Introduction to characteristic surfaces and bicharacteristics

Question

I am currently studying the propagation of contact discontinuities in systems of hyperbolic PDE (multidimensional and transient). I have found that the concept of characteristics is helpful in understanding these discontinuities in single dimension transient problems.

From my investigation online, I have seen that this concept cannot be extended "as is" to higher dimensions (i.e. more than two independent variables) simply. Thus, I have come across the concept of characteristic surfaces (as compared to characteristic lines) and bi-characteristics as a tool to study information propagation in hyperbolic PDE of general structure.

However, all the papers that I find on the topic do not give a full introduction and interpretation of those (characteristic surfaces and their computation).

Hence, can anyone recommend on a good introductory paper/book to study the above topics?

Answer 1

Perhaps a classical reference on such topics, still in print at nearly 100 years from its first publication and still widely cited (at least in engineering publications) is the treatise of Courant and Hilbert on mathematical physics, precisely the second volume [1] which deals explicitly and extensively with partial differential equations. This book perhaps surveys the "classical" theory of characteristics in the most complete way. To give an idea of the contents related to the question, I could say that

• the exposition starts with the introduction of the theory of characteristic curves ([1], §I.5.1, p. 29, §I.7.5, pp. 55-56, §II.1.1, pp. 62-64), then passes to the description of families of characteristic curves for the same kind of equation in $n>2$ variables ([1], §2, pp. 70-71). Then the concept of a characteristic $(n-1)$-dimensional manifold (again $n>2$, [1], §2, pp. 73-74 and in the appendix I to chapter II, §1.1-§1.2, pp. 132-135) is introduced and described. The concepts are extensively used (see for example paragraph §IV.I.3, "Geometry of characteristics of higher order operators") in several different contexts.
• The concept of bicharacteristics is analyzed for example in paragraph VI.1.3, pp. 558-559, p. 562, but relying on the work done in chapter II.

Courant and Hilbert's presentation of the topic may be defined non-systematic, since you cannot find a unique section or chapter dedicated to it: however, the concept is constantly recalled and used in several different contexts as a means to enlighten it. Considering also other characteristics of the book (forgive the pun) it is a must-have (or perhaps a must-look-in) for every analyst specializing in PDE as well as for "applied" mathematicians, physicists and engineers.

Reference

[1] Richard Courant, David Hilbert, Methods of mathematical physics. Volume II: Partial differential equations. Translated and revised from the German Original. Reprint of the 1st Engl. ed. 1962. (English) Wiley Classics Edition. New York-London-Brisbane: John Wiley & Sons/Interscience Publishers, pp. xxii+830 (1989), ISBN: 0-471-50439-4, MR1013360, Zbl 0729.35001.