Free Access
Math. Model. Nat. Phenom.
Volume 12, Number 1, 2017
Hamiltonian Systems
Page(s) 23 - 40
Published online 03 February 2017
  1. G.B. Airy. On the laws of the tides on the coasts of Ireland, as inferred from an extensive series of observations made in connexion with the Ordnance Survey of Ireland. Philos. Trans. R. Soc. London, 1–124, 1845.
  2. J.-L. Basdevant, Variational Principles in Physics. Springer-Verlag, New York, 2007.
  3. G.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967.
  4. H. Bateman. Notes on a Differential Equation Which Occurs in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 125 (1929), 598–618. [CrossRef]
  5. D.J. Benney, A.C. Newell. The propagation of nonlinear wave envelopes. J. Math. and Physics, 46 (1967), 133–139. [CrossRef]
  6. J.V. 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 (1872), 55–108.
  7. T.J. Bridges. Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc., 121 (1997), no. 1, 147–190. [CrossRef]
  8. T.J. Bridges, S. Reich. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A, 284 (2001), no. 4-5, 184–193. [CrossRef] [MathSciNet]
  9. L.J.F. Broer. On the Hamiltonian theory of surface waves,. Applied Sci. Res., 29 (1974), no. 6, 430–446. [CrossRef]
  10. R.K.-C. Chan, R.L. Street. A computer study of finite-amplitude water waves. J. Comp. Phys., 6 (1970), no. 1, 68–94. [CrossRef]
  11. Y. Chen, S. Song, H. Zhu. The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs. J. Comp. Appl. Math., 236 (2011), no. 6, 1354–1369. [CrossRef]
  12. M.. Chhay. Intégrateurs géométriques: application à la mécanique des fluides. PhD thesis (2008), Université de La Rochelle.
  13. M. Chhay, D. Dutykh, D. Clamond. On the multi-symplectic structure of the Serre-Green-Naghdi equations. J. Phys. A: Math. Gen, 49 (2016), no. 3, 03LT01. [CrossRef]
  14. M. Chhay, A. Hamdouni. On accuracy of invariant numerical schemes. Commun. Pure Appl. Anal., 10 (2011), no. 2, 761–783. [CrossRef] [MathSciNet]
  15. D. Clamond, D. Dutykh. Practical use of variational principles for modeling water waves. Phys. D, 241 (2012), no. 1, 25–36. [CrossRef] [MathSciNet]
  16. D. Clamond, D. Dutykh. Modeling water waves beyond perturbations. In E. Tobisch (Ed.), New Approaches to Nonlinear Waves, 908 (2016), 197–210. Springer, Heidelberg. [CrossRef]
  17. D. Clamond, D. Dutykh. Multi-symplectic structure of fully nonlinear weakly dispersive internal gravity waves. J. Phys. A: Math. Gen., 49 (2016), no. 31, 31LT01. [CrossRef]
  18. A.D.D. Craik. The origins of water wave theory. Ann. Rev. Fluid Mech., 36 (2004), 1–28. [CrossRef] [MathSciNet]
  19. T. de Donder. Théorie invariantive du calcul des variations. Gauthier-Villars (1930), Paris.
  20. 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 (1871), 147–154.
  21. D. Dutykh, M. Chhay, D. Clamond. Numerical study of the generalised Klein-Gordon equations. Physica D: Nonlinear Phenomena, 304–305 (2015), 23–33. [CrossRef]
  22. D. Dutykh, M. Chhay, F. Fedele. Geometric numerical schemes for the KdV equation. Comp. Math. Math. Phys., 53 (2013), no. 2, 221–236. [CrossRef]
  23. D. Dutykh, D. Clamond, P. Milewski, D. Mitsotakis. Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Math., 24 (2013), no. 5, 761–787. [CrossRef]
  24. A.I. Dyachenko, A.O. Korotkevich, V.E. Zakharov. Weak turbulence of gravity waves. JETP Lett., 77 (2003), 546–550. [CrossRef]
  25. K.B. Dysthe. Note on a modification to the nonlinear Schrödinger equation for application to deep water. Proc. R. Soc. Lond. A, 369 (1979), 105–114. [CrossRef]
  26. R. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 1: Mainly Mechanics, Radiation, and Heat. Addison Wesley (2005), 2 edition.
  27. H. Goldschmidt, S. Sternberg. The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier, 23 (1973), no. 1, 203–267. [CrossRef] [MathSciNet]
  28. O. Gramstad, K. Trulsen. Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. J. Fluid Mech., 670 (2011), 404–426. [CrossRef] [MathSciNet]
  29. F.S. Henyey. Hamiltonian description of stratified fluid dynamics. Phys. Fluids, 26 (1983), no. 1, 40–47. [CrossRef] [MathSciNet]
  30. J. Kijowki. Multiphase spaces and gauge in calculus of variations. Bull. Acad. Polon. des Sci., Série Sci. Math., Astr. et Phys., XXII (1974), 1219–1225.
