Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs

Abstract

Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e., those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three hour minicourse that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2-5, 2013, Evora, Portugal.

26 pages, short elementary review submitted for publication on the Proceedings of XXII IFWGP

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• Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities
• An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications
• Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge-Ampère Equations and their Characteristics
• Bicharacteristics and Fourier integral operators in Kasner spacetime
• Remarks on non-maximal integral elements of the Cartan plane in jet spaces
• An introduction to completely exceptional $2^{\text{nd}}$ order scalar PDEs
• A geometric framework to compare classical field theories and to transfer solutions between PDEs