Groupes de Lie et équations différentielles

Benno Fuchssteiner a écrit:

Since the discovery of the beautiful structure of the Korteweg-de Vries equation (KdV for short) in the 1960s, new frontiers in the treatment of nonlinear partial differential equations have been reached and numerous important techniques and methods have been developed. Apart from this–at least in this field–a new consciousness about historical perspectives has arisen. Many monographs treating recent developments in this field can be found but yet still missing was a book taking account of the historical context while reflecting the general spirit of research done in this area in a comprehensive way and at the same time presenting a broad view of some major parts of the recent developments. Ideally, the book should even do this in such a way that it may serve as a textbook for beginners.
Here is such a book, or at least a book which comes as close to these aims as possible at the present time. After the book of R. L. Anderson and H. Kh. Ibragimov [Lie-Bäcklund transformations in applications, SIAM, Philadelphia, Pa., 1979; MR0520395] which, although it was written ten years ago, still is an invaluable source, the present book is the second attempt to give an introduction to the classical treatment of applications of Lie groups to differential equations on one side and to serve on the other side as a textbook giving the beginner access to some beautiful mathematics developed over the last two decades.
The author succeeds very well in fulfilling one of his principal aims, namely to enable the reader to apply Lie groups as a basic computational tool. The book consists of two major parts. In the first part symmetry transformations in finite-dimensional spaces, i.e., only between dependent and independent variables of partial differential equations, are treated. These transformations are called "symmetries'' or sometimes "geometrical symmetries''. Here large parts of the fundamental work of Sophus Lie and other eminent mathematicians of that time are presented. But the material is presented from a somewhat enlarged viewpoint so that the ideas may serve as an introduction to the more general aspects of the second part. For this reason an introduction into contact transformations is omitted and consideration is concentrated mostly on what sometimes are called Lie-point transformations.
In the second part the author presents an introduction to what he calls "generalized symmetries'' or what sometimes in the literature is termed "Lie-Bäcklund symmetry'' (or transformation), or just "symmetry''. Roughly speaking, the difference between these two notions is that symmetries can be considered as transformations in finite-dimensional space, whereas the generalized symmetries are transformations in infinite-dimensional spaces.
With respect to presentation the general spirit of the book is reflected at the very beginning where the author quotes that "it is far easier to abstract a general mathematical theory from a single well-chosen example than to apply an existing abstract theory to a specific example''. Following this, the author presents numerous examples and exercises which are extremely beneficial for every reader.
Before entering into details let me mention what subjects cannot be found. The most important are: Inverse scattering theory and algebro-geometrical viewpoints. Furthermore, very little can be found about explicit solutions. Even the formula for the two-soliton of the KdV equation is not presented (although this is next to the most trivial group invariant solution if generalized symmetries are admitted).
Of course, no book can contain all these things, and the author makes up for this by his excellent notes and remarks presented at the end of each section. Although the author claims that little about manifolds is necessary he gives a fairly complete introduction into this subject and its differential aspects. This covers about 70 pages. No infinite-dimensional manifolds are treated; this is because of the extensive use made of the maximal rank condition. This absence of infinite-dimensional manifolds will cause some problems in the second part, i.e., Chapters 5 and 7.
In the second chapter, symmetry groups (i.e. Lie point transformations) and their infinitesimal treatment are presented. Since the method of "careless computation of formal variational derivatives on infinite-dimensional manifolds'' is avoided, the prolongation method has to be the basic tool in this chapter.
In Chapter 3 group invariant solutions and reductions are presented. In Chapter 4 conservation laws, mostly presented by their characteristics, are introduced. Of course, this chapter culminates in a presentation of Emmy Noether's result about the interrelation between conservation laws and symmetries. This result is obtained by relating the characteristics of a conservation law with what are called the characteristics of a symmetry, standing for an infinitesimal generator of a one-parameter symmetry group. A notable distinction from other presentations of Noether's theorem is that here it really is presented in full generality, and not just in the Hamiltonian case.
All these chapters make smooth and pleasant reading. They are rounded out by a variety of carefully worked out examples, and a lot of material is presented which may be difficult to find elsewhere. Some of the material is based on the author's own research.
In Chapter 5 generalized symmetries are admitted by allowing the coefficients of infinitesimal generators to depend also on derivatives of the dependent variable. These can be understood as infinitesimal generators of one-parameter groups acting on some (infinite-dimensional) manifold of smooth functions.
Since infinite-dimensional manifolds are mostly avoided, the basic technique for treating these quantities is again the prolongation method. Recursion operators for generating infinite hierarchies of symmetry generators (or their characteristics) are introduced; a subject which we partly credit to the author's pioneering research in this field.
As examples the popular exactly solvable systems (like KdV, mKdV, sine-Gordon, etc.) are presented. Then the suitable generalizations of the results given for the finite-dimensional case are presented: conservation laws, Noether's theorem and so on. In addition bi-Hamiltonian systems are introduced and their compatibility (in the sense of Gel'fand-Dorfman or Magri) is discussed.
Of course, the generation of infinitely many symmetries and conservation laws in case of compatible bi-Hamiltonian systems is presented. (Here one should have added that even when the two Hamiltonian structures are not compatible a bi-Hamiltonian system gives rise to infinitely many symmetry group generators, although the group then may be nonabelian.)
However, basing consideration about generalized symmetries solely on the prolongation method prevents the author from incorporating systems which are not differential equations, but are nevertheless completely integrable (for example, the Benjamin-Ono equation, the intermediate long wave equation and many others). Another difficulty arising from this approach can be seen in context with the treatment of the recursion operator where, for example, the inverse of differentiation causes some unnecessary complications.
For example in the KdV case one always has to ensure that the terms to which the recursion operator is applied are total derivatives, and, even more serious, the recursion operator cannot be applied to the scaling symmetry (page 322). But in fact it can be applied and leads to a hierarchy of symmetries depending explicitly on time and–at least in the multisoliton case–being intimately connected to the angle variables of the system.
Of course, this disadvantage is evened out by presenting an approach to integrable systems (on infinite-dimensional manifolds) which fulfills the usual requirements of mathematical rigor (which cannot be said of many of the original contributions to the field).
Apart from misprints, the reviewer discovered only one mistake in the book. The proof of Tu's important commutativity result (Theorem 5.20) closely follows Tu's original presentation by basing everything on the proposition that an evolution equation having a nontrivial generalized symmetry must have a quasilinear right-hand side (Theorem 5.22). A counterexample to this proposition is easily furnished by rewriting the Harry Dym equation in the form $u-t=u^3u_{xxx}$. For this equation the nontrivial symmetries (in fact infinitely many) can be found in a paper of M. Leo, R. A. Leo, G. Soliano and L. Solombrino [Phys. Rev. D (3) 27 (1983), no. 6, 1406–1408; MR0697872].
Altogether, a beautiful book, which offers a lot of new material and insight. It certainly has the chance to be a standard reference for some time to come.

