Studies in Non-Linear Stability Theory

Non-linear stability problems formulated in terms of non-linear partial differential equations have only recently begun to attract attention and it will probably take some time before our understanding of those problems reaches some degree of maturity. The passage from the more classical linear analysis to a non-linear analysis increases the mathematical complexity of the stability theory to a point where it may become discouraging, while some of the more usual mathematical methods lose their applicability. Although considerable progress has been made in recent years, notably in the field of fluid mechanics, much still remains to be done before a more permanent outline of the subject can be established. I have not tried to present in this monograph an account of what has been accomplished, since the rapidly changing features of the field make the periodical literature a more appropriate place for such a review. The aim of this book is to present one particular line of research, originally developed in a series of papers published in 'Journal de Mecanique' 1962-1963, in which I attempted to construct a mathematical theory for certain classes of non-linear stability problems, and to gain some understanding of the non-linear phenomena which are involved. The opportunity to collect the material in this volume has permitted a more coherent presentation, while various points of the analysis have been developed in greater detaiL I hope that a more unified form of the theory has thus been achieved.

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1. Introduction.- 1.1 The notion of stability.- 1.2 The nature of non-linear stability problems.- 1.3 Formal approach to stability theory.- 2. A Class of Problems in One-Dimensional Space.- 2.1 Preliminary remarks.- 2.2 Formulation.- 2.3 Behaviour and properties of the linearized solutions.- 2.4 Series expansion in the case of self-adjoint operators.- 2.5 Series expansion in the case of not self-adjoint operators.- 2.6 Interpretation of the series expansion in terms of the Green's function.- 2.7 The system of equations for the amplitude-functions.- 3. Behaviour of Solutions.- 3.1 Formal simplification of the system of equations.- 3.2 Stable and unstable stationary solutions.- 3.3 Effects of interactions. Forced solutions.- 3.4 Analysis of forced solutions.- 3.5 Instability to finite size perturbations.- 3.6 Other types of behaviour.- 4. Asymptotic Methods for Problems in One-Dimensional Space 2S.- 4.1 General outline.- 4.2 Weak stability or instability: the case aij(n) = 0.- 4.3 Weak stability or instability: the case a0 0(0) = 0.- 4.4 Weak stability and instability: the case a0 0(0) ? 0.- 4.5 Method of approximation for the case of simple developed instability.- 4.6 Behaviour of solutions as functions of time.- 5. Analysis of Some One-Dimensional Problems.- 5.1 Introductory remarks.- 5.2 Burgers' mathematical model of turbulence.- 5.3 Modification of Burgers' model. The problem of stability.- 5.4 Asymptotic expansions in Burgers' model.- 5.5 Another simple mathematical model.- 6. A Class of Problems in Two-Dimensional Space.- 6.1 Introductory remarks.- 6.2 Formulation.- 6.3 The problem of stability. Linearized theory.- 6.4 Fourier-analysis of the non-linear stability problem.- 6.5 Orthogonality relations.- 6.6 Initial conditions.- 7. Asymptotic Theory ofPeriodic Solutions.- 7.1 Basic equations and transformations.- 7.2 Forced solutions for the components ?m,m ? 1.- 7.3 Analysis of the component ?1.- 7.4 Further analysis of the forced solutions for ?m, m ? 1.- 7.5 The equations of the asymptotic approximation.- 7.6 Harmonic solutions.- 7.7 A simple example.- 8. Stability of Periodic Solutions.- 8.1 Introduction.- 8.2 Formulation of the stability problem.- 8.3 Analysis of small parameters.- 8.4 Perturbations in the region ?0(k) = 0 (1).- 8.5 Perturbations in the region ?0(k) = 0 (?2).- 8.6 Reduction of the system of equations.- 8.7 Forced solutions for ?? ? and $$ {psi _{2{k_0}}} pm varepsilon sigma $$.- 8.8 The equations for A0(k,) and A0(k,,).- 8.9 Solution of the stability problem for k0 = kcr.- 8.10 Regions of validity of the asymptotic results.- 8.11 Stability of periodic solutions in the case k0 ? kcr.- 8.12 Summary and interpretation of results.- 9. Periodic Solutions in Poiseuille Flow.- 9.1 Introduction.- 9.2 Formulation of the stability problem.- 9.3 Linearized stability theory.- 9.4 The adjoint linearized problem.- 9.5 Periodic solutions.- 9.6 Discussion of the results.