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
Chapter 6. Validity of the inviscid Burgers equation in
reaction-diffusion systems .......................... 65
6.1. An illustration: The Ginzburg-Landau equation .......... 65
6.2. Formal derivation of the conservation law .............. 66
6.3. Validity of the inviscid Burgers equation .............. 67
6.4. Proof of the theorems from §6.3 ........................ 68
Chapter 7. Modulations of wave trains near sideband
instabilities ....................................... 73
7.1. Introduction ........................................... 73
7.2. An illustration: The Ginzburg-Landau equation .......... 74
7.3. Validity of the Korteweg-de Vries and the Kuramoto-
Sivashinsky equation ................................... 75
7.4. Proof of Theorem 7.2 ................................... 78
7.5. Proof of Theorem 7.5 ................................... 79
Chapter 8. Existence and stability of weak shocks .............. 83
8.1. Proof of Theorem 4.10 .................................. 83
8.2. Proof of Theorem 4.12 .................................. 88
Chapter 9. Existence of shocks in the long-wavelength limit .... 93
9.1. A lattice model for weakly interacting pulses .......... 93
9.2. Proof of Theorem 9.2 ................................... 95
Chapter 10.Applications ........................................ 99
10.1.The FitzHugh-Nagumo equation ........................... 99
10.2.The weakly unstable Taylor-Couette problem ............ 100
Bibliography .................................................. 103
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