Abstract
We prove that slow modulations in time and space of periodic wave trains of the NLS equation can be approximated via solutions of Whitham’s equations associated with the wave train. The error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like existence and uniqueness theorem, and energy estimates.
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Bridges, T., Mielke, A.: A proof of the Benjamin–Feir instability. Arch. Ration Mech. Anal. 133(2), 145–198 (1995)
Busch, K., Schneider, G., Tkeshelashvili, L., Uecker, H.: Justification of the nonlinear Schrödinger equation in spatially periodic media. Z. Angew. Math. Phys. 57(6), 905–939 (2006)
Cercignani, C., Sattinger, D.H.: Scaling Limits and Models in Physical Processes. DMV Seminar, vol. 28. Birkhäuser, Basel (1998), vi+191 pp.
Collet, P., Eckmann, J.-P.: The time dependent amplitude equation for the Swift–Hohenberg problem. Commun. Math. Phys. 132(1), 139–153 (1990)
Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits. Commun. Part. Differ. Equ. 10, 787–1003 (1985)
Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. AMS Mem. (2006, accepted)
Dreyer, W., Herrmann, M., Mielke, A.: Micro-macro transition in the atomic chain via Whitham’s modulation equation. Nonlinearity 19(2), 471–500 (2006)
Kalyakin, L.A.: Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium. Math. USSR Sb. Surv. 60(2), 457–483 (1988)
Kamvissis, S., McLaughlin, K.D.T.-R., Miller, P.D.: Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation. Annals of Mathematics Studies, vol. 154. Princeton University Press, Princeton (2003)
Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations. In: Proc. Symp. Dedicated to Konrad Jörgens, Dundee, 1974. Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)
Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)
Melbourne, I., Schneider, G.: Phase dynamics in the complex Ginzburg–Landau equation. J. Differ. Equ. 199(1), 22–46 (2004a)
Melbourne, I., Schneider, G.: Phase dynamics in the real Ginzburg–Landau equation. Math. Nachr. 263/264 171–180 (2004b)
Mielke, A.: Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner–Husumi transforms. Arch. Ration. Mech. Anal. 181, 401–448 (2006)
Ovsjannikov, L.V.: Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification. In: Lecture Notes in Mathematics, vol. 503, pp. 416–437. Springer, Berlin (1976)
Schneider, G.: A new estimate for the Ginzburg–Landau approximation on the real axis. J. Nonlinear Sci. 4(1), 23–34 (1994a)
Schneider, G.: Error estimates for the Ginzburg–Landau approximation. Z. Angew. Math. Phys. 45(3), 433–457 (1994b)
Schneider, G.: Justification of modulation equations for hyperbolic systems via normal forms. Nonlinear Differ. Equ. Appl. (NODEA) 5, 69–82 (1998)
Schneider, G.: Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances. J. Differ. Equ. 216(2), 354–386 (2005)
Schneider, G., Wayne, C.E.: The long wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53(12), 1475–1535 (2000)
Schneider, G., Wayne, C.E.: The rigorous approximation of long-wavelength capillary-gravity waves. Arch. Ration Mech. Anal. 162, 247–285 (2002)
Serre, D.: Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999). xxii+263 pp. (Translated from the 1996 French original by I.N. Sneddon)
Serre, D.: Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems. Cambridge University Press, Cambridge (2000). xii+269 pp. (Translated from the 1996 French original by I.N. Sneddon)
van Harten, A.: On the validity of the Ginzburg–Landau equation. J. Nonlinear Sci. 1(4), 397–422 (1991)
Whitham, G.B.: A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283 (1965a)
Whitham, G.B.: Non-linear dispersive waves. Proc. R. Soc. Ser. A 283, 238–261 (1965b)
Whitham, G.B.: Linear and Nonlinear Waves. Pure and Applied Mathematics. Wiley, New York (1999). Reprint of the 1974 Original
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Communicated by J. Scheurle.
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Düll, WP., Schneider, G. Validity of Whitham’s Equations for the Modulation of Periodic Traveling Waves in the NLS Equation. J Nonlinear Sci 19, 453–466 (2009). https://doi.org/10.1007/s00332-009-9043-4
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DOI: https://doi.org/10.1007/s00332-009-9043-4