On the galilean invariance of some nonlinear dispersive wave equations

Studies in Applied Mathematics

Volumn 131, Issue 4, 2013, Pages 359-388

  Duran, Angel ^a   Dutykh, Denys ^b   Mitsotakis, Dimitrios ^c  

^a UNIVERSITY OF VALLADOLID   (Spain)

^b UNIVERSITY COLLEGE DUBLIN   (Ireland)

^c Departments of Cognitive Science and Neurosciences   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EULER EQUATIONS; NONLINEAR EQUATIONS; SOLITONS; SURFACE WATERS; WAVE EQUATIONS;

ASYMPTOTIC APPROXIMATION; EULER'S EQUATION; GALILEAN INVARIANCE; GALILEAN TRANSFORMATION; IDEAL FLUIDS; NONLINEAR DISPERSIVE WAVES; PROPERTY; SURFACE WATER WAVES; WATER-WAVE PROBLEM; WAVE MODELLING;

WATER WAVES;

EID: 84887015596     PISSN: 00222526     EISSN: 14679590     Source Type: Journal    
DOI: 10.1111/sapm.12015     Document Type: Article

References (71)

1. D. Dutykh, T. Katsaounis, and D. Mitsotakis, Dispersive wave runup on non-uniform shores, in Finite Volumes for Complex Applications VI - Problems & Perspectives, (J. Fort et al., Eds.), pp. 389-397, Springer Berlin Heidelberg, Prague, 2011.
2. D. Dutykh, T. Katsaounis, and D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, J. Comput. Phys 230:3035-3061, http://linkinghub.elsevier.com/retrieve/pii/S0021999111000118 ISSN 00219991 (2011).
3. I. Newton, Philosophiae Naturalis Principia Mathematica, 1687. Available at: http://books.google.ca/books?id=Tm0FAAAAQAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false.
4. P. J. Olver, Applications of Lie groups to differential equations, Vol. 107 (2nd ed. of Graduate Texts in Mathematics), Springer-Verlag, New York, 1993.
5. T. B. Benjamin and P. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech. 125:137-185 (1982).
6. C. I. Christov, An energy-consistent dispersive shallow-water model, Wave Motion 34:161-174 (2001).
7. A. J. C. de Saint-Venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sc. Paris 73:147-154 (1871).
8. D. Dutykh and D. Clamond, Shallow water equations for large bathymetry variations, J. Phys. A: Math. Theor. 44:332001 (2011).
9. F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, La Houille blanche 8:374-388 (1953).
10. C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Math. Phys. 10: 536-539 (1969).
11. A. E. Green, N. Laws, and P. M. Naghdi, On the theory of water waves, Proc. R. Soc. Lond. A 338:43-55 (1974).
12. A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78:237-246 (1976).
13. D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids 21:016601 (2009).
14. J. L. Bona and R. Smith, A model for the two-way propagation of water waves in a channel, Math. Proc. Camb. Phil. Soc. 79:167-182 (1976).
15. O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, J. WaterWays, Port, Coastal, Ocean Eng. 119:618-638 (1993).
16. J. L. Bona, M. Chen, and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci. 12:283-318 (2002).
17. J. A. Pava, J. L. Bona, and M. Scialom, Stability of cnoidal waves, Adv. Differ. Equ. 11:1321-1374 (2006).
18. J. D. Carter and R. Cienfuegos, The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations, Eur. J. Mech. B/Fluids 30:259-268 (2011).
19. P. Kim, Invariantization of numerical schemes using moving frames, BIT Numer. Math. 47:525-546 (2007).
20. P. Kim, Invariantization of the Crank-Nicolson method for Burgers' equation, Physica D 237:243-254 (2008).
21. M. Chhay and A. Hamdouni, On accuracy of invariant numerical schemes, Commun. Pure Appl. Anal. 10:761-783 (2011).
22. M. Chhay, E. Hoarau, A. Hamdouni, and P. Sagaut, Comparison of some Lie-symmetry-based integrators, J. Comp. Phys. 230(5):2174-2188 (2011).
23. J. J. Stoker, Water Waves, the Mathematical Theory with Applications, Wiley, New York, 1958.
24. G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons Inc., New York, 1999 ISBN 978-0471940906.
25. A. A. Petrov, Variational statement of the problem of liquid motion in a container of finite dimensions, Prikl. Math. Mekh. 28(4):917-922 (1964).
26. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9:190-194 (1968).
27. L. J. F. Broer, On the Hamiltonian theory of surface waves, Appl. Sci. Res. 29(6):430-446 (1974).
28. R. Salmon, Hamiltonian fluid mechanics, Ann. Rev. Fluid Mech. 20:225-256 (1988).
29. J. C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech. 27:375-397 (1967).
30. D. Clamond and D. Dutykh, Practical use of variational principles for modeling water waves, Phys. D: Nonlinear Phenom. 241:25-36, 1002.3019 (2012).
31. A. D. D. Craik, The origins of water wave theory, Ann. Rev. Fluid Mech. 36:1-28 (2004).
32. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17:55-108 (1872).
