The dynamics of modulated wave trains / Arjen Doelman [and three others].

Format:

Book

Author/Creator:

Doelman, A., author.

Series:

Memoirs of the American Mathematical Society ; Volume 199, Number 934.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 199, Number 934

Language:

English

Subjects (All):

Reaction-diffusion equations.

Approximation theory.

Burgers equation.

Physical Description:

1 online resource (122 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2009.

Language Note:

English

Summary:

This work examines the dynamics of weakly-modulated nonlinear wave trains. Areas covered by the authors include reaction-diffusion systems, the Ginsburg-Landau equation, Burgers equation, the Rankine-Hugoniot condition, the FitzHugh-Nagumo equation and hydrodynamic stability problems.

Contents:

""Contents""; ""Notation""; ""Chapter 1. Introduction""; ""1.1. Grasshopper's guide""; ""1.2. Slowly-varying modulations of nonlinear wave trains""; ""1.3. Predictions from the Burgers equation""; ""1.4. Verifying the predictions made from the Burgers equation""; ""1.5. Related modulation equations""; ""1.6. References to related works""; ""Chapter 2. The Burgers equation""; ""2.1. Decay estimates""; ""2.2. Fronts in the Burgers equation""; ""Chapter 3. The complex cubic Ginzburg�Landau equation""; ""3.1. Set-up""; ""3.2. Slowly-varying modulations of the k = 0 wave train: Results""

""3.3. Derivation of the Burgers equation""""3.4. The construction of higher-order approximations""; ""3.5. The approximation theorem for the wave numbers""; ""3.6. Mode filters, and separation into critical and noncritical modes""; ""3.7. Estimates of the linear semigroups""; ""3.8. Estimates of the residual""; ""3.9. Estimates of the errors""; ""3.10. Proofs of the theorems from Â3.2""; ""Chapter 4. Reaction-diffusion equations: Set-up and results""; ""4.1. The abstract set-up""; ""4.2. Expansions of the linear and nonlinear dispersion relations""

""4.3. Formal derivation of the Burgers equation""""4.4. Validity of the Burgers equation""; ""4.5. Existence and stability of weak shocks""; ""Chapter 5. Validity of the Burgers equation in reaction-diffusion equations""; ""5.1. From phases to wave numbers""; ""5.2. Bloch-wave analysis""; ""5.3. Mode filters, and separation into critical and noncritical modes""; ""5.4. Estimates for residuals and errors""; ""5.5. Proofs of the theorems from Â4.4""; ""Chapter 6. Validity of the inviscid Burgers equation in reaction-diffusion systems""; ""6.1. An illustration: The Ginzburgâ€?Landau equation""

""6.2. Formal derivation of the conservation law""""6.3. Validity of the inviscid Burgers equation""; ""6.4. Proof of the theorems from Â6.3""; ""Chapter 7. Modulations of wave trains near sideband instabilities""; ""7.1. Introduction""; ""7.2. An illustration: The Ginzburgâ€?Landau equation""; ""7.3. Validity of the Kortewegâ€?de Vries and the Kuramotoâ€?Sivashinsky equation""; ""7.4. Proof of Theorem 7.2""; ""7.5. Proof of Theorem 7.5""; ""Chapter 8. Existence and stability of weak shocks""; ""8.1. Proof of Theorem 4.10""; ""8.2. Proof of Theorem 4.12""

""Chapter 9. Existence of shocks in the long-wavelength limit""""9.1. A lattice model for weakly interacting pulses""; ""9.2. Proof of Theorem 9.2""; ""Chapter 10. Applications""; ""10.1. The FitzHugh�Nagumo equation""; ""10.2. The weakly unstable Taylor�Couette problem""; ""Bibliography""

Notes:

"Volume 199, Number 934 (fifth of 6 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0540-7