Rudolf Schmid

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity Dinfw,0supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. Dinfw,0supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.

Symplectic integration of Sine-Gordon type systems

We construct a class of symplectic integration schemes to general Sine–Gordon type systems. We also conduct several numerical tests for these symplectic schemes. Our numerical results demonstrate the effectiveness of these schemes for numerical computation of the solutions to the general Sine–Gordon type systems.

A symplectic algorithm for wave equations

Numerical schemes for finite-dimensional Hamiltonian system which preserve the symplectic structure are generalized to infinite-dimensional Hamiltonian systems and applied to construct finite difference schemes for the nonlinear wave equation. The numerical results show that these schemes compare favorably with conventional difference methods. Furthermore, the successful long-term tracking capability for these Hamiltonian schemes is remarkable and striking.

Local cohomology in gauge theories, BRST transformations and anomalies

Abstract We introduce a geometric framework needed for a mathematical understanding of the BRST symmetries and chiral anomalies in gauge field theories. We define the BRST bicomplex in terms of local cohomology using differential forms on the infinite jet bundle and consider variational aspects of the problem in this cohomological context. The adjoint representation of the structure group induces a representation of the infinite dimensional Lie algebra g of infinitesimal gauge transformations on the space of local differential forms, with respect to which the BRST bicomplex is defined using the Chevalley-Eilenberg construction. The induced coboundary operator of the associated cohomology H ∗ loc ( g ) is the BRST operator s. With this we derive the classical BRST transformations of the vector potential A and the ghost field η as s A = dη+[A, η] , and s η = - 1 2 [η, η] . Moreover the ghost field η is identified with the canonical Maurer-Cartan form of the infinite dimensional Lie group G of gauge transformations. We give a homotopy formula on the BRST bicomplex and with the introduction of Chern-Simon type forms we derive the associated descent equations and show that the non-Abelian anomalies, which satisfy the Wess-Zumino consistency condition, represent cohomology classes in H 1 loc ( g ) .

Limiting the complexity of limit sets in self-regulating systems☆☆☆

In the analysis of models of ecosystems one seeks to discover conditions that limit the complexity of the possible behavior of solutions since “intuitively” one feels that the full range of possible complex behaviors in systems of order three or more ought not to occur for simple ecological interactions. It has been shown by Hirsch [6,7] that the solutions of competitive and cooperative systems have limit sets which cannot be more complicated than invariant sets of systems of one lower dimension. In particular, autonomous 2-dimensional systems of these types have only “trivial” dynamics in the sense that all bounded solutions approach equilibrium asymptotically. In planar systems, for example, the absence of limit cycles makes the dynamics trivial in the sense that bounded solutions can have only critical points or orbits connecting critical points in their omega limit sets [3]. In this note we prove a theorem which appears to be useful in limiting the complexity of limit sets for a class of biologically important equations.

Bäcklund transformations induced by symmetries

Abstract We give a general method for the derivation of Backlund transformations induced by symmetries for soliton equations that are compatibility conditions of certain 2×2 linear systems. As an application of this method, we obtain new Backlund transformations for the discrete mKdV equation: ∂q n / ∂t =(1+ q 2 n ) ( q n +1 - q n −1 ), nonlinear Schrodinger equation: i ∂q n /∂t=q n+1 +q n−1 -2q n +q n q ∗ n (q n+1 +q n−1 ) , sine-Gordon equation: ∂q n +1 / ∂t - ∂q n / ∂t =2 h (sin q n +1 +sin q n ) and KdV equation: ∂q n / ∂t =exp(- q n −1 )-exp(- q n +1 ).

Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwells equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

The quadratic‐Hamiltonian theorem in infinite dimensions

It is shown that a smooth diffeomorphism on a symplectic Banach space is canonical if it is canonoid with respect to all dynamical systems whose Hamiltonian functions are quadratic.

On infinite‐dimensional variational principles with constraints

The Lagrange multiplier theorem is generalized for constrained functions on dual pairs of Banach spaces. Then a variational principle for dual pairs of Banach spaces is proven for the case when the constraint set is given by a symmetry and it is generalized to Banach manifolds.

A Finite Dimensional Completely Integrable System Associated with the WKI- and Heisenberg Hierarchies

Abstract We consider the following spectral problem (1) where u, v, w are smooth functions. It produces a hierarchy of evolution equations with an arbitrary function A m−1. This hierarchy includes the WKI [8] and Heisenberg [7] hierarchies by properly selecting the special function A mm−11. We derive this new evolution equations, and give the finite dimensional completely integrable systems (FDCIS) associated with theses equations.