Spatial patterns. Higher order models in physics and mechanics.

*(English)*Zbl 1076.34515
Progress in Nonlinear Differential Equations and their Applications 45. Boston, MA: Birkhäuser (ISBN 0-8176-4110-6/hbk). xv, 341 p. (2001).

Spatiotemporal patterns are ubiquitous in the physical and biological world, for example as waves on a water surface, pulses in optical fibres, periodic structures in metals, folds in rock formations, cloud formations in the sky, etc. Such phenomena are often described in terms of model equations, which are often simpler than the full equations for the system under investigation but capture its essential features, are more tractable to analyse and provide insights into the underlying mechanisms that are responsible for formation and evolution of complex patterns. Classical model equations are typically second-order, nonlinear partial differential equations, the best known of which bear such names as the Allen-Cahn, Burgers, Fisher-Kolmogorov, sine-Gordon, and Swift-Hohenberg equations. These partial differential equations are then reduced to ordinary differential equations describing the shape of, say, a stationary or travelling wave.

The focus of this book is on a family of such ordinary differential equations, which have the form \[ \frac{d^4 u}{dx^4} + q\,\frac{d^2 u}{dx^2} + f(u) = 0\tag{\(\text{CE}\)} \] and are called the canonical equations here. These equations have already been investigated extensively, if predominantly numerically in the context of a large number of applications. The main concern of the book is to provide a systematic mathematical analysis of the equations, in particular to describe the set \({\mathcal{B}}_q\) of bounded solutions on \(\mathbb{R}\) for different parameter values \(q\). The objective is to establish the existence and the qualitative properties of special solutions of a given type or shape, such as those associated with travelling pulses, fronts, kinks, or periodic wave trains, and to investigate how the complexity of these solutions depends on the parameter \(q\). The main mathematical tool is a method based on topological shooting, which is supplemented by the numerically calculated graphical presentation of many results.

The book divides into two main parts, following a lengthy introductory chapter in which the ideas and methods are outlined in terms of the special case, the simpler second-order Fisher-Kolmogorov equation \[ \frac{d^2 u}{dx^2} + u^3-u = 0.\tag{\(\text{FK}\)} \] A comprehensive overview of the results to follow is also presented. Part I focuses on (CE) with nonlinearity \(f(u) = u^3-u\), for which the corresponding potential function \(F(u)=\frac{1}{4}(u^2-1)^2\) is symmetric and has two wells located at \(u= +1\) and \(u=-1\). The spectrum of the linearisation about these wells plays a pivotal role here, with the character of the spectrum being different in each of the three intervals \(q \leq -\sqrt{8}\), \(-\sqrt{8}<q < \sqrt{8}\), and \(\sqrt{8}< q\), where the uniform solutions \(u = +1\) and \(u =-1\) are respectively saddles, saddle-foci and centers. In Chapter 2, it is shown for the parameter range \(q \leq -\sqrt{8}\), that the set \({\mathcal{B}}_q\) of bounded solutions for (CE) with nonlinearity \(f(u)= u^3-u\) is essentially the same as for the equation (FK). Beyond the threshhold \(q = -\sqrt{8}\) this set becomes extremely rich and the topological shooting method for its investigation is then developed in Chapter 3, where a series of general results about solution graphs, such as bounds and properties of critical points, are presented. These are then used in Chapters 4, 5 and 6, respectively, for a systematic investigation of periodic solutions, kinks and pulses, and chaotic solutions (which are defined in terms of symbolic dynamics). A variational approach is presented in Chapter 7, with symmetry arguments being used to characterise the qualitative properties of global minimisers.

In Part II, the authors consider the canonical equation with more general source functions \(f\), in particular examples associated with single-well potentials. There are three chapters. Chapter 8 still considers double-well potentials, but the potential function \(F\) now is not always symmetric. Stationary solutions of the Swift-Hohenberg equations are investigated in Chapter 9, resulting in the canonical equation \[ \frac{d^4 u}{dx^4} + 2\,\frac{d^2 u}{dx^2} + (1- \kappa)u + u^3 = 0, \] in which \(\kappa\) is the eigenvalue parameter, so the nonlinearity is associated with a single-well potential for \(\kappa < 1\) and a double-well potential for \(\kappa \geq 1\), both of which are symmetric. Finally, Chapter 10 considers a problem with a nonsymmetric single-well potential for which \(f(u) = e^u -1\).

