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 Bäcklund transformations of the equation. These are used to find 2-soliton solutions of the CH equation, as well as some novel singular solutions.
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.
This paper introduces a new class of methods, which we call Möbius 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 Möbius 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.
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.
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