Transformation theory of Hamiltonian PDE and the problem of water waves

  • Walter Craig
Part of the NATO Science for Peace and Security Series book series (NAPSB)

This set of lecture notes gives (i) a formal theory of Hamiltonian systems posed in infinite dimensions, (ii) a perturbation theory in the presence of a small parameter, adapted to reproduce some of the well-known formal computations of fluid mechanics, and (iii) a transformation theory of Hamiltonian systems and their symplectic structures. A series of examples is given, starting with a rather complete description of the problem of water waves, and, following a series of scaling and other simple transformations placed in the above context, a derivation of the well known equations of Boussinesq and Korteweg de Vries.




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  1. 1.
    M. Christ & J.-L. Journé. Polynomial growth estimates for multilinear singular integral oper-ators. Acta Math. 159 (1987), no. 1-2, 51-80.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    . R. Coifman & Y. Meyer. Nonlinear harmonic analysis and analytic dependence. Pseudodif-ferential operators and applications (Notre Dame, Ind., 1984), 71-78, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI (1985).Google Scholar
  3. 3.
    W. Craig & M. Groves. Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19 (1994), no. 4, 367-389.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    W. Craig, P. Guyenne & H. Kalisch. Hamiltonian long-wave expansions for free surfaces and interfaces. Comm. Pure Appl. Math. 58 (2005), no. 12, 1587-1641.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. Craig, P. Guyenne, D. P. Nicholls & C. Sulem. Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2055, 839-873.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    W. Craig, U. Schanz & C. Sulem. The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 5, 615-667.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W. Craig & C. Sulem. Numerical simulation of gravity waves. J. Comput. Phys. 108 (1993), no. 1, 73-83.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    . J. Hadamard. Leçons sur le calcul des variations. Hermann, Paris, 1910; paragraph §249.Google Scholar
  9. 9.
    J. Hadamard. Sur les ondes liquides. Rend. Acad. Lincei 5 (1916), no. 25, pp. 716-719.Google Scholar
  10. 10.
    D. Kaup. A higher-order water-wave equation and the method for solving it. Progr. Theoret. Phys. 54 (1975), no. 2, 396-408.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Sachs. On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. Phys. D 30 (1988), no. 1-2, 1-27.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968), pp. 1990-1994.Google Scholar
  13. 13.
    V. E. Zakharov. On stochastization of one-dimensional chains of nonlinear oscillators. Zh. Eksp. Teor. Fiz. 65 (1973), pp. 219-225.Google Scholar

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© Springer Science + Business Media B.V 2008

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  • Walter Craig

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