
SIGMA 8 (2012), 103, 54 pages arXiv:1203.5732
http://dx.doi.org/10.3842/SIGMA.2012.103
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”
Whitham's Method and DubrovinNovikov Bracket in SinglePhase and Multiphase Cases
Andrei Ya. Maltsev
L.D. Landau Institute for Theoretical Physics, 1A Ak. Semenova Ave., Chernogolovka, Moscow reg., 142432, Russia
Received April 23, 2012, in final form December 11, 2012; Published online December 24, 2012
Abstract
In this paper we examine in detail the procedure of averaging
of the local fieldtheoretic Poisson brackets proposed by B.A. Dubrovin
and S.P. Novikov for the method of Whitham.
The main attention is paid to
the questions of justification and the conditions of applicability of the
DubrovinNovikov procedure.
Separate consideration is given to special
features of singlephase and multiphase cases.
In particular, one of the
main results is the insensitivity of the procedure of bracket averaging
to the appearance of ''resonances'' which can arise in the multiphase
situation.
Key words:
quasiperiodic solutions; slow modulations; Hamiltonian structures.
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References
 Ablowitz M.J., Benney D.J., The evolution of multiphase modes for nonlinear
dispersive waves, Stud. Appl. Math. 49 (1970), 225238.
 Alekseev V.L., On nonlocal Hamiltonian operators of hydrodynamic type
connected with Whitham's equations, Russian Math. Surveys
50 (1995), 12531255.
 Arnol'd V.I., Geometrical methods in the theory of ordinary differential
equations, Grundlehren der Mathematischen Wissenschaften, Vol. 250,
2nd ed., SpringerVerlag, New York, 1988.
 Dobrokhotov S.Yu., Resonance correction of an adiabatically perturbed finitegap
almost periodic solution of the Kortewegde Vries equation,
Math. Notes 44 (1988), 551555.
 Dobrokhotov S.Yu., Resonances in asymptotic solutions of the Cauchy problem for
the Schrödinger equation with rapidly oscillating finitezone potential,
Math. Notes 44 (1988), 656668.
 Dobrokhotov S.Yu., Krichever I.M., Multiphase solutions of the
BenjaminOno equation and their averaging, Math. Notes
49 (1991), 583594.
 Dobrokhotov S.Yu., Maslov V.P., Finitegap almost periodic solutions in the
WKB approximation, J. Sov. Math. 15 (1980), 14331487.
 Dobrokhotov S.Yu., Maslov V.P., Multiphase asymptotics of nonlinear partial
differential equations with a small parameter, in Mathematical Physics
Reviews, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Vol. 3, Harwood
Academic Publ., Chur, 1982, 221311.
 Dobrokhotov S.Yu., Minenkov D.S., Remark on the phase shift in the
KuzmakWhitham ansatz, Theoret. Math. Phys. 166
(2011), 303316.
 Dubrovin B.A., Inverse problem for periodic finitezoned potentials in the
theory of scattering, Funct. Anal. Appl. 9 (1975), 6162.
 Dubrovin B.A., Theta functions and nonlinear equations, Russian Math.
Surveys 36 (1981), no. 2, 1192.
 Dubrovin B.A., Matveev V.B., Novikov S.P., Nonlinear equations of
Kortewegde Vries type, finiteband linear operators and Abelian
varieties, Russian Math. Surveys 31 (1976), no. 1, 59146.
 Dubrovin B.A., Novikov S.P., A periodic problem for the Kortewegde Vries
and SturmLiouville equations. Their connection with algebraic
geometry, Soviet Math. Dokl. 15 (1976), 15971601.
 Dubrovin B.A., Novikov S.P., Hamiltonian formalism of onedimensional systems
of the hydrodynamic type and the BogolyubovWhitham averaging method,
Soviet Math. Dokl. 27 (1983), 665669.
 Dubrovin B.A., Novikov S.P., Hydrodynamics of soliton lattices, Sov.
Sci. Rev. Sect. C 9 (1992), no. 4, 1136.
 Dubrovin B.A., Novikov S.P., Hydrodynamics of weakly deformed soliton lattices.
Differential geometry and Hamiltonian theory, Russian Math.
Surveys 44 (1989), no. 6, 35124.
 Dubrovin B.A., Novikov S.P., Periodic and conditionally periodic analogs of the
manysoliton solutions of the Kortewegde Vries equation, Soviet
Physics JETP 67 (1974), 10581063.
 Ferapontov E.V., Differential geometry of nonlocal Hamiltonian operators of
hydrodynamic type, Funct. Anal. Appl. 25 (1991), 195204.
 Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type:
differential geometry and applications, in Topics in Topology and
Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 170,
Amer. Math. Soc., Providence, RI, 1995, 3358.
 Ferapontov E.V., Nonlocal matrix Hamiltonian operators, differential geometry
and applications, Theoret. Math. Phys. 91 (1992), 642649.
 Ferapontov E.V., Restriction, in the sense of Dirac, of the Hamiltonian
operator δ^{ij}(d/dx) to a surface of the Euclidean space with a
plane normal connection, Funct. Anal. Appl. 26 (1992),
298300.
