We consider the weakly dissipative and weakly dispersive Burgers-Hopf-Korteweg-de-Vries equation with the diffusion coefficient ε and the dispersion rate δ in the range δ/ε → 0. We study the travelling wave connecting u(−∞) = 1 to u(+∞) = 0 and show that it converges strongly to the entropic shock profile as ε, δ → 0. Key-words Travelling waves, moderate dispersion, Korteweg de Vries equation, ...

Exact and approximate solutions of the initial-boundary value problem for the Korteweg-de Vries equation on the semi-infinite line are found. These solutions are found for both constant and time-dependent boundary values. The form of the solution is found to depend markedly on the specific boundary and initial value. In particular, multiple solutions and nonsteady solutions are possible. The an...

In this study we propose a space and time-accurate numerical method for Korteweg– de Vries equation. In deriving the computational scheme, Taylor generalized Euler time discretization is performed prior to wavelet based Galerkin spatial approximation. This leads to the implicit system which can also be solved by explicit algorithms. Korteweg– de Vries equation is also solved by a operator split...

We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an ap...

In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such a...

Recent studies of the evolution of weakly nonlinear long waves in shear flows have revealed that when the wave field contains a critical layer, a new nonlinear wave equation is needed to describe the wave evolution. This equation is of the same type as the well-known Korteweg-de Vries equation but has a more complicated nonlinear structure. Our main interest is in the steady solitary wave solut...

An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., N = 1 supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations. In connection with the extensively studied Korteweg-de Vries (KdV-) equation its co...

Abstract. We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asympto...

In recent publications [Chaos, Solitons Fractals 12 (2001) 2283; Int. J. Appl. Math. 3 (4) (2000) 361], we have dealt with the numerical solutions of the Korteweg–De-Vries (KDV) and modified Korteweg–De-Vries (MKDV) equations. We extend this study to a more general nonlinear equation, which is the general Korteweg–De-Vries (GKDV) equation, in which the previous studies is a special case of it. ...

‎in this paper‎, ‎we investigate a damped korteweg-de‎ ‎vries equation with forcing on a periodic domain‎ ‎$mathbb{t}=mathbb{r}/(2pimathbb{z})$‎. ‎we can obtain that if the‎ ‎forcing is periodic with small amplitude‎, ‎then the solution becomes‎ ‎eventually time-periodic.