Traveling wave solutions of parabolic systems.

*(English)*Zbl 1001.35060
Translations of Mathematical Monographs. 140. Providence, RI: American Mathematical Society. xii, 448 p. (1994).

From the preface: The theory of traveling wave solutions of parabolic equations is one of the fast developing areas of modern mathematics. The history of this theory begins with the famous mathematical work by A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov [Bull. Univ. État Moscou, Sér. Int., Sect. A: Math. et Mécan. 1, Fasc. 6, 1–25 (1937; Zbl 0018.32106)] and with works in chemical physics, the best known among them by Zel’dovich and Frank-Kamenetskii in combustion theory and by Semenov, who discovered branching chain flames.

Traveling wave solutions are solutions of special type. They can be usually characterized as solutions invariant with respect to translation in space. The existence of traveling waves appears to be very common in nonlinear equations, and, in addition, they often determine the behavior of the solutions of Cauchy-type problems.

From the physical point of view, traveling waves usually describe transition processes. Transition from one equilibrium to another is a typical case, although more complicated situations can arise. These transition processes usually forget their initial conditions and reflect the properties of the medium itself.

Among the basic questions in the theory of traveling waves we mention the problem of wave existence, stability of waves with respect to small perturbations and global stability, bifurcations of waves, determination of wave speed, and systems of waves (or wave trains). The case of a scalar equation has been rather well studied, basically due to applicability of comparison theorems of a special kind for parabolic equations and of phase space analysis for the ordinary differential equations. For systems of equations, comparison theorems of this kind are, in general, not applicable, and the phase space analysis becomes much more complicated. This is why systems of equations are much less understood and require new approaches. In this book, some of these approaches are presented, together with more traditional approaches adapted for specific classes of systems of equations and for a more complete analysis of scalar equations. From the authors’ point of view, it is very important that these mathematical results find numerous applications, first and fore most in chemical kinetics and combustion. The authors understand that the theory of travelling waves is far from being complete and hope that this book will help in development. An excellent 20-page bibliography (over 500 entries) with a large number of Russian sources is also given.

The book can be used by graduate students and researchers specializing in nonlinear differential equations, as well as by specialists in other areas (engineering, chemical physics, biology), where the theory of wave solutions of parabolic systems can be applied.

Traveling wave solutions are solutions of special type. They can be usually characterized as solutions invariant with respect to translation in space. The existence of traveling waves appears to be very common in nonlinear equations, and, in addition, they often determine the behavior of the solutions of Cauchy-type problems.

From the physical point of view, traveling waves usually describe transition processes. Transition from one equilibrium to another is a typical case, although more complicated situations can arise. These transition processes usually forget their initial conditions and reflect the properties of the medium itself.

Among the basic questions in the theory of traveling waves we mention the problem of wave existence, stability of waves with respect to small perturbations and global stability, bifurcations of waves, determination of wave speed, and systems of waves (or wave trains). The case of a scalar equation has been rather well studied, basically due to applicability of comparison theorems of a special kind for parabolic equations and of phase space analysis for the ordinary differential equations. For systems of equations, comparison theorems of this kind are, in general, not applicable, and the phase space analysis becomes much more complicated. This is why systems of equations are much less understood and require new approaches. In this book, some of these approaches are presented, together with more traditional approaches adapted for specific classes of systems of equations and for a more complete analysis of scalar equations. From the authors’ point of view, it is very important that these mathematical results find numerous applications, first and fore most in chemical kinetics and combustion. The authors understand that the theory of travelling waves is far from being complete and hope that this book will help in development. An excellent 20-page bibliography (over 500 entries) with a large number of Russian sources is also given.

The book can be used by graduate students and researchers specializing in nonlinear differential equations, as well as by specialists in other areas (engineering, chemical physics, biology), where the theory of wave solutions of parabolic systems can be applied.

##### MSC:

35K57 | Reaction-diffusion equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

80A30 | Chemical kinetics in thermodynamics and heat transfer |

92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |

80A25 | Combustion |

35B40 | Asymptotic behavior of solutions to PDEs |