The book is very well written in a very clear and readable style, which makes it accessible to a nonspecialist or graduate student. There are a large number of exercises which fill in details of proofs or provide illuminating examples or straightforward generalisations as well as a good number of open problems. There are also a large number of numerically computed graphs of branching curves and bifurcation curves throughout the book which provide insights into the mathematically formulated results. The book is a valuable contribution to the literature, for both the specialist and the nonspecialist reader.

The focus of this book is on a family of such ordinary differential equations, which have the form \[ \frac{d^4 u}{dx^4} + q\,\frac{d^2 u}{dx^2} + f(u) = 0\tag{\(\text{CE}\)} \] and are called the canonical equations here. These equations have already been investigated extensively, if predominantly numerically in the context of a large number of applications. The main concern of the book is to provide a systematic mathematical analysis of the equations, in particular to describe the set \({\mathcal{B}}_q\) of bounded solutions on \(\mathbb{R}\) for different parameter values \(q\). The objective is to establish the existence and the qualitative properties of special solutions of a given type or shape, such as those associated with travelling pulses, fronts, kinks, or periodic wave trains, and to investigate how the complexity of these solutions depends on the parameter \(q\). The main mathematical tool is a method based on topological shooting, which is supplemented by the numerically calculated graphical presentation of many results.

The book divides into two main parts, following a lengthy introductory chapter in which the ideas and methods are outlined in terms of the special case, the simpler second-order Fisher-Kolmogorov equation \[ \frac{d^2 u}{dx^2} + u^3-u = 0.\tag{\(\text{FK}\)} \] A comprehensive overview of the results to follow is also presented. Part I focuses on (CE) with nonlinearity \(f(u) = u^3-u\), for which the corresponding potential function \(F(u)=\frac{1}{4}(u^2-1)^2\) is symmetric and has two wells located at \(u= +1\) and \(u=-1\). The spectrum of the linearisation about these wells plays a pivotal role here, with the character of the spectrum being different in each of the three intervals \(q \leq -\sqrt{8}\), \(-\sqrt{8}<q < \sqrt{8}\), and \(\sqrt{8}< q\), where the uniform solutions \(u = +1\) and \(u =-1\) are respectively saddles, saddle-foci and centers. In Chapter 2, it is shown for the parameter range \(q \leq -\sqrt{8}\), that the set \({\mathcal{B}}_q\) of bounded solutions for (CE) with nonlinearity \(f(u)= u^3-u\) is essentially the same as for the equation (FK). Beyond the threshhold \(q = -\sqrt{8}\) this set becomes extremely rich and the topological shooting method for its investigation is then developed in Chapter 3, where a series of general results about solution graphs, such as bounds and properties of critical points, are presented. These are then used in Chapters 4, 5 and 6, respectively, for a systematic investigation of periodic solutions, kinks and pulses, and chaotic solutions (which are defined in terms of symbolic dynamics). A variational approach is presented in Chapter 7, with symmetry arguments being used to characterise the qualitative properties of global minimisers.

In Part II, the authors consider the canonical equation with more general source functions \(f\), in particular examples associated with single-well potentials. There are three chapters. Chapter 8 still considers double-well potentials, but the potential function \(F\) now is not always symmetric. Stationary solutions of the Swift-Hohenberg equations are investigated in Chapter 9, resulting in the canonical equation \[ \frac{d^4 u}{dx^4} + 2\,\frac{d^2 u}{dx^2} + (1- \kappa)u + u^3 = 0, \] in which \(\kappa\) is the eigenvalue parameter, so the nonlinearity is associated with a single-well potential for \(\kappa < 1\) and a double-well potential for \(\kappa \geq 1\), both of which are symmetric. Finally, Chapter 10 considers a problem with a nonsymmetric single-well potential for which \(f(u) = e^u -1\).

The book is very well written in a very clear and readable style, which makes it accessible to a nonspecialist or graduate student. There are a large number of exercises which fill in details of proofs or provide illuminating examples or straightforward generalisations as well as a good number of open problems. There are also a large number of numerically computed graphs of branching curves and bifurcation curves throughout the book which provide insights into the mathematically formulated results. The book is a valuable contribution to the literature, for both the specialist and the nonspecialist reader.

Reviewer: Peter E. Kloeden (MR1839555)

##### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

00A69 | General applied mathematics |

34C11 | Growth and boundedness of solutions to ordinary differential equations |

35B10 | Periodic solutions to PDEs |

35K55 | Nonlinear parabolic equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |