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...

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 ...

Wronskian determinants are used to construct exact solution to integrable equations. The crucial steps are to apply Hirota’s bilinear forms and explore linear conditions to guarantee the Plücker relations. Upon solving the linear conditions, the resultingWronskian formulations bring solution formulas, which can yield solitons, negatons, positions and complexitons. The solution process is illust...

We consider the evolution of small amplitude, long wavelength initial data by a polyatomic Fermi–Pasta–Ulam lattice differential equation whose material properties vary periodically. Using the methods of homogenization theory, we prove rigorous estimates that show that the solution breaks up into the linear superposition of two appropriately scaled and modulated counterpropagating waves, each o...

A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N0-graded left A-module with two covariant derivatives acting on it which, as a consequence of certain (e.g., nonlinear differential) equations, are flat and anticommute. As a consequence, there is an iterative construction of generalized conserved currents. We associate a gauged bi-different...

Abstract. We consider a Korteweg-de Vries equation perturbed by a noise term on a bounded interval with periodic boundary conditions. The noise is additive, white in time and “almost white in space”. We get a local existence and uniqueness result for the solutions of this equation. In order to obtain the result, we use the precise regularity of the Brownian motion in Besov spaces, and the metho...

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.