Chris Athorne

Solvable structures and hidden symmetries

We show that some examples in the literature of non-standard symmetry reductions of ordinary differential equations can be understood using the concept of a solvable structure of one-forms.

On linearization of the Ermakov system

Abstract It is shown that the nonlinear Ermakov system may be reduced to consideration of a pair of linear equations. Geometric aspects of the procedure along with analytic results pertaining to its inversion are noted. Graphical results are presented for a particular Ermakov system that arises in two-layer long wave theory.

A Z2 × R3 Toda system

Abstract We extend the classical theory of Darboux invariants from two to three dimensions in order to construct a three-dimensional Toda system. It is defined on a Z 2 lattice and has a Lax representation. The construction extends to arbitrary dimensions.

Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the -functions on the Jacobian of smooth hyperelliptic curves of genus ≤3.

On generalized Ermakov systems

Abstract The structure of a class of generalized Ermakov systems is seen to be that of an autonomous Hamiltonian system extended by a family of nonautonomous linear oscillators.

Kepler-Ermakov problems

A class of dynamical systems is presented which includes, as special cases, both the (autonomous) Ermakov system and central force problems of Kepler type with angular dependence of the force. It is shown that all members of this class are linearizable up to a pair of quadratures.

Algebraic invariants and generalized Hirota derivatives

Abstract We show that the Hirota derivative has a natural interpretation as a partial intertwining operator in the representation theory of sl(2, C ) which allows its use in the generation of algebraic invariants.

Identities for the classical genus two ℘ function

Abstract We present a simple method that allows one to generate and classify identities for genus two ℘ functions for generic algebraic curves of type (2, 6). We discuss the relation of these identities to the Boussinesq equation for shallow water waves and show, in particular, that these ℘ functions give rise to a family of solutions to Boussinesq.

Symmetries of linear ordinary differential equations

We discuss the Lie symmetry approach to homogeneous, linear, ordinary differential equations in an attempt to connect it with the algebraic theory of such equations. In particular, we pay attention to the fields of functions over which the symmetry vector fields are defined and, by defining a noncharacteristic Lie subalgebra of the symmetry algebra, are able to establish a general description of all continuous symmetries. We use this description to rederive a classical result on differential extensions for second-order equations.

The Hamiltonian structure of the (2+1)‐dimensional Ablowitz–Kaup–Newell–Segur hierarchy

By considering Hamiltonian theory over a suitable (noncommutative) ring the nonlinear evolution equations of the Ablowitz–Kaup–Newell–Segur (2+1) hierarchy are incorporated into a Hamiltonian framework and a modified Lenard scheme.