Peter Olver a écrit:

From the preface: "For the second edition, I have corrected a number of misprints and inadvertent mathematical errors that found their way into the original version [1986; MR0836734]. More substantial changes are the inclusion of a simpler proof of Theorem 4.26 due to L. Martinez Alonzo [Lett. Math. Phys. 3 (1979), no. 5, 419–424; MR0545031] and the omission of the false (at least in the form stated in the first edition) Theorem 5.22 on the commutativity of generalized symmetries. Also, I have corrected some of the exercises and added several new ones. Hopefully this now eliminates all of the major (and almost all of the minor) mistakes. The one substantial addition to the second edition is a short presentation of the calculus of pseudo-differential operators and their use in Shabat's theory of formal symmetries, which provides a powerful, algorithmic method for determining the integrability of evolution equations.
"The years since the appearance of the original edition of the book have witnessed a remarkable explosion of research, both pure and applied, into symmetry group methods in differential equations, proceeding at a pace well beyond my expectations. Innumerable papers, as well as several substantial textbooks devoted to the subject of symmetry and differential equations, have appeared in the literature. The former are too numerous to try to list here, although I have added a few of the more notable contributions to the list of references and have correspondingly updated the historical notes at the end of each chapter."

homo topi a écrit:

Sinon, qu'est-ce que tu appelles résoudre l'équation de la chaleur "sur un groupe" ?