Contact geometry and nonlinear differential equations. (English) Zbl 1122.53044
Encyclopedia of Mathematics and Its Applications 101. Cambridge: Cambridge University Press (ISBN 978-0-521-82476-7/hbk). xxi, 496 p. (2007).
The book is devoted to the geometry of nonlinear partial and ordinary differential equations.
The first part of the book has three chapters and is devoted to the geometric study of symmetries and integrals for differential equations. They present the theory of distributions that fit the aim of the book, by studying complete integrability, integrability, curvature, integral manifolds and symmetries. Chapter 2 studies ordinary differential equations, with a special emphasize on their symmetries. The advantage of using the geometric approach in studying differential equations is presented in chapter 3. Two methods are presented here: the symmetry reduction and the Lie superposition principle.
The second part of the book is devoted to symplectic algebra and contains five chapters. Chapter 4 presents the main facts concerning symplectic structures on a vector space. It is continued with the special features of exterior algebra on symplectic vector spaces presented in chapter 5. The next three chapters of the second part deal with symplectic classification problems for 2-forms in dimension 4 (chapter 6) and arbitrary dimension (chapter 7) and 3-forms in dimension 6 (chapter 8).
The third part of the book has 7 chapters and is devoted to Monge-Ampère equations (MAE). The first two chapters of this part (chapters 9 and 10) present the geometry of symplectic and contact manifolds with all the geometric structures that are needed in the next chapters to study Monge-Ampère equations. Chapter 11 studies Monge-Ampère equations with the associated Monge-Ampère operators. The following two chapters study symmetries and contact transformations of MAE as well as conservation laws. Chapter 14 deals with the geometry of MAE on two-dimensional manifolds, by studying associated non-holonomic geometric structures, integrability problems, symplectic MAEs and the Cauchy problem for hyperbolic MAE. Chapter 15 studies geometric aspects of systems of first-order PDE on two-dimensional manifolds.
The fourth part of the book is divided into three chapters that present applications from non-linear acoustics, nonlinear thermal conductivity and meteorology.
The last part of the book deals with symplectic or contact classifications of Monge-Ampère equations in two or three-dimensional manifolds using the geometric approach developed in the book.
The first part of the book has three chapters and is devoted to the geometric study of symmetries and integrals for differential equations. They present the theory of distributions that fit the aim of the book, by studying complete integrability, integrability, curvature, integral manifolds and symmetries. Chapter 2 studies ordinary differential equations, with a special emphasize on their symmetries. The advantage of using the geometric approach in studying differential equations is presented in chapter 3. Two methods are presented here: the symmetry reduction and the Lie superposition principle.
The second part of the book is devoted to symplectic algebra and contains five chapters. Chapter 4 presents the main facts concerning symplectic structures on a vector space. It is continued with the special features of exterior algebra on symplectic vector spaces presented in chapter 5. The next three chapters of the second part deal with symplectic classification problems for 2-forms in dimension 4 (chapter 6) and arbitrary dimension (chapter 7) and 3-forms in dimension 6 (chapter 8).
The third part of the book has 7 chapters and is devoted to Monge-Ampère equations (MAE). The first two chapters of this part (chapters 9 and 10) present the geometry of symplectic and contact manifolds with all the geometric structures that are needed in the next chapters to study Monge-Ampère equations. Chapter 11 studies Monge-Ampère equations with the associated Monge-Ampère operators. The following two chapters study symmetries and contact transformations of MAE as well as conservation laws. Chapter 14 deals with the geometry of MAE on two-dimensional manifolds, by studying associated non-holonomic geometric structures, integrability problems, symplectic MAEs and the Cauchy problem for hyperbolic MAE. Chapter 15 studies geometric aspects of systems of first-order PDE on two-dimensional manifolds.
The fourth part of the book is divided into three chapters that present applications from non-linear acoustics, nonlinear thermal conductivity and meteorology.
The last part of the book deals with symplectic or contact classifications of Monge-Ampère equations in two or three-dimensional manifolds using the geometric approach developed in the book.
Reviewer: Ioan Bucataru (Iaşi)
MSC:
| 53D10 | Contact manifolds (general theory) |
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |
| 53D05 | Symplectic manifolds (general theory) |
| 37J55 | Contact systems |
| 58A15 | Exterior differential systems (Cartan theory) |
| 35Q80 | Applications of PDE in areas other than physics (MSC2000) |
| 34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