  31. G.J. Komen, L. Cavalieri, M. Donelan, K. Hasselmann, S. Hasselmann, P.A.E.M. Janssen. Dynamics and Modelling of Ocean Waves. Cambridge University Press, Cambridge (1996),
  32. A.O. Korotkevich, A.N. Pushkarev, D. Resio, V.E. Zakharov. Numerical verification of the weak turbulent model for swell evolution. Eur. J. Mech. B/Fluids, 27 (2008), no. 4, 361–387. [CrossRef] [MathSciNet]
  33. R.A. Kraenkel, J. Leon, M.A. Manna, Theory of small aspect ratio waves in deep water. Physica D, 211 (2005), 377–390. [CrossRef] [MathSciNet]
  34. D. Krupka. A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections. Journal of Mathematical Analysis and Applications, 49 (1975), no. 1, 180–206. [CrossRef] [MathSciNet]
  35. D. Krupka. A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance. Journal of Mathematical Analysis and Applications, 49 (1975), no. 2, 469–476. [CrossRef] [MathSciNet]
  36. J.-L. Lagrange. Mécanique analytique. Hallet-Bachelier, Paris, 3 edition, (1853).
  37. T. Lepage. Sur les champs géodésiques du calcul des variations. Bull. Acad. Roy. Belg., Cl. Sci, 27 (1936), 716–729, 1036–1046.
  38. A. Lew, J. Marsden, M. Ortiz, M. West, An overview of variational integrators. In Finite Element Methods: 1970s and beyond (CIMNE, 2003), p. 18, Barcelona, Spain, (2004).
  39. J.C. Luke. A variational principle for a fluid with a free surface. J. Fluid Mech., 27 (1967), 375–397.
  40. J.E. Marsden, G.W. Patrick, S. Shkoller. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys., 199 (1998), no. 2, 351–395. [CrossRef] [MathSciNet]
  41. J.W. Miles. On Hamilton’s principle for water waves. J. Fluid Mech., 83 (1977), 153–158. [CrossRef] [MathSciNet]
  42. B. Moore, S. Reich. Multi-symplectic integration methods for Hamiltonian PDEs. Future Generation Computer Systems, 19 (2003), no. 3, 395–402. [CrossRef]
  43. A.A. Petrov. Variational statement of the problem of liquid motion in a container of finite dimensions. Prikl. Math. Mekh., 28 (1964, no. 4, 917–922.
  44. F. Serre. Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille blanche, 8 (1953), 830–872. [CrossRef] [EDP Sciences]
  45. J.-M. Souriau. Structure of Dynamical Systems: a Symplectic View of Physics. Birkhäuser, Boston, MA, (1997). [CrossRef]
  46. J.J. Stoker, Water Waves: The mathematical theory with applications. Interscience, New York, (1957).
  47. G.G. Stokes, Supplement to a paper on the theory of oscillatory waves. Mathematical and Physical Papers, 1 (1880), 314–326.
  48. S.A. Thorpe. The Turbulent Ocean. Cambridge University Press, Cambridge, (2005). [CrossRef]
  49. V. Volterra. Sopra una estensione della teoria Jacobi-Hamilton del calcolo delle variazioni. Rend. Cont. Acad. Lincei, ser. IV, VI (1890), 127–138.
  50. V. Volterra. Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni. Rend. Cont. Acad. Lincei, ser. IV, VI (1890), 42–54.
  51. H. Weyl. Geodesic Fields in the Calculus of Variation for Multiple Integrals. Annals of Mathematics, 36 (1935), no. 3, 607–629. [CrossRef] [MathSciNet]
  52. H.C. Yuen, B.M. Lake. Nonlinear dynamics of deep-water gravity waves. Adv. App. Mech., 22 (1982), 67–229. [CrossRef]
  53. V.E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., 9 (1968), 190–194. [CrossRef]
  54. V.E. Zakharov. Turbulence in Integrable Systems. Studies in Applied Mathematics, 122 (2009), no. 3, 219–234. [CrossRef] [MathSciNet]
  55. V.E. Zakharov, E.A. Kuznetsov. Hamiltonian formalism for nonlinear waves. Usp. Fiz. Nauk, 167 (1997), 1137–1168. [CrossRef]
  56. V.E. Zakharov, V.S. Lvov, G. Falkovich. Kolmogorov Spectra of Turbulence I Wave Turbulence.. Series in Nonlinear Dynamics, Springer-Verlag, Berlin, (1992). [CrossRef]

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