It is well known that the Boussinesq equation is the bidirectional equivalent of the celebrated Korteweg-de Vries equation. Here we consider Boussinesq-type versions of two classical unidirectional integrable equations. A procedure is presented for deriving multisoliton solutions of one of these equations – a bidirectional Kaup–Kupershmidt equation. These solitons have the unusual property that...

In this paper, we consider the controllability of the Korteweg-de Vries equation in a bounded interval when the control operates via the right Dirichlet boundary condition, while the left Dirichlet and the right Neumann boundary conditions are kept to zero. We prove that the linearized equation is controllable if and only if the length of the spatial domain does not belong to some countable cri...

A method is given for finding the shifts in position of the solitons for the case of nonzero reflection coefficient. Expressions for boost generators in terms of scattering data play a prominent role in the analysis. Phase-shift formulas which show the effect of the radiation component on the soliton motion are deduced for the nonlinear Schrodinger equation, the Korteweg —de Vries equation, and...

Abstract We prove local existence and uniqueness of solutions of the focusing modified Korteweg de Vries equation ut+u 2 ux+uxxx = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to x and that solves a gener...

Elzaki transform and Adomian polynomial is used to obtain the exact solutions of nonlinear fifth order Korteweg-de Vries (KdV) equations. In order to investigate the effectiveness of the method, three fifth order KdV equations were considered. Adomian polynomial is introduced as an essential tool to linearize all the nonlinear terms in any given equation because Elzaki transform cannot handle n...

We introduce modal de Vries algebras and develop a duality between the category of modal de Vries algebras and the category of coalgebras for the Vietoris functor on compact Hausdorff spaces. This duality serves as a common generalization of de Vries duality between de Vries algebras and compact Hausdorff spaces, and the duality between modal algebras and modal spaces.

A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel'fand-Levitan type, the classic example being the Korteweg-de Vries equation. An ordinary differential equation is of Painlev type if the only singularities of its solutions in the complex plane are poles. It is shown that, under cer...

An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. 1. Introduction Our aim in this note is to display the close similarity between the well-known Cole{Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura trans...

R esum e. La propagation unidirectionnelle d'ondes de faible amplitude et de grande longueur d'onde est d ecrite, dans de nombreux syst emes physiques, par l' equation de Korteweg-de Vries. L'objet de ce travail est de proposer un probl eme mixte bien pos e lorsque le do-maine spatial est born e. Plus pr ecis ement nous etablissons l'existence de solutions locales en temps pour des donn ees ini...