This book presents developments in the geometric approach to nonlinear partial differential equations (PDEs). The expositions discuss the main features of the approach, and the theory of symmetries and the conservation laws based on it. The book combines rigorous mathematics with concrete examples. Nontraditional topics, such as the theory of nonlocal symmetries and cohomological theory of conservation laws, are also included.

The volume is largely self-contained and includes detailed motivations, extensive examples and exercises, and careful proofs of all results. Readers interested in learning the basics of applications of symmetry methods to differential equations of mathematical physics will find the text useful. Experts will also find it useful as it gathers many results previously only available in journals.

Readership

Advanced graduate students and mathematicians working in nonlinear PDEs and their applications, mathematical and theoretical physics, and mechanics.

Reviews

"A valuable feature about the present text is that it provides an overview of a number of results that originally appeared in Russian journals and are often difficult to track down. Moreover, the text is a useful reference for the practitioner in the field not only due to the descriptions of the various relevant algorithms but also due to the numerous examples involving symmetry algebras and conservation laws of particular equations, which, because of the book's emphasis on higher-order and potential symmetries, can often not be found in other standard references in the field. In all, this text provides a useful and readable introduction to the recent developments in the theory of symmetries and conservation laws of differential equations."

-- Mathematical Reviews

"Rigorous mathematics and concrete examples illustrate the geometric approach to the study of nonlinear PDEs."

-- American Mathematical Monthly

Table of Contents

• Ordinary differential equations
• First-order equations
• The theory of classical symmetries
• Higher symmetries
• Conservation laws
• Nonlocal symmetries
• From symmetries of partial differential equations towards secondary ("quantized") calculus
• Bibliography
• Index