 Flaschka H., Forest M.G., McLaughlin D.W., Multiphase averaging and the inverse
spectral solution of the Kortewegde Vries equation, Comm. Pure
Appl. Math. 33 (1980), 739784.
 Haberman R., Standard form and a method of averaging for strongly nonlinear
oscillatory dispersive traveling waves, SIAM J. Appl. Math.
51 (1991), 16381652.
 Haberman R., The modulated phase shift for weakly dissipated nonlinear
oscillatory waves of the Kortewegde Vries type, Stud. Appl.
Math. 78 (1988), 7390.
 Hayes W.D., Group velocity and nonlinear dispersive wave propagation,
Proc. R. Soc. Lond. Ser. A 332 (1973), 199221.
 Its A.R., Matveev V.B., Hill's operator with finitely many gaps, Funct.
Anal. Appl. 9 (1975), 6566.
 Its A.R., Matveev V.B., Schrödinger operators with finitegap spectrum and
Nsoliton solutions of the Kortewegde Vries equation,
Theoret. Math. Phys. 23 (1975), 343355.
 Krichever I.M., Perturbation theory in periodic problems for twodimensional
integrable systems, Sov. Sci. Rev. Sect. C 9 (1992), no. 2,
1103.
 Krichever I.M., The averaging method for twodimensional "integrable"
equations, Funct. Anal. Appl. 22 (1988), 200213.
 Krichever I.M., The "Hessian" of integrals of the Kortewegde Vries
equation and perturbations of finitegap solutions, Sov. Math. Dokl.
270 (1983), 757761.
 Luke J.C., A perturbation method for nonlinear dispersive wave problems,
Proc. R. Soc. Lond. Ser. A 292 (1966), 403412.
 Maltsev A.Ya., Conservation of Hamiltonian structures in Whitham's averaging
method, Izv. Math. 63 (1999), 11711201.
 Maltsev A.Ya., Deformations of the Whitham systems in the almost linear case,
in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc.
Transl. Ser. 2, Vol. 224, Amer. Math. Soc., Providence, RI, 2008, 193212,
arXiv:0709.4618.
 Maltsev A.Ya., The averaging of nonlocal Hamiltonian structures in Whitham's
method, Int. J. Math. Math. Sci. 30 (2002), 399434,
solvint/9910011.
 Maltsev A.Ya., Whitham systems and deformations, J. Math. Phys.
47 (2006), 073505, 18 pages, nlin.SI/0509033.
 Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly
nonlocal Poisson brackets, Phys. D 156 (2001), 5380,
nlin.SI/0006030.
 Maltsev A.Ya., Pavlov M.V., On Whitham's averaging method, Funct.
Anal. Appl. 29 (1995), 619, nlin.SI/0306053.
 Mokhov O.I., Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic
type related to metrics of constant curvature, Russian Math. Surveys
45 (1990), 218219.
 Newell A.C., Solitons in mathematics and physics, CBMSNSF Regional
Conference Series in Applied Mathematics, Vol. 48, Society for Industrial
and Applied Mathematics (SIAM), Philadelphia, PA, 1985.
 Novikov S.P., Geometry of conservative systems of hydrodynamic type. The
averaging method for fieldtheoretic systems, Russian Math. Surveys
40 (1985), no. 4, 8598.
 Novikov S.P., The periodic problem for the Kortewegde Vries equation,
Funct. Anal. Appl. 8 (1974), 236246.
 Novikov S.P., Manakov S.V., Pitaevski L.P., Zakharov V.E., Theory of
solitons. The inverse scattering method, Contemporary Soviet Mathematics,
Plenum, New York, 1984.
 Pavlov M.V., Elliptic coordinates and multiHamiltonian structures of
hydrodynamictype systems, Russian Acad. Sci. Dokl. Math.
50 (1995), 374377.
 Pavlov M.V., MultiHamiltonian structures of the Whitham equations,
Russian Acad. Sci. Dokl. Math. 50 (1995), 220223.
 Schmidt W.M., Diophantine approximation, Lecture Notes in Mathematics,
Vol. 785, SpringerVerlag, Berlin  Heidelberg  New York, 1980.
 Tsarev S.P., On Poisson bracket and onedimensional systems of hydrodynamic
type, Soviet Math. Dokl. 31 (1985), 488491.
 Vorob'ev Y.M., Dobrokhotov S.Yu., Completeness of the system of eigenfunctions of
a nonelliptic operator on the torus, generated by a Hill operator with a
finitezone potential, Funct. Anal. Appl. 22 (1988),
137139.
 Whitham G.B., A general approach to linear and nonlinear dispersive waves
using a Lagrangian, J. Fluid Mech. 22 (1965), 273283.
 Whitham G.B., Linear and nonlinear waves, Pure and Applied Mathematics,
WileyInterscience, New York, 1974.
 Whitham G.B., Nonlinear dispersive waves, Proc. R. Soc. Lond. Ser. A
283 (1965), 238261.

