Class Of Nonlinear Evolution Equations Research Articles

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2+1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

Asymptotic behavior of a class of nonlinear evolution equations

In this paper, using a new method (or technology), we prove the existence and upper semi-continuity of the global attractors A ω for a class of nonlinear evolution equations in D ( A ) × D ( A ) , where the nonlinear term f satisfies a critical exponential growth condition.

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

In this paper, we introduce a function set Ω m . There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω m . A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.

A new expansion and new families of exact traveling solutions to Hirota equation

In this paper, we construct a frame of soliton solutions and periodic solutions to nonlinear equations. And we successfully solve Hirota equation. Generally, the proposed scheme can be used to a wide class of nonlinear evolution equations.

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u x ) where cη2 ≤ G(η) ≤ Cη2 for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

UDC 517.95 Abstract: We obtain a criterion of global strong solvability for one class of nonlinear evolution equa- tions in Hilbert space.

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

Conditions for the existence of a global strong solution to a class of nonlinear evolution equations in a Hilbert space

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.

Numerical Solutions of a Class of Nonlinear Evolution Equations with Nonlinear Term of Any Order

In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt + auxx + bu + cup + du2p-1 = 0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions: numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa–Holm equation, Kolmogorov–Petrovskii–Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the ( 2 + 1 ) -dimensional asymmetric version of the Nizhnik–Novikov–Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.

Optimal L ∞ error estimates for finite element Galerkin methods for nonlinear evolution equations

Finite element Galerkin solutions for three classes of nonlinear evolution equations are considered. The existence, uniqueness and convergence of the fully discrete Crank-Nicolson scheme are discussed. At last a linearized Galerkin approximation is presented, which is also second order accurate in time fully discrete scheme.

A new high-order algorithm for a class of nonlinear evolution equation**This project is partly supported by the National Natural Science Foundation of China (no 10471067), Natural Science Foundation of Jiangsu Provincial Education Department (2005101TSJB156), Jiangsu Provincial Government Scholarship for Overseas Studies and Jiangsu Provincial NSF (no BK2006215).

We derive a new finite difference scheme which is easily extended to fourth-order accurate in both temporal and spatial dimensions. It is shown through a discrete Fourier analysis that the method is unconditionally stable for a 2D problem. It requires only a regular seven-point difference stencil similar to that used in the standard second-order algorithms, such as the Crank–Nicolson algorithm. Numerical experiments are conducted to test its high accuracy and efficiency of the new algorithm.

Nonlinear Fourier transforms, integrability and nonlocality in multidimensions

We present a nonlinear Fourier transform pair in any number of dimensions. This pair cannot be expressed in closed form but is defined through the solution of certain linear integral equations. The derivation of this result is based on the formulation of a nonlocal d-bar problem. The above nonlinear Fourier transform pair can be used for the solution of the Cauchy problem of a large class of nonlinear evolution equations in any number of spatial dimensions. Unfortunately, these nonlinear equations are highly nonlocal. Nevertheless, for the case of two dimensions they do reduce to the Davey–Stewartson equation which is a prototypical example of an integrable nonlinear evolution partial differential equation.

Stability for a class of nonlinear pseudo-differential equations

We study a class of nonlinear evolutionary equations generated by a pseudo-differential operator with the elliptic principal symbol and with nonlinearities of the form G ( u x ) where c η 2 ≤ G ( η ) ≤ C η 2 for large | η | . We demonstrate existence of a universal absorbing set, and a compact attractor, and show that the attractor is of a finite Hausdorff dimension. The stabilization mechanism is similar to the nonlinear saturation well known for the Kuramoto–Sivashinsky equation.

Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations

In this work we formally derive travelling wave solutions for the Burgers, Fisher, Huxley equations and combined forms of these equations. The tanh–coth method is used to determine these sets of travelling wave solutions. Kinks and periodic wave solutions are constructed. The proposed scheme can be used to a wide class of nonlinear evolution equations.

New traveling wave solutions for nonlinear evolution equations

The generalized Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the new exact traveling wave solutions. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in mathematical physics. As a result, many exact traveling wave solutions are obtained which include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach

In this paper, we propose a modified generalized transformation for constructing analytic solutions to nonlinear differential equations. This improved unified ansätze is utilized to acquire exact solutions that are general solutions of simpler equations that are either integrable or possess special solutions. The ansätze is constructed via the choice of an integrable differential operator or a basis set of functions. The technique is implemented to obtain several families of exact solutions for a class of nonlinear evolution equations with nonlinear term of any order. In particular, the Klein–Gordon, the Sine–Gordon and Landau–Ginburg–Higgs equations are chosen as examples to illustrate the method.

Solving solitary wave solutions of nonlinear evolution equations with the homotopy analysis method

We solved the Burgers equation by using the homotopy analysis method (HAM) and obtained its approximate kink solitary wave solutions. The solutions agree very well with the exact solutions. The results indicate that the HAM is valid for finding the solitary wave solutions of a class of nonlinear evolution equations. We also made some efforts to extend the HAM. As an example, we got the bell-like solitary wave solutions of KP equation by this approach. Our extended method can be used to find the approximate solitary wave solutions for more nonlinear evolution equations in an easy way.

A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order

With the aid of symbolic computation, a new algebraic method, named generalize Riccati equation rational expansion (GRERE) method, is devised for constructing travelling-wave solutions for nonlinear evolution equations (NEEs) with nonlinear terms of any order. Compared with most existing tanh methods for finding travelling-wave solutions, the proposed method not only recovers the results by most known algebraic methods, but also provides new and more general solutions. For illustration, we apply the generalized method to a class of nonlinear evolution equations with nonlinear term of any order. As a result, we obtain several new kinds of exact solutions. This approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order.

The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations

The extended Fan sub-equation method with a computerized symbolic is used to construct the exact travelling wave solutions for a class of nonlinear evolution equations in mathematical physics. The key idea of this method is to take full advantage of general elliptic equations involving five parameters which have more new solutions and whose degeneracies can lead to special sub equation involving three parameters like Riccati equation, first kind elliptic equation, auxiliary ordinary equation, and generalized Riccati equation. As a result, many exact travelling wave solutions are obtained which include a more general soliton solutions, triangular solutions and double periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.