Time-fractional KdV equation: formulation and?solution using variational methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg?Cde Vries (KdV) equation. Then the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann?CLiouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler?CLagrange equation. Via Agrawal??s method, one can easily derive the time-fractional KdV equation from this Euler?CLagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He??s variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.     

Solitary wave and similarity solutions of the combined KdV equation

In this paper, we discuss a property of solitary wave solutions of the combined KdV equation. Meantime, we point out that the combined KdV equation can be reduced to the Painlevé equation. Furthermore, utilizing special transformations of similarity variables, we derive a kind of new partial differential equations.     

Finite dimensional dynamics for evolutionary equations

We suggest a new method for studying finite dimensional dynamics for evolutionary differential equations. We illustrate this method for the case of the KdV equation. As a side result we give constructive solutions of the boundary problem for the Schrodinger equations whose potentials are solutions of stationary KdV equations and their higher generalizations.     

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

Approximate conservation laws of perturbed partial differential equations

This paper presents a general result on approximate conservation laws of perturbed partial differential equations. A method of constructing approximate conservation laws to systems of perturbed partial differential equations is given, which is based on approximate Noether symmetries of approximate and standard adjoint systems of the original system. The relationship between the Noether symmetry operators of approximate and standard adjoint system is established. As a result, the approach is applied to the perturbed wave equation and the perturbed KdV equation.     

Darboux transformation of generalized coupled KdV soliton equation and its odd-soliton solutions

Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.     

THE UNIQUENESS AND EXISTENCE OF SOLUTION OF THE CHABACTERISTIC PROBLEM ON THE GENERALIZED KdV EQUATION

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一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.     

1-Soliton solution of KdV6 equation

Nonlinear Dynamics - This paper applies three forms of integration tools to integrate KdV6 equation that represents a nonholonomic deformation of the well-known KdV equation, which models...     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Axially symmetric long waves on the surface of a varying-depth basin

The axially symmetric Korteweg-de Vries (KdV) equation for the case of a constant-depth basin was obtained in [1]. In the present paper we derive the axially symmetric KdV equation for a varying-depth basin. Conditions are shown for the equation obtained, under which the asymptotic behavior of its solution is described by an equation of the form $$u_t + uu_x + u_{xxx} = 0,$$ whose asymptotic behavior is well known [2].     

Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

We prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized Korteweg-de Vries equations in the subcritical case (this includes in particular the KdV equation and the modified KdV equation). This result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation, which has its own interest. The proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized KdV equation. Accepted December 1, 2000?Published online April 3, 2001     

Negative-order integrable modified KdV equations of higher orders

Nonlinear Dynamics - A variety of negative-order integrable modified KdV (mKdV) equations of higher orders is constructed. The inverse profile of the recursion operator of the modified KdV equation...     

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Her-mite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection- diffusion problem) and KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.