Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.

A birational transformation is one which leaves invariant an ordinary differential eqution, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the mater Painlevé equation P6. Then we improve...

There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra ~gl(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral...

In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we...

We consider surfaces arising from integrable partial differential equations and from their deformations. Symmetries of the equation, gauge transformation of the corrsponding Lax pair and spectral parameter transformations are the deformations which lead infinitely many integrable surfaces. We also...

Painlevé equations belong to the class y +a1 y 3 +3a2 y 2 +3a3 y +a4 = 0, where ai = ai(x, y). This class of equations is invariant under the general point transformation x = (X, Y ), y = (X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We...

A series of rational solutions are presented for an extended Lotka-Volterra eqution. These rational solutions are obtained by using Hirota's bilinear formalism and Bäcklund transformation. The crucial step is the use of nonlinear superposition fomula. The so-called extended Lotka-Volterra equation...

In this paper, we obtain some exact solutions of Derivative Reaction-Diffusion (DRD) system and, as by-products, we also show some exact solutions of DNLS via Hirota bilinearization method. At first, we review some results about two by two AKNS-ZS system, then introduce Hirota bilinearization method...

We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue...

Adjoint symmetry constraints are presented to manipulate binary nonlinearization, and shown to be a slight weaker condition than symmetry constraints in the case of Hamiltonian systems. Applications to the multicomponent AKNS system of nonlinear Schrödinger equations and the multi-wave interaction...

Hirota's bilinear technique is applied to some integrable lattice systems related to the Bäcklund transformations of the 2DToda, Lotka-Volterra and relativistic LotkVolterra lattice systems, which include the modified Lotka-Volterra lattice system, the modified relativistic Lotka-Volterra lattice system,...

Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijehuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined...

It is known that many integrable systems can be reduced from self-dual Yang-Mills equations. The formal solution space to the self-dual Yang-Mills equations is given by the so called ADHM construction, in which the solution space are graded by vector spaces with dimensionality concerning topological...

We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the charateristics of nonlocal vector fields.

In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one...

Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N-soliton solutions to our nonlocal equations corrspond to self-adjoint reflectionless analytic...

It is shown here that the possibility of the existence of new (2 + 1) dimensional intgrable equations of the modified KdV equation using the Painlevé test.

This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry.

We demonstrate, through the fourth Painlevé and the modified KdV equations, that the attempt at linearizing the mirror systems (more precisely, the equation satisfied by the new variable introduced in the indicial normalization) near movable poles can naturally lead to the Schlesinger transformations...

We consider an important class of deformations of the genus zero bihamiltonian struture defined on the loop space of semisimple Frobenius manifolds, and present results on such deformations at the genus one and genus two approximations.