In this study, the localfractional variational iterationmethod (LFVIM) and the localfractional series expansion method (LFSEM) are utilized to obtain approximate solutions for Korteweg-de Vries equation (KdVE) within local fractionalderivative operators (LFDOs). The efficiency of the considered methods is illustrated by some examples. The results reveal that the suggested algorithms are very ef...

In the course of the years the names of Korteweg and de Vries have come to be closely associated. The equation which is named after them plays a fundamental role in the theory of nonlinear partial differential equations. What are the origins of the doctoral dissertation of De Vries and of the Korteweg-de Vries paper? Bastiaan Willink, a distant relative of both of these mathematicians, has soug...

Complexiton solutions (or complexitons for short) are exact solutions newly introduced to integrable equations. Starting with the solution classification for a linear differential equation, the Korteweg-de Vries equation and the Toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations. The crucial step in the solution process is to app...

Abstract: New exact solutions of some important nonlinear partial differential equations in one and two dimensions are obtained by using the direct algebraic method. The applicability of the method is demonstrated by applying it for the equal width wave (EW) equation, modified equal width wave (MEW) equation, improved Korteweg de Vries (IKdV) equation, modified regularized long wave (MRLW) equa...

A bridge going from Wronskian solutions to generalized Wronskian solutions of the Korteweg-de Vries equation is built. It is then shown that generalized Wronskian solutions can be viewed as Wronskian solutions. The idea is used to generate positons, negatons and their interaction solutions to the Korteweg-de Vries equation. Moreover, general positons and negatons are constructed through the Wro...

Wave group dynamics is studied in the framework of the extended Korteweg-de Vries equation. The nonlinear Schrodinger equation is derived for weakly nonlinear wave packets, and the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coe cient of the cubic nonlinear term in the extended Korteweg-de Vries equation, and for...

We consider an extended Korteweg-de Vries (eKdV) equation, the usual Korteweg-de Vries equation with inclusion of an additional cubic nonlinearity. We investigate the statistical behaviour of flat-top solitary waves described by an eKdV equation in the presence of weak dissipative disorder in the linear growth/damping term. With the weak disorder in the system, the amplitude of solitary wave ra...

In this present study the double layers structure model of extended Korteweg-de Vries(K-dV) equation will be obtained with the help of the reductive perturbation method, which admits a double layer structure in current plasma model. Then by using of new analytical method we obtain the new exact solitary wave solutions of this equation. Double layer is a structure in plasma and consists of two p...

We study the Boussinesq equation from the point of view of a multipletime reductive perturbation method. As a consequence of the elimination of the secular producing terms through the use of the Korteweg–de Vries hierarchy, we show that the solitary–wave of the Boussinesq equation is a solitary–wave satisfying simultaneously all equations of the Korteweg–de Vries hierarchy, each one in an appro...

Large amplitude multiphase solutions of the periodic Korteweg-de Vries equation are excited and controlled by a small forcing. The approach uses passage through an ensemble of resonances and subsequent multiphase self-locking of the system with eikonal-type perturbations. The synchronization of each phase in the Korteweg-de Vries wave is robust, provided the corresponding driving amplitude exce...