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.

We solve the Vlasov equation for the longitudinal distribution function and nd 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 Hig...

Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325–1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of...

It is known that a transform of Liouville type allows one to pass from an equation of the Korteweg-de Vries (K-dV) hierarchy to a corresponding equation of the Camassa-Holm (CH) hierarchy [2, 39]. We give a systematic development of the correspondence between these hierarchies by using the coefficients of asymptotic expansions of certain Green’s functions. We illustrate our procedure with some ...

In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of no...

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that using the classical spectral element method, i.e. a high order continuous Galerkin approximation, it is possible to develop satisfactory schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invar...