Variable-coefficient KdV Equation Research Articles

Equatorial Rossby Solitary Wave Under the External Forcing

A simple shallow-water model with influence of external forcing ona β-plane is applied to investigate the nonlinear equatorialRossby waves in a shear flow. By the perturbation method, theextended variable-coefficient KdV equation under an externalforcing is derived for large amplitude equatorial Rossby wave in ashear flow. And then various periodic-like structures for theseequatorial Rossby waves are obtained with the help of Jacobielliptic functions. It is shown that the external forcing plays animportant role in various periodic-like structures.

A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation

A general mapping deformation method is applied to a generalized variable coefficient KdV equation. Many new types of exact solutions, including solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions and other exact excitations are obtained by the use of a simple algebraic transformation relation between the generalized variable coefficient KdV equation and a generalized cubic nonlinear Klein–Gordon equation.

A new generalized expansion method and its application in finding explicit exact solutions for a generalized variable coefficients KdV equation

A new generalized expansion method and its application in finding explicit exact solutions for a generalized variable coefficients KdV equation

A new generalized expansion method and its application in finding explicit exact solutions for a generalized variable coefficients KdV equation

A generalized expansion method is proposed to uniformly construct a series of exact solutions for general variable coefficients non-linear evolution equations. The new approach admits the following types of solutions (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic and solitary wave solutions and (f) Jacobi and Weierstrass doubly periodic wave solutions. The efficiency of the method has been demonstrated by applying it to a generalized variable coefficients KdV equation. Then, new and rich variety of exact explicit solutions have been found.

Soliton-like solutions for a (2+1) -dimensional nonintegrable KdV equation and a variable-coefficient KdV equation

Based on a Riccati equation and a symbolic computation system-Maple, a generalized Riccati equation expansion method is presented for constructing soliton-like solutions and periodic form solutions for some nonlinear evolution equations (NEEs) or NEEs with variable coefficients. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. We study a (2+1)-dimensional general nonintegrable KdV equation, a KdV equation with variable coefficients. As a result, rich new families of exact solutions, including the non-travelling wave's and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained. When setting the arbitrary functions in some solutions be equal to special constants or special functions, the solitary wave solutions can be recovered.

Improved tanh-function method and the new exact solutions for the general variab le coefficient KdV equation and MKdV equation

In this paper, by using the extended tanh-function method,the general variable coefficient KdV and MKdV equations are reduced to first-order variable coefficient nonlinear ordinary differential equations,and then the new exact solutions for these equations,which include exact soliton-like,rational formal and triangle function solutions,are obtained through solving these ordinary differential equations-

An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications

By using the homogeneous balance principle, an auto-Backlund transformation (BT) to the generalized KdV equation with variable coefficients is derived. The auto-BT only involves one quadratic homogeneity equation to be solved. Solving the homogeneity equation by use of the ε-expansion method and using the auto-BT, generally speaking, we can obtain an exact solution containing N-solitary wave of the generalized KdV equation with variable coefficients. As an illustrative example, we obtain an exact solution containing 2 solitary wave of the equation in detail. Since the generalized KdV (GKdV) equation, cylindrical KdV equation and variable coefficient KdV equation are all the special cases of the generalized KdV equation with variable coefficients, the corresponding results of these equations are also given respectively.

Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations

Based on the idea of the homogeneous balance (HB) method, we study the Bäcklund transformation and similarity reductions of general variable coefficient KdV equation. It is found that the corresponding results are coincide with those by Weiss–Tabor–Carnevale (WTC) method and Clarkson–Kruskal (CK) method, respectively. The close connections among the HB, WTC and CK methods are exposured in theory. In the meantime, a Lax pair, a symmetry, two conservation laws and a analytic solution for the general variable coefficient KdV equation are given.

Exact solutions for general variable-coefficient KdV equation

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable-coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable-coefficient KdV equation is given.

New bäcklund transformation and exact solutions for variable coefficient KdV equation

In this paper, with the aid of Lax pairs, a new Backlund transformation for the variable coefficient KdV equation is found. Based on the Backlund transformation, only if integration is needed, a series of exact solutions can be obtained. This method is important for finding more new and physical-signficant solutions.

