Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit Abstract In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillat...

In this study, we aim to construct a traveling wave solution for nonlinear partial differential equations. In this regards, a cosine-function method is used to find and generate the exact solutions for three different types of nonlinear partial differential equations such as general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKDV) and general equal width wave equ...

The Benjamin-Bona-Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg-de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. Using a computerized symbolic computation tech...

The exponential decay rate of L−norm related to the Korteweg-de Vries equation with localized damping posed on the whole real line is established. In addition, using classical arguments we determine that the solutions associated to the fully damped Korteweg-de Vries equation do not decay in H− level for arbitrary initial data.

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...

In this paper, with the aid of symbolic computation we improve the extended F-expansion method described in Chaos, Solitons and Fractals 22, 111 (2004) to solve the (2 +1)-dimensional Korteweg de Vries equation. Using this method, we derive many exact non-travelling wave solutions. These are more general than the previous solutions derived with the extended F-expansion method. They include the ...

A scheme is developed for the numerical study of the Korteweg-de Vries (KdV) and the Korteweg-de Vries Burgers (KdVB) equations with initial conditions by a homotopy approach. Numerical solutions obtained by homotopy analysis method are compared with exact solution. The comparison shows that the obtained solutions are in excellent agreement.

Solitons are ubiquitous and exist in almost every area from sky to bottom. For solitons to appear, the relevant equation of motion must be nonlinear. In the present study, we deal with the Korteweg-deVries (KdV), Modied Korteweg-de Vries (mKdV) and Regularised LongWave (RLW) equations using Homotopy Perturbation method (HPM). The algorithm makes use of the HPM to determine the initial expansion...

A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to Picard, is illustrated in the concrete case of the Lamé-Ince potentials −s(s+1)P(z), s ∈ N (P(.) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKdV) hierarchy for the first time.

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...