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Mathematics of dispersive water waves. (English) Zbl 1093.37511
Summary: A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q30 Navier-Stokes equations
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Broer, L.J.F.: Approximate equations for long water waves. Appl. Sci. Res.31, 377–395 (1975) · Zbl 0326.76017 · doi:10.1007/BF00418048
[2] Kaup, D.J.: A higher-order water-wave equation and the method for solving it. Prog. Theor. Phys.54, 396–408 (1975) · Zbl 1079.37514 · doi:10.1143/PTP.54.396
[3] Matveev, V.B., Yavor, M.I.: Solutions presque périodiques et aN-solitons de l’équation hydrodynamique non linéaire de Kaup. Ann. Inst. H. Poincare, Sect. A,XXXI, No. 1, 25–41 (1979) · Zbl 0435.76016
[4] Wilson, G.: Commuting flows and conservation laws for Lax equations. Math. Proc. Cambr. Phil. Soc.86, 131–143 (1979) · Zbl 0427.35024 · doi:10.1017/S0305004100000700
[5] Manin, Yu. I.: Algebraic aspects of non-linear differential equations. Itogi Nauki Tekh., Ser. Sovr. Prob. Math.11, 5–152 (1978) [J. Sov. Math.11, 1–122 (1979)]
[6] Kupershmidt, B.A.: Discrete Lax equations and differential-difference calculus. E.N.S. Lecture Notes, Paris (1982); Revue Asterisque, Vol. 123, Paris (1985) · Zbl 0565.58024
[7] Gel’fand, I.M., Dorfman, I.Ya.: Hamiltonian operators and infinite-dimensional Lie algebras. Funct. Anal. Appl.15, 23–40 (1981) (Russian); 173–187 (English) · Zbl 0479.18003 · doi:10.1007/BF01082287
[8] Kupershmidt, B.A.: On dual spaces of differential Lie algebras. Physica7D, 334–337 (1983) · Zbl 0544.58009
[9] Kupershmidt, B.A.: On algebraic models of dynamical systems. Lett. Math. Phys.6, 85–89 (1982) · Zbl 0494.58021 · doi:10.1007/BF00401731
[10] Kupershmidt, B.A.: Normal and universal forms in integrable hydrodynamical systems. In Proc. NASA Ames-Berkeley 1983 conf. on nonlinear problems in optimal control and hydrodynamics. Hunt, R.L., Martin, C., eds. Math. Sci. Press (1984) · Zbl 0594.76003
[11] Benney, D.J.: Some properties of long nonlinear waves. Stud. Appl. Math. L11 (1), 45–50 (1973) · Zbl 0259.35011
[12] Kupershmidt, B.A., Manin, Yu.I.: Long-wave equation with free boundaries. I. Conservation Laws. Funct. Anal. Appl.11: 3, 31–42 (1977) (Russian); 188–197 (English) · Zbl 0364.35043
[13] Kupershmidt, B.A., Manin, Yu.I.: Equations of long waves with a free surface. II. Hamiltonian structure and higher equations. Funct. Anal. Appl.12: 1, 25–37 (1978) (Russian); 20–29 (English) · Zbl 0405.58049
[14] Kupershmidt, B.A.: Modern Hamiltonian formalism (to appear)
[15] Kupershmidt, B.A., Wilson, G.: Modifying Lax equations and the second Hamiltonian structure. Invent. Math.62, 403–436 (1981) · Zbl 0464.35024 · doi:10.1007/BF01394252
[16] Kupershmidt, B.A.: Deformations of Hamiltonian structures. M.I.T. preprint (1978) · Zbl 0405.58049
[17] Kupershmidt, B.A., Wilson, G.: Conservation laws and symmetries of generalized Sine-Gordon equations. Commun. Math. Phys.81, 189–202 (1981) · Zbl 0487.35078 · doi:10.1007/BF01208894
[18] Wilson, G.: The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras. Ergod. Th. Dynam. Syst.1, 361–380 (1981) · Zbl 0495.58008
[19] Drinfel’d, V.G., Sokolov, V.V.: Equations of KdV type and simple Lie algebras. Dokl. Akad. Nauk SSSR258, 11–16 (1981) [Sov. Math. Dokl.23, 457–462 (1981)]
[20] Drinfel’d, V.G., Sokolov, V.V.: Lie algebras and the Korteweg-de-Vries type equations. Itogi Nauki Tekh., Ser. Sovr. Prob. Math., Vol. 24. Moscow (1984)
[21] Knörrer, H.: Private communication, Bonn (June, 1981)
[22] Kupershmidt, B.A.: Involutivity of conservation laws for a fluid of finite depth and Benjamin-Ono equations. Libertas Math.1, 189–202 (1981) · Zbl 0479.76030
[23] Gibbons, J.: Related integrable hierarchies. I. Two nonlinear Schrödinger equations. DIAS preprint STP-81-06 (Dublin, 1981)
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