On the asymptotic linearization of acoustic waves
Authors: Athanassios S. Fokas and Laihan Luo
Journal: Trans. Amer. Math. Soc. 360 (2008), 6403-6445
MSC (2000): Primary 35B40, 35C20, 35G25, 35Q53; Secondary 76B03, 76B15, 76M99
Published electronically: July 24, 2008
MathSciNet review: 2434293
Full-text PDF Free Access
Abstract: The initial value problem of a certain generalization of the nonlinear, dispersive wave equations with dissipation is rigorously studied. The solutions of the equations can be found exactly up to in certain norms. The essential use is made of the fact that this equation is asymptotically linearizable to i.e., the equations can be mapped to an equation which differs from a linearizable equation only in terms which are of An application of the equations to unidirectional small amplitude acoustic waves is discussed. The general methodology used here can also be applied to other asymptotically linearizable equations.
- 1. C. J. Amick, J. L. Bona, and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989), no. 1, 1–49. MR 1012198, https://doi.org/10.1016/0022-0396(89)90176-9
- 2. Jerry L. Bona and Laihan Luo, More results on the decay of solutions to nonlinear, dispersive wave equations, Discrete Contin. Dynam. Systems 1 (1995), no. 2, 151–193. MR 1355869, https://doi.org/10.3934/dcds.1995.1.151
- 3. A. S. Fokas and V. E. Zakharov (eds.), Important developments in soliton theory, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1993. MR 1280466
- 4. Roger K. Dodd, J. Chris Eilbeck, John D. Gibbon, and Hedley C. Morris, Solitons and nonlinear wave equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. MR 696935
- 5. A. S. Fokas and B. Fuchssteiner, The hierarchy of the Benjamin-Ono equation, Phys. Lett. A 86 (1981), no. 6-7, 341–345. MR 637024, https://doi.org/10.1016/0375-9601(81)90551-X
- 6. A. S. Fokas and Q. M. Liu, Asymptotic integrability of water waves, Phys. Rev. Lett. 77 (1996), no. 12, 2347–2351. MR 1406987, https://doi.org/10.1103/PhysRevLett.77.2347
- 7. Thanasis Fokas and Laihan Luo, On the asymptotic integrability of a generalized Burgers equation, Mathematical problems in the theory of water waves (Luminy, 1995) Contemp. Math., vol. 200, Amer. Math. Soc., Providence, RI, 1996, pp. 85–98. MR 1410502, https://doi.org/10.1090/conm/200/02511
- 8. Yuji Kodama, On integrable systems with higher order corrections, Phys. Lett. A 107 (1985), no. 6, 245–249. MR 778808, https://doi.org/10.1016/0375-9601(85)90207-5
- 9. Q.M. Liu, Ph.D. Dissertation, Clarkson University, 1995.
- 10. G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- C.J. Amick, J.L. Bona and M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989), pp. 1-49. MR 1012198 (90k:35210)
- J.L. Bona and L. Luo, More Results on the Decay of Solutions to Nonlinear, Dispersive Wave Equations, Discrete and Continuous Dynamical Systems 1 (1995), pp. 151-193. MR 1355869 (96h:35177)
- P. Deift, A.R. Its and X. Zhou, in Important Developments in Soliton Theory, A.S. Fokas and V.E. Zakharov (eds.), Springer-Verlag, 1993. MR 1280466 (95a:58002)
- R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solutions and Nonlinear Wave Equations, Academic Press, London, 1982. MR 696935 (84j:35142)
- A.S. Fokas and B. Fuchssteiner, The Hierarchy of the Benjamin-Ono Equation, Phys. Lett. A 86 (1981), pp. 341-345. MR 637024 (83f:58039)
- A.S. Fokas and Q.M. Liu, Asymptotic Integrability of Water Waves, Phys. Rev. Lett. 77 (1996), pp. 2347-2351. MR 1406987 (97f:76009)
- A.S. Fokas and L. Luo, On the Asymptotic Integrability of a Generalized Burgers Equation, in Hydrodynamic Waves, Contemporary Mathematics, Vol. 200 (Dias, Ghidaglia and Saut, eds.), American Mathematical Society, Providence, RI (1996), pp. 85-98. MR 1410502 (98b:35156)
- Y. Kodama, On Integrable Systems with Higher Order Corrections, Phys. Lett. A 107 (1985), pp. 245-249; and Nearly Integrable Systems, Physica D 16 (1985), pp. 14-26. MR 778808 (86i:58065)
- Q.M. Liu, Ph.D. Dissertation, Clarkson University, 1995.
- G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)
Athanassios S. Fokas
Affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, United Kingdom
Affiliation: Department of Mathematics, New York Institute of Technology, 1855 Broadway, New York, New York 10023
Keywords: Nonlinear, dissipation, dispersive, wave, asymptotic linearization
Received by editor(s): November 14, 2006
Published electronically: July 24, 2008
Additional Notes: This work was partially supported by the EPSRC, GR/J71885.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.