We prove special decay properties of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of pertu...

In this paper, the reduced differential transform method (RDTM) is applied to various nonlinear evolution equations, Korteweg–de Vries Burgers' (KdVB) equation, Drinefel’d–Sokolov–Wilson equations, coupled Burgers equations and modified Boussinesq equation. Approximate solutions obtained by the RDTM are compared with the exact solutions. The present results are in good agreement with the exact ...

We apply a multiple–time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity–free perturbation theory, we show that the well known N–soliton dynamics of the shallow water wave equation, in the particular case of α = 2β, can be reduced to the N–soliton solution that satisfies si...

Who remembers 'Hirota's method'? In the early days of solitons, although the Korteweg-de Vries equation had been solved by the 'inverse scattering method' most solutions to integrable non-linear equations were found by simpler more direct methods. Outstanding among these was a method due mainly to Hirota, which involved casting the equation into a 'bilinear form' and then applying intelligent g...

The modulational stability of both the Korteweg-de Vries (KdV) and the Boussinesq wavetralns is investigated using Whltham’s variational method. It is shown that both KdV and Boussinesq wavetrains are modulationally stable. This result seems to confirm why it is possible to transform the KdV equation into a nonlinear Schr’dinger equation with a repulsive potential. A brief discussion of Whltham...

Abstract. In this paper we consider some dissipative versions of the modified Korteweg de Vries equation ut+uxxx+ |Dx| u+uux = 0 with 0 < α ≤ 3. We prove some well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s > 1/4−α/4 on the basis of the [k; Z]−multiplier norm estimate obtained by Tao in [9] for KdV equation. 2000 Mathematics Subject Classification: 35Q5...

We solve the Vlasov equation for the longitudinal distribution function and find stationary wave patterns when the distribution in the energy error is Maxwellian. In the long wavelength limit a stability criterion for linear waves has been obtained and a Korteweg-de VriesBurgers equation for the relevant hydrodynamic quantities has been derived. Paper presented at Workshop on Instabilities of H...

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation ut + 6uux + ǫ uxxx = 0 in a critical scaling regime where x approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to ...

In this Letter, we consider a coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values using the Adomian’s decomposition method (or ADM). In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Th...

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating ...