In this paper, we establish exact solutions for the timefractional Klein-Gordon equation, and the time-fractional HirotaSatsuma coupled KdV system. The Hes semi-inverse and the Kudryashov methods are used to construct exact solutions of these equations. We apply Hes semi-inverse method to establish a variational theory for the time-fractional Klein-Gordon equation, and the timefractional Hirota...

In this Letter, we analyze the structure of linearization operators of the Korteweg–de Vries (KdV) hierarchy equations expanded around single-soliton solutions. We uncover the remarkable property that these linearization operators can be factored into the integro-differential operator which generates this hierarchy and the linearization operator of the KdV equation. An important consequence of ...

The matrix 2x2 spectral differential equation of the second order is considered on x in (−∞, +∞). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter soluti...

Abstract: In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applie...

A local discontinuous Galerkin method for solving Korteweg-de Vries (KdV) type equations with non-homogeneous boundary effect is developed. We provide a criterion for imposing appropriate boundary conditions for general KdV type equations. The discussion is then focused on the KdV equation posed on the negative half plane, which arises in the modelling of transition dynamics in the plasma sheat...

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In...