04458nam 2200613 450 991078885490332120170822144133.01-4704-0540-7(CKB)3360000000465118(EBL)3114122(SSID)ssj0000888894(PQKBManifestationID)11480048(PQKBTitleCode)TC0000888894(PQKBWorkID)10866172(PQKB)10354562(MiAaPQ)EBC3114122(RPAM)15574912(PPN)195418239(EXLCZ)99336000000046511820150417h20092009 uy 0engur|n|---|||||txtccrThe dynamics of modulated wave trains /Arjen Doelman [and three others]Providence, Rhode Island :American Mathematical Society,2009.©20091 online resource (122 p.)Memoirs of the American Mathematical Society,0065-9266 ;Volume 199, Number 934"Volume 199, Number 934 (fifth of 6 numbers)."0-8218-4293-5 Includes bibliographical references.""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 Ginzburgâ€?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 FitzHughâ€?Nagumo equation""; ""10.2. The weakly unstable Taylorâ€?Couette problem""; ""Bibliography""Memoirs of the American Mathematical Society ;Volume 199, Number 934.Reaction-diffusion equationsApproximation theoryBurgers equationReaction-diffusion equations.Approximation theory.Burgers equation.515.3534Doelman A.MiAaPQMiAaPQMiAaPQBOOK9910788854903321The dynamics of modulated wave trains3836163UNINA