Lie's Structural Approach to PDE Systems

Description

Here is a lucid and comprehensive introduction to the differential geometric study of partial differential equations (PDE). The first book to present substantial results on local solvability of general and nonlinear PDE systems without using power series techniques, it describes a general approach to PDE systems based on ideas developed by Lie, Cartan and Vessiot. The central theme is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail. This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.

**

cited in Joël Merker's translation of Lie's Theory of Transformation Groups

This monograph is devoted to an approach to foundations of the theory of PDE. The study is restricted to local solvability, and a major idea is to regard PDE theory from the point of view of differential geometry. Most topics treated here can be found in the classical works of Lie, Cartan and Vessiot. However, a great effort has been made to present the ideas in a unified and very simple manner.
Introductory Chapter 1 contains a summary of the main text.
In Chapter 2 it is explained how any decent PDE system S can be considered as a submanifold of an appropriate jet bundle. The latter is equipped with its canonical contact Pfaffian system, the restriction of which to S makes S a manifold with a Pfaffian system P (Cartan) or, dually, a manifold with a vector field system V (Lie and Vessiot). The problem of solving the PDE system S then transforms into that of finding integral manifolds of P (or V) of a prescribed dimension. The simplest case, when V is complete with respect to Lie brackets, is solved using the Frobenius theorem.
The general case when there arises the derived vector field system V′⫌V is solved in Chapter 3 by means of Cartan's local existence theorem. The key idea is to first look for maximal involutions, where an involution is a subsystem I of V satisfying [I,I]⊆V. Then these involutions are specialized to complete subsystems W of V with [W,W]⊆W. Thereupon the Frobenius theorem yields the wanted integral manifolds. The step from involutions to complete subsystems is based upon a repeated application of the Cauchy-Kovalevskaya theorem and requires analyticity.
Let n be the number of independent variables of a given PDE system S. If the general n-dimensional involution In satisfies ω1∧⋯∧ωn|In≠0, then V is said to be involutive with respect to given 1-forms ω1,…,ωn. In this case Cartan's procedure yields integral manifolds of the kind wanted. Any vector field system V can be prolonged to a vector field system V(1) on a higher-dimensional manifold, and this in turn can be prolonged to V(2), and so on. Moreover there is a one-to-one correspondence between the integral manifolds of V and those of V(k) for k=1,2,3,…. Chapter 4 sketches the prolongation theorem of Cartan and Janet, which says that by a finite number of prolongations it is possible to conclude either that some V(m) is involutive with respect to given 1-forms ω1,…,ωn, in which case the wanted integral manifolds are given by Cartan's existence theorem, or that V does not admit any integral manifold on which ω1∧⋯∧ωn≠0.
According to Drach, any PDE system is equivalent to either a first- or a second-order PDE system in one dependent variable. Chapter 5 is devoted to a preliminary study of a single second-order PDE in one dependent variable, and in particular to a presence of singular vector fields, i.e., vector fields in V commuting modulo V with a greater number of vector fields than the average one does. There turn out to be either exactly two singular subsystems of V, or none at all. If there are such, the PDE is called hyperbolic when they are different, and parabolic when they coincide. This gives rise to the notion of Monge characteristic subsystems of the vector field system V: a subsystem M of V is Monge if M is singular, M∩I≠0 for any maximal involution I of V, and M∩W is complete for any maximal complete subsystem W of V. The integral manifolds of M∩W are called Monge characteristics. A special case is the Cauchy characteristic subsystem C(V)={X∈V∣[X,V]⊆V}.
In Chapter 6 the integration of vector field systems satisfying dimV′=dimV+1 is considered, which includes first-order PDE systems in one dependent variable as a special case.
By Drach's classification there then remains to consider second-order PDE systems in one dependent and n independent variables; the remainder of the monograph is devoted to the cases n=2 and 3. The main method is to look for the Monge systems and their first integrals.
Chapter 7 discusses higher-order contact transformations and prolongations of local diffeomorphisms to jet bundles.
In Chapter 8 the general solution of the defining PDE system is supposed to depend on a finite number of parameters only, in which case the Lie pseudogroup turns into a local Lie group. The main properties of this group are presented.
The study of hyperbolic second-order PDE systems requires the classification of Lie groups of dimension ≤3, and it is derived in Chapter 9.
Cartan's local existence theorem ultimately depends on the Cauchy-Kovalevskaya theorem. The latter requires analyticity, and should therefore be avoided if possible. For instance, the integration of vector field systems V with dimV′−dimV=0 or 1 is reduced to solving ODE systems only. Lie and his followers were not satisfied with general ODE systems either. An example is Lie's study of complete vector field systems admitting a nontrivial symmetry group, in which case one achieves a reduction to so-called Lie equations. Later Lie and Vessiot found that such systems can be characterized as ODE systems having the property that the general solution may be expressed as a certain function of the number of particular solutions. All this is explained in Chapter 10, which also contains Vessiot's generalization to 2-dimensional Lie vector field systems.
After these preparations, the discussion of second-order PDEs in one dependent and two independent variables is started in Chapter 11. One item is the characterization of those vector field systems that arise from second-order PDEs, and another is a sketch of Darboux's method for finding solutions by means of first integrals of the Monge systems.
Chapter 12 gives an account of Vessiot's theory of hyperbolic PDEs in one dependent and two independent variables, for which each of Monge systems admits at least two independent first integrals. Making use of these first integrals, the corresponding vector field systems are brought to a finite number of canonical forms. Remarkably, the classification thus obtained ultimately depends on the classification of 2- and 3-dimensional Lie groups.
The Goursat equations, which are a particular case of the Vessiot's theory, are further investigated in Chapter 13.
While Vessiot's theory of hyperbolic PDEs consists in a straightforward reduction to canonical forms, Cartan uses his solution of the equivalence problem in order to classify parabolic PDEs. The equivalence problem is this: given two manifolds M1 and M2 of the same dimension and having local structures S1 and S2, respectively (in our case vector fields or Pfaffian systems), is it possible to find local diffeomorphisms transforming the one structure into another? The family of self-equivalences of (Mk,Sk) forms the symmetry group of (Mk,Sk), k=1,2, and it is such symmetry groups that are the original reason for the introduction of the concept of Lie pseudogroup. Cartan's idea for solving the equivalence problem consists of two steps: first determine all local diffeomorphisms making the symmetry groups isomorphic, and then, among them, determine those which also are local equivalences. In most applications the first step alone suffices, and one is reduced to studying equivalences of Lie pseudogroups.
Chapter 14 sketches Cartan's theory of Lie pseudogroups, which is a preparation for the equivalence problem. The latter is dealt with in Chapter 15.
Cartan's results on classification of parabolic PDEs are presented in Chapters 16 and 17. First it shown that parabolic PDEs for which the Monge system admits at least two independent first integrals are equivalent to systems of two second-order PDEs admitting a Cauchy characteristic vector field. The existence of one Cauchy characteristic vector field makes it possible to reduce the integration problem to one less dimension.
Chapter 18 summarizes Cartan's work on second-order PDE systems in one dependent and three independent variables. The idea is to simplify the structure equations of the corresponding Pfaffian equations as far as possible, and then regard systems with the same reduced structures as structurally equivalent, while renouncing the detailed study of local equivalence. The reduced structure equations reveal for instance that all PDE systems consisting of at least two PDEs do admit singular subsystems, and the latter are then used in order to solve the integration problem by means of the method of Monge. The most surprising result is that all systems of two PDEs can be solved by a reduction to ODE systems.
The book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations; it will be a valuable resource for graduate students and researchers in related fields.

Reviewed by Victor V. Zharinov