Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Staeckel separable systems of Benenti type. We further show that solutions of these Staeckel systems generate a large class of finite-gap and rational solutions of cKdV hierarchies. Most of these solutions are new.Comment: 15 page

Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

We present a multiparameter generalization of the St\"ackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized St\"ackel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized St\"ackel transform.Comment: 21 pages, LaTeX 2e, no figures; major revision; Propositions 2 and 7 and several new references adde

A complete list of conservation laws for non-integrable compacton equations of K(m,m)K(m,m) type

In 1993, P. Rosenau and J. M. Hyman introduced and studied Korteweg-de-Vries-like equations with nonlinear dispersion admitting compacton solutions, $u_t+D_x^3(u^n)+D_x(u^m)=0$, $m,n>1$, which are known as the $K(m,n)$ equations. In the present paper we consider a slightly generalized version of the $K(m,n)$ equations for $m=n$, namely, $u_t=aD_x^3(u^m)+bD_x(u^m)$, where $m,a,b$ are arbitrary real numbers. We describe all generalized symmetries and conservation laws thereof for $m\neq -2,-1/2,0,1$; for these four exceptional values of $m$ the equation in question is either completely integrable ($m=-2,-1/2$) or linear ($m=1$) or trivial ($m=0$). It turns out that for $m\neq -2,-1/2,0,1$ there are only three symmetries corresponding to $x$- and $t$-translations and scaling of $t$ and $u$, and four nontrivial conservation laws, one of which expresses the conservation of energy, and the other three are associated with the Casimir functionals of the Hamiltonian operator $\mathfrak{D}=aD_x^3+bD_x$ admitted by our equation. Our result, \textit{inter alia}, provides a rigorous proof of the fact that the K(2,2) equation has just four conservation laws found by P. Rosenau and J. M. Hyman

Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited

We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x + (4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no recursion operator or master symmetry was known so far, and prove that the system (*) admits infinitely many local generalized symmetries that are constructed using a nonlocal {\em two-term} recursion relation rather than from a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and definitions adde

Maximal superintegrability of Benenti systems

For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.Comment: 5 pages, LaTeX 2e, to appear in J. Phys. A: Math. Ge

Classical R-matrix theory for bi-Hamiltonian field systems

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.Comment: review article, 39 page