Ian A. B. Strachan

A geometry for multidimensional integrable systems

Abstract A deformed differential calculus is developed based on an associative ★-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears correspond to taking the dispersionless limit in these hierarchies.

Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.

The Moyal bracket and the dispersionless limit of the KP hierarchy

A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sate approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, under which the Moyal bracket collapses to the Poisson bracket, is particularly simple.

Hypercomplex integrable systems

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one of which is in Cauchy-Kovaleskaya form which enables a formal general solution to be given. Various other properties of the field equations and their solutions are studied, such as their symmetry properties and the associated hierarchy of conservation laws.

The Moyal algebra and integrable deformations of the self-dual Einstein equations

Abstract A two-dimensional chiral model is studied whose gauge potentials are valued in the Moyal algebra. This algebra has recently been shown to be the most general two-index Lie algebra, and contains (either as special cases or as subalgebras) the finite classical algebras as well as the algebras of volume preserving diffeomorphisms of a two-surface. The equations so obtained are a deformation of the self-dual Einstein equations, and solutions may be constructed (using a number of standard constructions) as a power series in the deformation parameter.

Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e. the metric which defines the Hamiltonian structure has vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.The bi-Hamiltonian structure of certain multicomponent integrable systems, generalizations of the dispersionless Toda hierarchy, is studied for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e., the metric that defines the Hamiltonian structure has a vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extended to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, is given.

Jordan Manifolds and Dispersionless KdV Equations

Abstract We apply the Darboux theory of integrability to polynomial ODE’s of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3–dimensional Lotka–Volterra systems with polynomial invariant algebraic solutions linear and quadratic and determine numerous cases of integrability.

Symmetries and solutions of Getzler's equation forCoxeter and extended affine Weyl Frobenius manifolds

The G-function associated with the semisimple Frobenius manifold Cn/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G-function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold [FORMULA], where [FORMULA] is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space [FORMULA]), together with the general form of the G-function in terms of data on caustics. Symmetries of the G-function are also studied.The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1}) where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general form of the G-function in terms of data on caustics. Symmetries of the G function are also studied.

The symmetry structure of the anti‐self‐dual Einstein hierarchy

An important example of a multi‐dimensional integrable system is the anti‐self‐dual Einstein equations. By studying the symmetries of these equations, a recursion operator is found and the associated hierarchy constructed. Owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them. In addition, the Lax pair for this hierarchy is constructed.An important example of a multi-dimensional integrable system is the anti-self-dual Einstein equations. By studying the symmetries of these equations, a recursion operator is found and the associated hierarchy constructed. Owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them. In addition, the Lax pair for this hierarchy is constructed.

Deformations of the Monge/Riemann hierarchy and approximately integrable systems

Dispersive deformations of the Monge equation ut=uux are studied using ideas originating from topological quantum field theory and the deformation quantization program. It is shown that, to a high order, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system. Quasi-Miura transformations are also constructed for such deformed equations.