This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.E. Zakharov (ed.), What is Integrability?, Springer, Heidelberg, 1990.
A. Degasperis, Resource Letter Sol-1: Solitons, Am. J. Phys. 66, 486–497, 1998.
T. Taniuti and N. Yajima, Perturbation method for a nonlinear wave modulation. I, J. Math. Phys. 10, 1369–1372, 1969; Suppl. Prog. Theor. Phys. 55, 1974 (special issue devoted to the Reductive Perturbation Method for Nonlinear Wave Propagation).
P.L. Kelley, Self-Focusing of Optical Beams, Phys Rev. Lett. 15, 1005, 1965.
D.J. Benney and A.C. Newell, J. Math. and Phys. (now Stud. Appl. Math.) 46, 133–139, 1967.
V.E. Zakharov, Instability of self-focusing of light Soviet Phys. JETP 26, 994–998, 1968.
H. Hasimoto and H. Ono, Nonlinear Modulation of Gravity Waves, J. Phys. Soc. Japan 33, 805, 1972.
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett. 23, 142, 1973.
F. Calogero and W. Eckhaus, Necessary conditions for integrability, Inverse Probl. 3, L27–L32, 1987.
F. Calogero, Why are certain nonlinear PDEs both widely applicable and integrable?. In [1], pp. 1–62.
F. Calogero and A. Degasperis, Spectral Transform and Solitons. Vol. I, North Holland, Amsterdam, 1982.
M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge U.P., Cambridge, 1991.
V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 34, 62–69, 1972 [Russian original: Zh. Eksp. Teor. Fiz. 61, 118–134, 1971].
V.E. Zalharov and E.A. Kuznetsov, Multiscale Expansion in the Theory of Systems Integrable by the Inverse Scattering Transform, Physica 18D, 455–463, 1986.
C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19, 1095–1097, 1967.
J.A. Murdock, Perturbations, John Wiley & Sons, Inc., New York, 1991.
F. Calogero and A. Degasperis, Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform, J. Math.Phys. 22, 23–31, 1981.
F.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, Inc., New York, 1974.
R.A. Kraenkel, M.A. Manna, and J.G. Pereira, The Korteweg-de Vries hierarchy and long water-waves, J. Math. Phys. 36, 307, 1995.
Y. Kodama and A.V. Mikhailov, Obstacle to asymptotic integrability In: Algebraic Aspects of Integrable Systems: in memory of Irene Dorfman, A.S. Fokas and I.M. Gel’fand (eds.), Birkhauser, Boston, 173–204, 1996.
V.E. Zakharov and E.I. Schulman, Integrability of Nonlinear Systems and Perturbation Theory. In [1], pp. 185–250.
D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234, 253–285, 2003.
B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov – Witten invariants, arXiv:math/0108160v1 [math.DG].
Y. Hiraoka and Y. Kodama, Normal form and solitons, Chap. 5 of this book and arXiv:nlin.SI/020602.
F. Calogero, A. Degasperis, and J. Xiaoda, Nonlinear Schrödinger-type equations from multiscale reduction of PDEs. I. Systematic derivation, J. Math. Phys. 41, 6399–6443, 2000.
F. Calogero, A. Desgasperis, and J. Xiaoda, Nonlinear Schrödinger-type equations from multiscale reduction of PDEs. II. Necessary conditions of integrability for real PDEs, J. Math. Phys. 42, 2635–2652, 2001.
N. Yajima and M. Oikawa, Formation and Interaction of Sonic-Langmuir Solitons, Prog. Theor. Phys. 56, 1719–1739, 1976.
A. Degasperis, S.V. Manakov, and P.M. Santini, Multiple-scale perturbation beyond the Nonlinear Schrödinger Equation. I, Physica D100, 187–211, 1997.
A. Degasperis and M. Procesi, Asymptotic Integrability. In: Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta (eds.), World Scientific, Singapore, 23–37, 1999.
R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71, 1661–1664, 1993.
A. Degasperis, D.D. Holm, and A.N.W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133, 1463–1474, 2002.
Y. Matsuno, Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit, Inverse Problems 21, 1553–1570, 2005.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Degasperis, A. (2009). Multiscale Expansion and Integrability of Dispersive Wave Equations. In: Mikhailov, A.V. (eds) Integrability. Lecture Notes in Physics, vol 767. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88111-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-88111-7_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88110-0
Online ISBN: 978-3-540-88111-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)