33. D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39:422-443 (1895).
34. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 2004.
35. D. H. Peregrine, Long waves on a beach, J. Fluid Mech. 27:815-827 (1967).
36. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math. 27:97-133 (1974).
37. P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21:467-490 (1968).
38. R. M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18:412-459 (1976).
39. J. L. Bona, Solitary waves and other phenomena associated with model equations for long waves, Fluid Dyn. Trans. 10:77-111 (1980).
40. T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A 272:47-78 (1972).
41. D. H. Peregrine, Calculations of the development of an undual bore, J. Fluid Mech. 25:321-330 (1966).
42. H. R. Dullin, G. A. Gottwald, and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D 190:1-14, http://linkinghub.elsevier.com/retrieve/pii/S01672-78903004536, ISSN 01672789, (2004).
43. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71(11):1661-1664 (1993).
44. R. Fetecau and D. Levy, Aproximate model equations for water waves, Comm. Math. Sci. 3:159-170 (2005).
45. A. Degasperis and M. Procesi, Asymptotic Integrability, in Symmetry and Perturbation Theory (A. Degasperis and G. Gaeta, eds.) World Scientific, pp. 23-37, 1999.
46. P. J. Olver, Personal web page (2012). Available at: http://www.math.umn.edu/~olver/. Accessed June 1, 2010.
47. R. L. Pego and M. I. Weinstein, Convective linear stability of solitary waves for Boussinesq equations, Stud. Appl. Maths. 99:311-375 (1997).
48. M. Chen, Solitary-wave and multi pulsed traveling-wave solution of Boussinesq systems, Applic. Analysis 7:213-240 (2000).
49. E. Barthélémy, Nonlinear shallow water theories for coastal waves, Surv. Geophys. 25:315-337 (2004).
50. D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear ID Serre equations. (2013). doi:10.1017/S0956792513000168.
51. R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 455:63-82 (2002).
52. Y. A. Li, Linear stability of solitary waves of the Green-Naghdi equations, Commun. Pure Appl. Math. 54:501-536 (2001).
53. Y. A. Li, Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations, J. Nonlinear Math. Phys. 9:99-105 (2002).
54. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, Society for Industrial and Applied Mathematics, 2010 ISBN 978-0-89871-705-1.
55. V. I. Petviashvili, Equation of an extraordinary soliton, Sov. J. Plasma Phys. 2(3):469-472 (1976).
56. D. Pelinovsky and Y. A. Stepanyants, Convergence of Petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J. Num. Anal. 42:1110-1127 (2004).
57. T. I. Lakoba and J. Yang, A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, J. Comp. Phys. 226:1668-1692 (2007).
58. R. Grimshaw, The solitary wave in water of variable depth. Part 2., J. Fluid Mech. 46:611-622 (1971).
59. V. A. Dougalis and D. E. Mitsotakis, Theory and numerical analysis of Boussinesq systems: a review, in Effective Computational Methods in Wave Propagation, (N. A. Kampanis, V. A. Dougalis, and J. A. Ekaterinaris, Eds.), pp. 63-110, CRC Press, Boca Raton, 2008.
60. V. Dougalis, A. Duran, M. A. Lopez-Marcos, and D. E. Mitsotakis, A numerical study of the stability of solitary waves of Bona-Smith family of Boussinesq systems, J. Nonlinear Sci. 17:595-607 (2007).
61. M. Tanaka, The stability of solitary waves, Phys. Fluids 29(3):650-655 (1986).
62. J. Fenton, A ninth-order solution for the solitary wave, J. Fluid Mech. 53(2):257-271 (1972).
63. M. Longuet-Higgins and J. Fenton, On the mass, momentum, energy and circulation of a solitary wave. II, Proc. R. Soc. A 340(1623):471-493 (1974).
64. Y. A. Li, J. M. Hyman, and W. Choi, A numerical study of the exact evolution equations for surface waves in water of finite depth, Stud. Appl. Maths. 113:303-324 (2004).
65. N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett 15:240-243 (1965).
66. J. L. Bona, W. G. Pritchard, and L. R. Scott, Solitary-wave interaction, Phys. Fluids 23:438-441 (1980).
67. J. L. Bona, W. G. Pritchard, and L. R. Scott, An Evaluation of a Model Equation for Water Waves, Phil. Trans. R. Soc. Lond. A 302:457-510 (1981).
68. K. O. Abdulloev, I. L. Bogolubsky, and V. G. Makhankov, One more example of inelastic soliton interaction, Phys. Lett. 56A:1976 (1976).
69. J. Courtenay Lewis and J. A. Tjon, Resonant production of solitons in the RLW equation, Phys. Lett. 73A:275-279 (1979).
70. W. Craig, P. Guyenne, J. Hammack, D. Henderson, and C. Sulem, Solitary water wave interactions, Phys. Fluids 18(5):57106, (2006).
71. D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of Boussinesq systems of the Bona-Smith family, Appl. Numer. Math. 30:314-336 (2010).