. A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a biHamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to new integrable hierarchies endowed with nonlinear dispersion that supports compactons, or cusped and/or peaked solitons. A general algorithm for effecting this new duality between classical solitons and their non-smooth counterparts is illustrated by the construction of dual versions of the modified Korteweg-deVries equation, the nonlinear Schrodinger equation, the integrable Boussinesq system used to model the two way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These new hierarchies include a remarkable variety of new, interesting integrable nonlinear differential equations. PACS numbers: 03.40.Kf, 47.20.Ky, 52.35.Sb, 63.20.Ry y Supported in Part by NSF Grants DMS 92--04192 and 95--00931. z ...

this paper, we will present such an algorithm and its implementation in the computer algebra system AXIOM.

Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder–Weyl (DW). The bi-vertical (n + 1)-form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset it gives rise to an invariantly defined map between horizontal forms playing the role of dynamical variables and the so-called vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Z-graded Lie algebra on the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a “higher-order ” and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian n-form H ˜ vol ( ˜ vol is the space-time volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. 1

Abstract. This paper outlines a new general construction, named “multi-space”, that forms the proper geometrical foundation for the numerical analysis of differential equations — in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations. Application of the theory of moving frames leads to a general framework for constructing symmetry-preserving numerical approximations to differential invariants and invariant differential equations.

There are three different actions of the unimodular Lie group SL(2) on a two-dimensional space. In every case, we show how an ordinary differential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation. A particular example is the Chazy equation, whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation. The reduction method leads to an alternative formula in terms of solutions to the Lamé equation, resulting in a surprising transformation between the Lam'e and hypergeometric equations. Finally, we discuss the Painlevé analysis of the singularities of solutions to the Chazy equation.

A systematic way of construction of (2+1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the so-called central extension procedure and classical R-matrix applied to the Poisson algebras of formal Laurent series. Results are illustrated with the known and new (2+1)dimensional dispersionless systems. (To appear in J. Phys. A: Math. Gen.) 1

The Painlevé property, one century later Abstract. The “Painlevé analysis ” is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations. To achieve our goal, we will not start the exposition with a more or less precise “Painlevé test”. On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe. The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincaré. A summary can be found at the beginning of each chapter. The Painlevé property, one century later, ed. R. Conte, CRM series in mathematical

In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.

The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u t = F (t, x, u, u x )u xx + G(t, x, u, u x ). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use, relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial di#erential equations under study.

Abstract. Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k = 1, 2, · · · , n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k> n. In this 2 paper we prove the existence of solutions for the remaining cases 2 ≤ k ≤ n, assuming that 2 the equation is variational. 1.