In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a γ law for the pressur...

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...

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, ...

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...

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...

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...

When a system supports two distinct long-wave modes with nearly coincident phase speeds, the weakly nonlinear and linear dispersion unfolding generically leads to two coupled Korteweg-de Vries equations. In this paper, we review the derivation of such systems in stratified fluids, extending previous studies by allowing for background shear flows. Coupled Korteweg-de Vries systems have very rich...

In the first part of this paper, we prove the existence of a global strong solution for the Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a γ law for the p...

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...