We make two observations concerning the generalised Korteweg de Vries equation ut + uxxx = μ(|u|u)x. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for L-critical equation (p = 5) automatically implies an analogous scattering result for the L-critical nonlinear Schrödinger equation iut+uxx = μ|u|4u. Secondly, in the defocusing case μ > 0...

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg– de Vries (KdV) equation on a periodic domain and with initial condition in FLα,p spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of w...

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An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave U(x, t) under a stochastic time-dependent force is developed. The dynamics of the soliton wave U(x, t) is described by the Korteweg-de Vries Equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude . The ta...

In this work, we numerically investigate the influence of a homogeneous noise on the evolution of solitons for the Korteweg–de Vries equation. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and describe different types of behaviours. ©1999 Elsevier Science B.V. All rights reserved.

Fourth order analogue to the second Painlevé equation is studied. This equation has its origin in the modified Korteveg de Vries equation of the fifth order when we look for its self similar solution. All power and non power expansions of the solutions for the fouth order analogue to the second Painlevé equation near points z = 0 and z = ∞ are found by means of the power geometry method. The ex...

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 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...