Contents
Notation 1
Chapter 1. Introduction 3
1.1. Grasshopper’s guide 3
1.2. Slowly-varying modulations of nonlinear wave trains 4
1.3. Predictions from the Burgers equation 7
1.4. Verifying the predictions made from the Burgers equation 8
1.5. Related modulation equations 12
1.6. References to related works 13
Chapter 2. The Burgers equation 15
2.1. Decay estimates 15
2.2. Fronts in the Burgers equation 17
Chapter 3. The complex cubic Ginzburg–Landau equation 19
3.1. Set-up 19
3.2. Slowly-varying modulations of the k = 0 wave train: Results 20
3.3. Derivation of the Burgers equation 23
3.4. The construction of higher-order approximations 24
3.5. The approximation theorem for the wave numbers 25
3.6. Mode filters, and separation into critical and noncritical modes 25
3.7. Estimates of the linear semigroups 29
3.8. Estimates of the residual 30
3.9. Estimates of the errors 31
3.10. Proofs of the theorems from §3.2 34
Chapter 4. Reaction-diffusion equations: Set-up and results 39
4.1. The abstract set-up 39
4.2. Expansions of the linear and nonlinear dispersion relations 41
4.3. Formal derivation of the Burgers equation 43
4.4. Validity of the Burgers equation 45
4.5. Existence and stability of weak shocks 48
Chapter 5. Validity of the Burgers equation in reaction-diffusion equations 53
5.1. From phases to wave numbers 53
5.2. Bloch-wave analysis 55
5.3. Mode filters, and separation into critical and noncritical modes 58
5.4. Estimates for residuals and errors 61
5.5. Proofs of the theorems from §4.4 63
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