Nonlinear PDEs: A Dynamical Systems Approach

This novel introduction to nonlinear partial differential equations (PDEs) uses dynamical systems methods and reduction techniques to get more insight into the physical phenomena underlying the equations. The presentation itself is unusual since its pattern is often to begin with an example and a specific equation, and then to develop the relevant mathematical tools. The combination of rigor with simultaneous attention to associated real physical systems makes it particularly appealing.

The book has four fairly distinct pieces. The first introduces nonlinear dynamics in finite dimensions, and this focuses on ordinary differential equations (ODEs). The second moves to countably many dimensions and argues that a PDE on a bounded domain can, under many circumstances, be reduced to a system of countably many ODEs. The authors develop this first on an interval and then consider the Navier-Stokes equations on a torus. In the third part, they look at PDEs on the line and begin their discussion of the concept of modulation equations. Finally, the fourth part develops a more complete and rigorous modulation theory with a full set of examples and applications.

Modulation equations, sometimes called envelope or amplitude equations, are approximate and often explicitly solvable model equations that are usually derived via asymptotic analysis and are used to represent more complicated physical systems. For example, the Korteweg-de Vries equation is a leading order approximation to the equation describing long wavelength water waves.

Modulation equations can be most useful for problems that can’t be directly simulated and where analysis is needed to reduce the dimensionality of a system. Situations like these arise in spatially extended domains where the wavelength of a typical solution is much smaller than the underlying physical domain. In such cases reduction to a simpler model governed by a modulation equation is sometimes possible. Justification of the reduction is the subject of the fourth part of the book.

Modulation equations arise via perturbative methods from more complex PDEs. The question then is to determine what can (or can not) be deduced from the original full system from these reduced PDEs. The authors consider three equations (Ginzburg-Landau, Nonlinear Schrödinger, and Korteweg-de Vries) as universal modulation equations, each associated with a variety of different systems. They derive and justify the approximations that lead to these equations in three parallel chapters. The final two chapters address aspects of existence and stability theory for special solutions of nonlinear PDEs on unbounded domains in the context of modulation theory.

This book is not suitable for a student’s first exposure to PDEs. It does not treat any of the usual basic topics of linear PDEs such as the standard wave and heat equations, the idea of characteristics, or even the classification of linear PDEs. Used as a text for a follow up course it could be quite successful. It is a challenging work and the latter parts approach the frontiers of current research.

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Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.