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Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Advanced Studies in Contemporary Math, Volume 1): Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M.

Introduction

Questions related to the structure and properties of nonlinear differential operators abound in various branches of modern mathematics - pure as well as applied. They are so numerous, and the methods of solution created for them so multiform, that the picture of an oriental bazaar comes to mind; the idea of regularizing this picture within the framework of a unified theory seems to be purely Utopian. R. Courant, for example writes: "Questions related to partial differential equations of order higher than the first are so varied that the construction of a unified general theory does not appear possible" (p. 159 [0.13]). This point of view, undoubtedly, correctly reflects the state of affairs that prevailed fifteen or twenty years ago. However, times change, and today we definitely have grounds for reasonable optimism.

The development of the language and the notable simplification of techniques in such branches of mathematics as differential geometry and commutative algebra now provide the mathematician with the long-awaited possibility of thinking about differential equations in adequate terms. The category theory style of thought and, in particular, the habit of drawing commutative diagrams help him find his way through the huge amount of experimental material that has accumulated since the time of Sophus Lie, after whom practically all active theoretical research of a truly general character in nonlinear equations was suspended.

The classical period in the history of nonlinear differential equations, which can be represented by G. Monge and S. Lie, was concluded when the main ideas concerning "first-order" problems were clarified. These include the theory of one first-order equation (Monge-Cauchy-Lie) and the parallel Hamilton-Jacobi theory. The former, in the hands of Lie, acquired the remarkably harmonious form of contact geometry, while the latter transformed itself much later into symplectic geometry. Further, the Pfaff problem, i.e., the equation <o = 0 where u> is a differential form of the first order, was solved in general (Monge-Darboux-Goursat). The final result is now known as Darboux's lemma. Finally, the "Frobenius theorem" was discovered. At the same time the foundations of Galois theory for differential equatons, or the Picard-Vessiot theory (Liouville-Picard-Vessiot), were developed (in our day this has become differential algebra). Other important developments, due to Lie and his pupils, were the theory of differential invariants, the theory of Lie equations, and the general theory of symmetries (finite and infinitesimal) of differential equations. To this we should add fragments of the theory of intermediate integrals, the setting and preliminary study of the "general integral" problem for higher order equations (Ampere - Cauchy-Darboux), the Cauchy-Kovalevsky theorem, the foundations of the general theory of characteristics, Backlund transformations and Lie's interpretation of numerous methods of explicit integration of differential equations in terms of symmetry. To this list we should perhaps add some of the technical achievements of the classical period (e.g., results of the Stokes formula type) as well as important observations and results obtained in the calculus of variations, mathematical physics, etc. (e.g., Noether's theorem or the theory of strong discontinuities due to Hugoniot-Renkin), which essentially also belong to the general theory of differential equations.

The prevailing role of geometrical methods in this period should be noted. It is no accident that the men who symbolize it - Monge and Lie - were, above all, geometers, while Darboux's classical work, Theorie generate des surfaces, was much more concerned with differential equations than with the theory of surfaces as such. The classical heritage in nonlinear differential equations did not fare very well in the subsequent set-theoretic period: most of it either was forgotten or did not reach the user (working in mechanics and mathematical and theoretical physics). Thus, in our day, textbooks in differential equations do not usually contain an exposition of the theory of first-order equations as contact geometry, while some specialists in the theory are still convinced that the solution of the Cauchy problem for quasilinear equations of the first order is impossible since the characteristics intersect. Another typical example is the theory of similitude and dimension, developed by L. Sedov and G. Birkhoff, which is actually a very special case of the theory of Lie transformations published much earlier. This fact was noticed only recently and used by L. Ovsyannikov [0.16].

E. Cartan was apparently the first to understand the importance of an invariant language in the situation that had developed ("I wanted to develop a theory which would include notions and operations independent of any change of variables . . .; to do this it was necessary to replace partial derivatives by differentials, which have an invariant meaning." [2.1]. The calculus of differential forms, which he developed in this connection, yielded deep results concern- concerning "general solutions" and compatibility. Further, the language of differential forms turned out to have many other applications and, in keeping with current abstract algebraic trends, has now become an attribute of all of mathematics. Cartan's works bridged the gap between classical and modern results. It should also be noted that two of his theories have been developed significantly in the last 20 years.

The first arose from the general compatibility problem and is now known as formal theory. Under the combined efforts of Spencer, Quillen, and Goldschmidt, it has now acquired, in many of its aspects, the form of a finished product. The second - the theory of structures and Lie equations - is still rapidly developing and is associated with Spencer, Sternberg, Guillemin, Goldschmidt, Malgrange, and others.

All these new works use the language and techniques of algebraic topology and modern differential geometry. Note that many fundamental concepts (which belong, in fact, to the general theory of differential operators), including the notion of the jet introduced by S. Ehresmann, first took shape within the framework of differential geometry.

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