Jeremy Schiff

THE CAMASSA HOLM EQUATION : CONSERVED QUANTITIES AND THE INITIAL VALUE PROBLEM

Abstract Using a Miura–Gardner–Kruskal type construction, we show that the Camassa–Holm equation has an infinite number of local conserved quantities. We explore the implications of these conserved quantities for global well-posedness.

The Camassa-Holm equation: a loop group approach

Abstract A map is presented that associates with each element of a loop group a solution of an equation related by a simple change of coordinates to the Camassa-Holm (CH) equation. Certain simple automorphisms of the loop group give rise to Backlund transformations of the equation. These are used to find 2-soliton solutions of the CH equation, as well as some novel singular solutions.

Kähler-Chern-Simons theory and symmetries of anti-self-dual gauge fields

Abstract Kahler-Chern-Simons theory, which was proposed as a generalization of ordinary Chern-Simons theory, is explored in more detail. The theory describes anti-self-dual instantons on a four-dimensional Kahler manifold. The phase space is the space of gauge potentials, whose symplectic reduction by the constraints of anti-self-duality leads to the moduli space of instantons. We show that infinitesimal Backlund transformations, previously related to “hidden symmetries” of instantons, are canonical transformations generated by the anti-self-duality constraints. The quantum wave functions naturally lead to a generalized Wess-Zumino-Witten action, which in turn has associated chiral current algebras. The dimensional reduction of the anti-self-duality equations leading to integrable two-dimensional theories is briefly discussed in this framework.

A Natural Approach to the Numerical Integration of Riccati Differential Equations

This paper introduces a new class of methods, which we call Mobius schemes, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmannian of m-dimensional subspaces of an (n+m)-dimensional vector space. Since the Grassmannians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilities is an artifact of the coordinate system, but since Mobius schemes are based on the natural geometry, they are able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.

A Kahler-{Chern-Simons} Theory and Quantization of Instanton Moduli Spaces

Abstract A five-dimensional field theory is introduced which is an analogue of three-dimensional Chern-Simons theory. The reduced phase space in the theory is a moduli space of instantons in four-dimensional euclidean gauge theory, with a symplectic structure induced by the Donaldson μ -map. Issues related to quantization are discussed.

Communication: Quantum mechanics without wavefunctions

We present a self-contained formulation of spin-free non-relativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications-theoretical, computational, and interpretational-are discussed.

Zero curvature formulations of dual hierarchies

Zero curvature formulations are given for the ‘‘dual hierarchies’’ of standard soliton equation hierarchies, recently introduced by Olver and Rosenau, including the physically interesting Fuchssteiner–Fokas–Camassa–Holm hierarchy.

The supersymmetric Camassa-Holm equation and geodesic flow on the superconformal group

We study a family of fermionic extensions of the Camassa–Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently Hamiltonian, describing geodesic flow with respect to an H1 metric on the group of superconformal transformations in two dimensions, (b) equations which are Hamiltonian with respect to a different Hamiltonian structure and (c) supersymmetric equations. Classes (a) and (b) have no intersection, but the intersection of classes (a) and (c) gives a system with interesting integrability properties. We demonstrate the Painleve property for some simple but nontrivial reductions of this system, and also discuss peakon-type solutions.

Integrability of Chern-Simons Higgs and Abelian Higgs vortex equations in a background metric

The equations for Chern–Simons–Higgs and Abelian Higgs vortices are reformulated in a certain form of background metric. Painleve analysis is applied to determine integrability of the equations, and explicit solutions for cylindrically symmetric Chern–Simons–Higgs vortices are found for a specific choice of the metric.

Complex trajectory method in time-dependent WKB.

We present a significant improvement to a complex time-dependent WKB (CWKB) formulation developed by Boiron and Lombardi [J. Chem. Phys. 108, 3431 (1998)] in which the time-dependent WKB equations are solved along classical trajectories that propagate in complex space. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper, we show that this limitation can be overcome by superposing the contributions of crossing trajectories. Secondly, we demonstrate that the approximation improves when incorporating higher order terms in the expansion. Thirdly, equations of motion for caustics and Stokes lines are implemented to help overcome Stokes discontinuities. These improvements could make the CWKB formulation a competitive alternative to current time-dependent semiclassical methods.