A Family of the Inertial Manifolds for a Class of Generalized Kirchhoff-Type Coupled Equations

The paper considers the long-time behavior for a class of generalized high-order Kirchhoff-type coupled equations, under the corresponding hypothetical conditions, according to the Hadamard graph transformation method, obtain the equivalent norm in space , and we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectral interval condition.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how con- straints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact in- terface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This pa- per describes the field affected by the gravity constraints. Applying the multi-scale anal- ysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analy- sis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic per- turbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface.

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.

Exact solutions for four coupled complex nonlinear differential equations

In this paper, exact solutions are derived for four coupled complex nonlinear different equations by using simplified transformation method and algebraic equations.

Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.

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.

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.

New finite-gap solutions for the coupled Burgers equations engendered by the Neumann systems

On the tangent bundle TSN-1 of the unit sphere SN-l, this paper reduces the coupled Burgers equations to two Neumann systems by using the nonlinearization of the Lax pair, whose Liouville integrability is displayed in the scheme of the r-matrix technique. Based on the Lax matrix of the Neumann systems, the Abel-Jacobi coordinates are appropriately chosen to straighten out the restricted Neumann flows on the complex torus, from which the new finite-gap solutions expressed by Riemann theta functions for the coupled Burgers equations are given in view of the Jacobi inversion.

Exact solutions for the coupled Klein-Gordon-Schrǒdinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations （K-G-S equations） by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

EXPLICIT SOLUTIONS TO THE COUPLED KdV EQUATIONS WITH VARIABLE COEFFICIENTS

By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

The periodic solutions for coupled integrable dispersionless equations

By using the Jacobi elliptic-function method, this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.

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.

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation. In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-bright- rogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.