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On Whitham’s averaging method. (English. Russian original) Zbl 0843.35018

Funct. Anal. Appl. 29, No. 1, 6-19 (1995); translation from Funkts. Anal. Prilozh. 29, No. 1, 7-24 (1995).
The paper presents a proof that two different methods allowing to derive the Witham’s approximation for a given system of nonlinear Lagrangian PDEs are equivalent. The Witham’s equations are a system of first-order quasilinear equations which describe a slow evolution in time and space (only the one-dimensional case is considered) of parameters of a periodic traveling-wave solution (“cnoidal wave”) of the underlying PDEs. The classical Witham’s derivation is based on performing averaging not in terms of the underlying equations, but rather in terms of their Lagrangian, into which a trial wave form in the form of the cnoidal wave with slowly varying parameters is inserted. An alternative derivation, due to Dubrovin and Novikov, starts from the Hamiltonian structure (the Poisson bracket) of the underlying PDEs.
In the present work, it is shown that both derivations are tantamount to each other, and they finally produce the same Poisson bracket for the Witham’s equations. It is also demonstrated that the energy and momentum conservation in the original equations produce, on averaging, the “number of waves” conservation law in terms of the Witham’s equations. If the underlying equations have additional conservation laws, they will produce certain conservation laws of the Witham’s system, which can also be derived by means of the averaging procedure.

MSC:

35F25 Initial value problems for nonlinear first-order PDEs
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References:

[1] G. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). · Zbl 0373.76001
[2] B. A. Dubrovin and S. P. Novikov, ?Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,? Dokl. Akad. Nauk SSSR,270, No. 4, 781-785 (1983). · Zbl 0553.35011
[3] B. A. Dubrovin and S. P. Novikov, ?Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,? Uspekhi Mat. Nauk,44, No. 6 (270), 29-98 (1989). · Zbl 0712.58032
[4] S. P. Novikov and A. Ya. Maltsev, ?The Liouville form of averaged Poisson brackets,? Uspekhi Mat. Nauk,48, No. 1 (289), 155-156 (1993). · Zbl 0807.58016
[5] M. V. Pavlov, ?The Hamiltonian structure of Whitham’s equations,? Uspekhi Mat. Nauk,49, No. 1 (295), 219-220 (1994).
[6] M. V. Pavlov, ?Double Lagrangian representation of the KdV equation,? Dokl. Ross. Akad. Nauk,339, No. 2, 157-161 (1994).
[7] S. P. Tsarev, ?On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type,? Dokl. Akad. Nauk SSSR,282, No. 3, 534-537 (1985). · Zbl 0605.35075
[8] I. M. Krichever, ?Averaging method for two-dimensional ?integrable? equations,? Funkts. Anal. Prilozhen.,22, No. 3, 37-52 (1988).
[9] I. M. Krichever, ?Spectral theory of two-dimensional operators and its applications,? Uspekhi Mat. Nauk,44, No. 2, 121-184 (1989).
[10] M. J. Ablowitz and D. J. Benney, ?The evolution of multiphase modes for nonlinear dispersive waves,? Stud. Appl. Math.,49, No. 3, 225-238 (1970). · Zbl 0203.41001
[11] W. D. Hayes, ?Group velocity and nonlinear dispersive wave propagation,? Proc. Roy. Soc. London,332, 199-221 (1973). · Zbl 0271.76006
[12] H. Flaschka, M. G. Forest, and D. W. McLaughlin, ?Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation,? Comm. Pure Appl. Math.,33, No. 6, 739-784 (1980). · Zbl 0454.35080
[13] J. C. Luke, ?A perturbation method for nonlinear dispersive wave problems,? Proc. Roy. Soc. London Ser. A,292, No. 1430, 403-412 (1966). · Zbl 0143.13603
[14] S. Yu. Dobrokhotov and V. P. Maslov, Finite-Gap Almost Periodic Solutions in the WKB Approximation. Contemporary Problems in Mathematics [in Russian], Vol. 15, Itogi Nauki i Tekhniki, VINITI, Moscow (1980). · Zbl 0446.35008
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