Fractional Korteweg-de Vries Equation Research Articles

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

<p style='text-indent:20px;'>We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

Averaging Principle for Multiscale Stochastic Fractional Schrödinger–Korteweg-de Vries System

This work concerns the averaging principle for multiscale stochastic fractional Schrodinger–Korteweg-de Vries system. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, the multiscale system can be reduced to a single stochastic fractional Korteweg-de Vries equation (averaged equation) with a modified coefficient, the slow component of multiscale system towards to the solution of the averaged equation in moment. In order to prove the averaging principle, there are two key points: uniform bounds for the solution and Holder continuity of time variable for the slow component. For this, we need a crucial property—smoothing effect of the fractional Korteweg-de Vries semigroup, but the traditional method can’t help us obtain this property. Such analysis does not follow in a straightforward manner from results already available in the mathematical literature. On the contrary, it requires the introduction of some new ideas and techniques. We try to overcome this difficulty by the duality method, the interpolation arguments, the energy estimate method and refined inequality technique.

On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV)(0.1){∂tu−Dxα∂xu+u∂xu=0,x,t∈R,0<α<1,u(x,0)=u0(x). It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This property was found originally in solutions of Korteweg-de Vries (KdV) equation and it has been also verified in other dispersive equations as the Benjamin-Ono (BO) equation.Recently, it has been shown that the solutions of the dispersive generalized Benjamin-Ono (DGBO) equation, this is α∈(2,3) in (0.1); also satisfy the propagation of regularity phenomena. This is achieved by introducing a commutator decomposition to handle the dispersive part in the equation. Following the approach used in the DGBO, we prove that the solutions of the fKdV also satisfies the propagation of regularity phenomena. Consequently, this type of regularity travels with infinite speed to its left as time evolves.

On abundant new solutions of two fractional complex models

We use an analytical scheme to construct distinct novel solutions of two well-known fractional complex models (the fractional Korteweg–de Vries equation (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation). A new fractional definition is used to covert the fractional formula of these equations into integer-order ordinary differential equations. We obtain solitons, rational functions, the trigonometric functions, the hyperbolic functions, and many other explicit wave solutions. We illustrate physical explanations of these solutions by different types of sketches.

New variational characterization of periodic waves in the fractional Korteweg–de Vries equation

Periodic waves in the fractional Korteweg–de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of the quadratic form of energy subject to fixed cubic part of energy and the zero mean. This new variational characterization allows us to unfold the existence region of travelling periodic waves and to give a sharp criterion for spectral stability of periodic waves with respect to perturbations of the same period. The sharp stability criterion is given by the monotonicity of the map from the wave speed to the wave momentum similarly to the stability criterion for solitary waves.

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries (KdV) equation with conformable derivatives. With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.

Two novel linear-implicit momentum-conserving schemes for the fractional Korteweg-de Vries equation

We propose two conservative linear-implicit schemes for the space fractional Korteweg-de Vries (fKdV) equation. One is the linear-implicit Crank–Nicolson scheme and the other is the linear-implicit leap-frog scheme. In order to obtain a high order discretization in the space direction, we adopt the Fourier pseudospectral method. The Crank–Nicolson scheme and leap-frog scheme are used for temporal discretization, and those two schemes are efficient in practical computations because of their linear property. Furthermore, we analyse the uniqueness, boundness, convergence of the two schemes. Numerical experiments are presented to validate the theoretical analysis.

Numerical Solution of Time Fractional Stochastic Korteweg–de Vries Equation via Implicit Meshless Approach

In this article, a numerical scheme is implemented to approximate the solution of time fractional stochastic Korteweg–de Vries (KdV) equation. The structure of the proposed method is such that it first employ finite difference technique to transform the time fractional stochastic KdV equation into elliptic stochastic differential equations (SDEs). Then resulting elliptic SDEs has been estimated via a meshless method based on radial basis functions (RBFs). It should be mentioned that every RBFs with sufficient smoothness can be applied. In this paper, we use Gaussian RBFs which are infinitely smoothness to approximate the functions in the resulting elliptic SDEs. The most important advantage of proposed method respect to traditional numerical method is that the noise terms are directly simulated at the collocation points in each step time. Finally, some test problems are presented to investigate the performance and accuracy of the new method.

Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method

This paper introduces an approximate-analytical method (AAM) for solving nonlinear fractional partial differential equations (NFPDEs) in full general forms. The main advantage of the paper is to apply the proposed AAM to solve the fractional Korteweg-de Vries (KdV) equations. Moreover, the analytical travelling wave solutions for the fractional KdV equation and the modified fractional KdV equation are successfully obtained. The numerical solutions are also obtained in the forms of tables and graphs. The fractional partial derivatives are considered in Caputo sense.

Enhanced Existence Time of Solutions to the Fractional Korteweg--de Vries Equation

We consider the fractional Korteweg-de Vries equation $u_t + u u_x - |D|^\alpha u_x = 0$ in the range of $-1<\alpha<1$ , $\alpha\neq0$. Using basic Fourier techniques in combination with the modified energy method we extend the existence time of classical solutions with initial data of size $\varepsilon$ from $\frac{1}{\varepsilon}$ to a time scale of $\frac{1}{\varepsilon^2}$. This analysis, which is carried out in Sobolev space $H^N(\mathbb{R})$, $N \geq 3$, answers positively a question posed by Linares, Pilod and Saut (SIAM J. Math. Anal. 46 (2014), no. 2, 1505-1537).

A local fractional homotopy perturbation method for solving the local fractional Korteweg-de Vries equations with non-homogeneous term

In this paper, a local fractional homotopy perturbation method is presented to solve the boundary and initial value problems of the local fractional Korteweg-de Vries equations with non-homogeneous term. In order to demonstrate the validity and reliability of the method, two types of the Korteweg-de Vries equations with non-homogeneous term are proposed.

Nonlinear time fractional Korteweg-de Vries equations for the interaction of wave phenomena in fluid-filled elastic tubes

This work investigates the wave-wave interactions by considering two-sided time fractional Korteweg-de Vries equations (TFKdVEs). The generalized $\exp(-\Phi(\xi))$ -expansion method (GEEM) along with Khalil's fractional derivatives is employed to divulge several types of scattered wave solutions of TFKdVEs. It is found that the fractional parameters significantly modified the interaction of nonlinear wave dynamics. The results obtained may be useful for clarifications of the interaction between two waves not only in non-conservative fluid-filled elastic tubes but also in shallow water and plasma physics.

On the numerical evaluation for studying the fractional KdV, KdV-Burgers and Burgers equations

This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev polynomials of the third kind. This system is solved by using the FDM. Some theorems about the convergence analysis are stated and proved. A numerical simulation and a comparison with the previous work are presented.

An efficient method to compute solitary wave solutions of fractional Korteweg–de Vries equations

ABSTRACTConsidered here is an efficient technique to compute approximate profiles of solitary wave solutions of fractional Korteweg–de Vries equations. The numerical method is based on a fixed-point iterative algorithm along with extrapolation techniques of acceleration. This combination improves the performance in both the velocity of convergence and the computation of profiles for limiting values of the fractional parameter. The algorithm is described and numerical experiments of validation are presented. The accuracy attained by the procedure can be used to investigate additional properties of the waves. This approach is illustrated here by analysing the speed–amplitude relation.

Analytical solutions of conformable time, space, and time-space fractional KdV equations

In this study, we consider the Korteweg-de Vries (KdV) equation for solitary waves in the domain of conformable fractional calculus. By means of this fractional theory, we obtain exact solutions for time, space, and time-space fractional KdV equations and demonstrate our results graphically according to the fractional order of the related equations. Furthermore, we report that the fractional order in the solution of the time fractional KdV equation is associated with viscosity of the medium by comparing three-dimensional picture presentations of solutions of time fractional KdV and Burgers-KdV equations.

New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

The Laplace series solution for local fractional Korteweg-de Vries equation

In this paper, we consider a new application of the local fractional Laplace series expansion method to handle the local fractional Korteweg-de Vries equation. The obtained solution with non-differentiable type shows that the technology is accurate and efficient.

The local fractional series expansion solution for local fractional Korteweg-de Vries equation

In this paper, the local fractional series expansion method is used to find the series solution for the local fractional Korteweg-de Vries equation.

Higher regularity of global attractors of a weakly dissipative fractional Korteweg de Vries equation

In this paper, we study the long time behavior of solutions to a weakly dissipative fractional Korteweg de Vries (KdV) equation on the real line R. The main difficulty lies in that the dissipative term is a nonlocal operator. We overcome this difficulty by the commutator estimates and product estimates associated with fractional Laplacian. The asymptotical compactness of solution semigroup is proved by the tail estimates. Finally, we conclude the existence of (H2(R), H5(R)) global attractor of the weakly dissipative fractional KdV equation.