliton solutions are presented.Soliton propagation and collision are graphically discussed with the group-velocity dispersion,third-and fourth-order dispersions,which can affect the solitons’velocities but have no effect on the shapes.Elastic collisions between the two dark solitons and among the three dark solitons are displayed,while the elasticity cannot be influenced by the above three coefficients.

Localized Properties of Rogue Wave for a Higher-Order Nonlinear Schr?dinger Equation

In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higherorder nonlinear Schr6dinger equation （HONLS） by the Darboux transformation and confirm the decomposition rule of the rogue wave solutions up to fourth-order. These solutions have two parameters a and ;3 which denote the contribution of the higher-order terms （dispersions and nonlinear effects） included in the HONLS equation. Two localized properties, i.e., length and width of the first-order rogue wave solution are expressed by above two parameters, which show analytically a remarkable influence of higher-order terms on the rogue wave. Moreover, profiles of the higher-order rogue wave solutions demonstrate graphically a strong compression effect along t-direction given by higher-order terms.

Darboux Transformations, Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrdinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrdinger(HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and secondorder rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation on the rogue waves is discussed with the help of graphical simulation.

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.

Solitons and Other Solutions for the Generalized KdV-mKdV Equation with Higher-order Nonlinear Terms

The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.

Modulation instability,rogue waves and conservation laws in higher-order nonlinear Schrodinger equation

In this paper,the modulation instability(MI),rogue waves(RWs)and conseryation laws of the coupled higher-order nonlinear Schrodinger equation are investigated.According to MI and the 2×2 Lax pair,Darboux-dressing transformation with an asymptotic expansion method,the existence and properties of the one-,second-,and third-order RWs for the higher-order nonlinear Schrodinger equation are constructed.In addition,the main characteristics of these solutions are discussed through some graphics,which are draw widespread attention in a variety of complex systems such as optics,Bose-Einstein condensates,capillary fow,superfluidity,fluid dynamics,and finance.In addition,infinitely-many conservation laws are established.

Breathers and Rogue Waves Derived from an Extended Multi-dimensional N-Coupled Higher-Order Nonlinear Schrdinger Equation in Optical Communication Systems

In this paper, an extended multi-dimensional N-coupled higher-order nonlinear Schr¨odinger equation (NCHNLSE), which can describe the propagation of the ultrashort pulses in wavelength division multiplexing (WDM)systems, is investigated. By the bilinear method, we construct the breather solutions for the extended (1+1),(2+1) and(3+1)-dimensional N-CHNLSE. The rogue waves are derived as a limiting form of breathers with the aid of symbolic computation. The effect of group velocity dispersion (GVD), third-order dispersion (TOD) and nonlinearity on breathers and rogue waves solutions are discussed in the optical communication systems.