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. [Google Scholar]
  2. J.-L. Basdevant, Variational Principles in Physics. Springer-Verlag, New York, 2007. [Google Scholar]
  3. G.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967. [Google Scholar]
  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] [Google Scholar]
  5. D.J. Benney, A.C. Newell. The propagation of nonlinear wave envelopes. J. Math. and Physics, 46 (1967), 133–139. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  7. T.J. Bridges. Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc., 121 (1997), no. 1, 147–190. [CrossRef] [Google Scholar]
  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] [Google Scholar]
  9. L.J.F. Broer. On the Hamiltonian theory of surface waves,. Applied Sci. Res., 29 (1974), no. 6, 430–446. [CrossRef] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  12. M.. Chhay. Intégrateurs géométriques: application à la mécanique des fluides. PhD thesis (2008), Université de La Rochelle. [Google Scholar]
  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] [Google Scholar]
  14. M. Chhay, A. Hamdouni. On accuracy of invariant numerical schemes. Commun. Pure Appl. Anal., 10 (2011), no. 2, 761–783. [CrossRef] [MathSciNet] [Google Scholar]
  15. D. Clamond, D. Dutykh. Practical use of variational principles for modeling water waves. Phys. D, 241 (2012), no. 1, 25–36. [CrossRef] [MathSciNet] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  18. A.D.D. Craik. The origins of water wave theory. Ann. Rev. Fluid Mech., 36 (2004), 1–28. [CrossRef] [MathSciNet] [Google Scholar]
  19. T. de Donder. Théorie invariantive du calcul des variations. Gauthier-Villars (1930), Paris. [Google Scholar]
  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. [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  24. A.I. Dyachenko, A.O. Korotkevich, V.E. Zakharov. Weak turbulence of gravity waves. JETP Lett., 77 (2003), 546–550. [CrossRef] [Google Scholar]
  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] [Google Scholar]
  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. [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  29. F.S. Henyey. Hamiltonian description of stratified fluid dynamics. Phys. Fluids, 26 (1983), no. 1, 40–47. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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), [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  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] [Google Scholar]
  36. J.-L. Lagrange. Mécanique analytique. Hallet-Bachelier, Paris, 3 edition, (1853). [Google Scholar]
  37. T. Lepage. Sur les champs géodésiques du calcul des variations. Bull. Acad. Roy. Belg., Cl. Sci, 27 (1936), 716–729, 1036–1046. [Google Scholar]
  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). [Google Scholar]
  39. J.C. Luke. A variational principle for a fluid with a free surface. J. Fluid Mech., 27 (1967), 375–397. [Google Scholar]
  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] [Google Scholar]
  41. J.W. Miles. On Hamilton’s principle for water waves. J. Fluid Mech., 83 (1977), 153–158. [CrossRef] [MathSciNet] [Google Scholar]
  42. B. Moore, S. Reich. Multi-symplectic integration methods for Hamiltonian PDEs. Future Generation Computer Systems, 19 (2003), no. 3, 395–402. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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] [Google Scholar]
  45. J.-M. Souriau. Structure of Dynamical Systems: a Symplectic View of Physics. Birkhäuser, Boston, MA, (1997). [CrossRef] [Google Scholar]
  46. J.J. Stoker, Water Waves: The mathematical theory with applications. Interscience, New York, (1957). [Google Scholar]
  47. G.G. Stokes, Supplement to a paper on the theory of oscillatory waves. Mathematical and Physical Papers, 1 (1880), 314–326. [Google Scholar]
  48. S.A. Thorpe. The Turbulent Ocean. Cambridge University Press, Cambridge, (2005). [CrossRef] [Google Scholar]
  49. V. Volterra. Sopra una estensione della teoria Jacobi-Hamilton del calcolo delle variazioni. Rend. Cont. Acad. Lincei, ser. IV, VI (1890), 127–138. [Google Scholar]
  50. V. Volterra. Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni. Rend. Cont. Acad. Lincei, ser. IV, VI (1890), 42–54. [Google Scholar]
  51. H. Weyl. Geodesic Fields in the Calculus of Variation for Multiple Integrals. Annals of Mathematics, 36 (1935), no. 3, 607–629. [CrossRef] [MathSciNet] [Google Scholar]
  52. H.C. Yuen, B.M. Lake. Nonlinear dynamics of deep-water gravity waves. Adv. App. Mech., 22 (1982), 67–229. [CrossRef] [Google Scholar]
  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] [Google Scholar]
  54. V.E. Zakharov. Turbulence in Integrable Systems. Studies in Applied Mathematics, 122 (2009), no. 3, 219–234. [CrossRef] [MathSciNet] [Google Scholar]
  55. V.E. Zakharov, E.A. Kuznetsov. Hamiltonian formalism for nonlinear waves. Usp. Fiz. Nauk, 167 (1997), 1137–1168. [CrossRef] [Google Scholar]
  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] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.