In this paper we construct a family of commuting multidimen-sional differential operators of order 3, which is closely related to the KdV hierarchy. We find a common eigenfunction of this family and an algebraic relation between these operators. Using these operators we associate a hy-perelliptic curve to any solution of the stationary KdV equation. A basic generating function of the solutions ...

In this paper we describe a non-local moving frame along a curve of pure spinors in O(2m, 2m)/P , and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generaliza...

For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund–Darboux transformations and the corresponding discrete system are considered for Kupershmidt’s super KdV equation....

We give a unified view of the relation between the SL(2) KdV, the mKdV, and the UrKdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bs...

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...

In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation al...

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on a integro-differential expression. This expression belongs to a class generated by means of Virasoro vector fields acting on the KdV field. The “almost” rational KdV fields are described in terms of a geometrical locus of complex points. A class of solutions of this locus has recently appea...

We show how the supersymmetric KdV equation can be obtained from the self–duality condition on Yang–Mills fields in four dimension associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of Susy KdV equations from such a condition. We formulate the Susy KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self–duality cond...

The goal of this paper is to calculate of order reduction of the KdV typeequation and the non-isospectral KdV type equation using the μ-symmetrymethod. Moreover we obtain μ-conservation law of the non-isospectral KdVtype equation using the variational problem method.

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence...