- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in Mathematical Physics
Volume 2011 (2011), Article ID 854719, 4 pages
Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation
IADM, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Received 4 March 2011; Accepted 16 March 2011
Academic Editor: Pavel Exner
Copyright © 2011 Guido Schneider. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- L. A. Kalyakin, “Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium,” Matematicheskiĭ Sbornik. Novaya Seriya, vol. 132(174), no. 4, pp. 470–495, 592, 1987.
- P. Kirrmann, G. Schneider, and A. Mielke, “The validity of modulation equations for extended systems with cubic nonlinearities,” Proceedings of the Royal Society of Edinburgh A, vol. 122, no. 1-2, pp. 85–91, 1992.
- G. Schneider, “Justification of modulation equations for hyperbolic systems via normal forms,” NoDEA. Nonlinear Differential Equations and Applications, vol. 5, no. 1, pp. 69–82, 1998.
- G. Schneider, “Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances,” Journal of Differential Equations, vol. 216, no. 2, pp. 354–386, 2005.
- G. Schneider, “Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation,” Journal of Differential Equations, vol. 147, no. 2, pp. 333–354, 1998.
- W.-P. Düll and G. Schneider, “Justification of the nonlinear Schrödinger equation for a resonant Boussinesq model,” Indiana University Mathematics Journal, vol. 55, no. 6, pp. 1813–1834, 2006.
- V. E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of a deep fluid,” Journal of Applied Mechanics and Technical Physics, vol. 4, pp. 190–194, 1968.
- N. Totz and S. Wu, “A rigorous justification of the modulation approximation to the 2d full water wave problem,” Preprint, 2011.
- R. S. Johnson, “On the modulation of water waves on shear flows,” Proceedings of the Royal Society of London A, vol. 347, pp. 537–546, 1976.
- V. E. Zakharov and E. A. Kuznetsov, “Multiscale expansions in the theory of systems integrable by the inverse scattering transform,” Physica D, vol. 18, no. 1–3, pp. 455–463, 1986.
- E. R. Tracy, J. W. Larson, A. R. Osborne, and L. Bergamasco, “On the nonlinear Schrödinger limit of the Korteweg-de Vries equation,” Physica D, vol. 32, no. 1, pp. 83–106, 1988.
- J. P. Boyd and G.-Y. Chen, “Weakly nonlinear wavepackets in the Korteweg-de Vries equation: the KdV/NLS connection,” Mathematics and Computers in Simulation, vol. 55, no. 4–6, pp. 317–328, 2001.
- P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1989.