Exact Solutions of a Variable-Coefficient KdV Equation Arising in a Shallow Water

This paper studies a variable-coefficient KdV (vcKdV) equation arising in a shallow water. For a practically acceptable water bottom, we find that it can be transformed into the cylindrical KdV equation. As a result, several exact bounded solutions are obtained. One of the solutions has the shape of a solitary wave with a shelf behind, which confirms approximate analytical and numerical results in the literature.

Auto-Bäcklund transformation and analytic solutions for general variable-coefficient KdV equation

We find analytic solutions for the general variable-coefficient KdV equation of the form u t + f( t) uu x + g( t) u xxx =0. We make use of both the application of the truncate Painlevé expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed analytic solutions with a constraint on f( t) and g( t).

Bäcklund transformation, lax pairs, symmetries and exact solutions for variable coefficient KdV equation

The homogeneous balance method is extended to seek for Backlund transformation, Lax pairs, non-local symmetries of variable coefficient KdV equation (VCKdVE). Then based on the Backlund transformation and general solutions of a fourth-order nonlinear ordinary differential equation, five kinds of exact solutions of VCKdVE are derived. The soliton-like solution also belongs to these solutions.

Exact solutions of the variable coefficient kdV and sg type equations

In this paper,the variable cofficient KdV equation with dissipative loss and nonuniformity terms and the variable coefficient SG equation with nonuniformity term are studied. The exact solutions of the KdV and SG equations are obtained. In particular,the soliton solutions of two equations are found.

On KdV type equations

The variable coefficient KdV equation and KdV-Burger's equation are analyzed using the decomposition method.

Direct Similarity Solution Method and Comparison with the Classical Lie Symmetry Solutions

Abstract We study the general applicability of the Clarkson–Kruskal’s direct method, which is known to be related to symmetry reduction methods, for the similarity solutions of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or ansatz before performing any explicit reduction expansions. We shall apply the method to both scalar and vector NEEs in either 1+1 or 2+1 dimensions, including in particular, a variable coefficient KdV equation and the 2+1 dimensional Khokhlov–Zabolotskaya equation. Explicit solutions that are beyond the classical Lie symmetry method are obtained, with comparison discussed in this connection.

Symmetries, conservation laws and Hamiltonian structures of the non-isospectral and variable coefficient KdV and MKdV equations

An infinite number of form-invariant symmetries is obtained and a one-to-one correspondence between symmetries and conservation laws is established for the non-isospectral and variable coefficient generalizations of both the KdV and the MKdV equations. Two families of symmetries and their Lie algebraic structures are constructed. Some interesting facts about their Hamiltonian structures are presented.

Scattering of Solitons from the Density Gradient of a Relativistic Bounded Plasma with Variable Streaming

We study a relativistic bounded plasma consisting of hot ions and electrons with density gradient and variable streaming. It is observed that the nonlinear wave generated inside is governed by a variable coefficient KdV equation, which we analyse numerically. Furthermore, the solitary wave is scattered at the discontinuity due to inhomogeneity, and the characteristics of the left and right moving wave become different. Such scattering usually takes place in the solar wind, interplanetary shocks, and in the environment of a comet where, as the relativistic effect gives rise to streaming, the non-uniformity of the plasma leads to a density variation. By evaluating a suitable form lof such variations we compute the amplitude, velocity and width of the soliton before and after the scattering. The amplitude of the reflected wave is seen to decrease as in previous observations.

Symmetries of the Variable Coefficient KdV Equation and Three Hierarchies of the Integro differential Variable Coefficient KdV Equation

By using a simple method to factorize the recursion operator, the inverse recursion operator of the variable coefficient KdV equation is exhibited explicitly. Three new sets of symmetries of the variable coefficient KdV equation are given in addition to the known K symmetries and τ symmetries. Starting from these three sets of symmetries, we obtained three hierarchies of the variable coefficient KdV integro-differential equations.

Symmetries and Infinite Dimensional Lie Algebra of the Variable Coefficient KDV Equation11The project supported by the Natural Science Foundation of Zhejiang Province and NSF of China.

With the help of the conditional strong symmetry, we obtain the symmetries, symmetries, infinite dimensional Lie algebra and the corresponding infinite conservation quantities of the variable coefficient KdV equation with and being three arbitrary functions of .