Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 103, 54 pages      arXiv:1203.5732      https://doi.org/10.3842/SIGMA.2012.103
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Whitham's Method and Dubrovin-Novikov Bracket in Single-Phase 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 field-theoretic 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 Dubrovin-Novikov procedure. Separate consideration is given to special features of single-phase 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 multi-phase situation.

Key words: quasiperiodic solutions; slow modulations; Hamiltonian structures.

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References

  1. Ablowitz M.J., Benney D.J., The evolution of multi-phase modes for nonlinear dispersive waves, Stud. Appl. Math. 49 (1970), 225-238.
  2. Alekseev V.L., On non-local Hamiltonian operators of hydrodynamic type connected with Whitham's equations, Russian Math. Surveys 50 (1995), 1253-1255.
  3. Arnol'd V.I., Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 250, 2nd ed., Springer-Verlag, New York, 1988.
  4. Dobrokhotov S.Yu., Resonance correction of an adiabatically perturbed finite-gap almost periodic solution of the Korteweg-de Vries equation, Math. Notes 44 (1988), 551-555.
  5. Dobrokhotov S.Yu., Resonances in asymptotic solutions of the Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential, Math. Notes 44 (1988), 656-668.
  6. Dobrokhotov S.Yu., Krichever I.M., Multi-phase solutions of the Benjamin-Ono equation and their averaging, Math. Notes 49 (1991), 583-594.
  7. Dobrokhotov S.Yu., Maslov V.P., Finite-gap almost periodic solutions in the WKB approximation, J. Sov. Math. 15 (1980), 1433-1487.
  8. 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, 221-311.
  9. Dobrokhotov S.Yu., Minenkov D.S., Remark on the phase shift in the Kuzmak-Whitham ansatz, Theoret. Math. Phys. 166 (2011), 303-316.
  10. Dubrovin B.A., Inverse problem for periodic finite-zoned potentials in the theory of scattering, Funct. Anal. Appl. 9 (1975), 61-62.
  11. Dubrovin B.A., Theta functions and non-linear equations, Russian Math. Surveys 36 (1981), no. 2, 11-92.
  12. Dubrovin B.A., Matveev V.B., Novikov S.P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Russian Math. Surveys 31 (1976), no. 1, 59-146.
  13. Dubrovin B.A., Novikov S.P., A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry, Soviet Math. Dokl. 15 (1976), 1597-1601.
  14. Dubrovin B.A., Novikov S.P., Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method, Soviet Math. Dokl. 27 (1983), 665-669.
  15. Dubrovin B.A., Novikov S.P., Hydrodynamics of soliton lattices, Sov. Sci. Rev. Sect. C 9 (1992), no. 4, 1-136.
  16. Dubrovin B.A., Novikov S.P., Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys 44 (1989), no. 6, 35-124.
  17. Dubrovin B.A., Novikov S.P., Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Soviet Physics JETP 67 (1974), 1058-1063.
  18. Ferapontov E.V., Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funct. Anal. Appl. 25 (1991), 195-204.
  19. 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, 33-58.
  20. Ferapontov E.V., Nonlocal matrix Hamiltonian operators, differential geometry and applications, Theoret. Math. Phys. 91 (1992), 642-649.
  21. Ferapontov E.V., Restriction, in the sense of Dirac, of the Hamiltonian operator δij(d/dx) to a surface of the Euclidean space with a plane normal connection, Funct. Anal. Appl. 26 (1992), 298-300.
  22. Flaschka H., Forest M.G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33 (1980), 739-784.
  23. Haberman R., Standard form and a method of averaging for strongly nonlinear oscillatory dispersive traveling waves, SIAM J. Appl. Math. 51 (1991), 1638-1652.
  24. Haberman R., The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg-de Vries type, Stud. Appl. Math. 78 (1988), 73-90.
  25. Hayes W.D., Group velocity and nonlinear dispersive wave propagation, Proc. R. Soc. Lond. Ser. A 332 (1973), 199-221.
  26. Its A.R., Matveev V.B., Hill's operator with finitely many gaps, Funct. Anal. Appl. 9 (1975), 65-66.
  27. Its A.R., Matveev V.B., Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation, Theoret. Math. Phys. 23 (1975), 343-355.
  28. Krichever I.M., Perturbation theory in periodic problems for two-dimensional integrable systems, Sov. Sci. Rev. Sect. C 9 (1992), no. 2, 1-103.
  29. Krichever I.M., The averaging method for two-dimensional "integrable" equations, Funct. Anal. Appl. 22 (1988), 200-213.
  30. Krichever I.M., The "Hessian" of integrals of the Korteweg-de Vries equation and perturbations of finite-gap solutions, Sov. Math. Dokl. 270 (1983), 757-761.
  31. Luke J.C., A perturbation method for nonlinear dispersive wave problems, Proc. R. Soc. Lond. Ser. A 292 (1966), 403-412.
  32. Maltsev A.Ya., Conservation of Hamiltonian structures in Whitham's averaging method, Izv. Math. 63 (1999), 1171-1201.
  33. 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, 193-212, arXiv:0709.4618.
  34. Maltsev A.Ya., The averaging of nonlocal Hamiltonian structures in Whitham's method, Int. J. Math. Math. Sci. 30 (2002), 399-434, solv-int/9910011.
  35. Maltsev A.Ya., Whitham systems and deformations, J. Math. Phys. 47 (2006), 073505, 18 pages, nlin.SI/0509033.
  36. Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly non-local Poisson brackets, Phys. D 156 (2001), 53-80, nlin.SI/0006030.
  37. Maltsev A.Ya., Pavlov M.V., On Whitham's averaging method, Funct. Anal. Appl. 29 (1995), 6-19, nlin.SI/0306053.
  38. Mokhov O.I., Ferapontov E.V., Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature, Russian Math. Surveys 45 (1990), 218-219.
  39. Newell A.C., Solitons in mathematics and physics, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 48, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985.
  40. Novikov S.P., Geometry of conservative systems of hydrodynamic type. The averaging method for field-theoretic systems, Russian Math. Surveys 40 (1985), no. 4, 85-98.
  41. Novikov S.P., The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl. 8 (1974), 236-246.
  42. 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.
  43. Pavlov M.V., Elliptic coordinates and multi-Hamiltonian structures of hydrodynamic-type systems, Russian Acad. Sci. Dokl. Math. 50 (1995), 374-377.
  44. Pavlov M.V., Multi-Hamiltonian structures of the Whitham equations, Russian Acad. Sci. Dokl. Math. 50 (1995), 220-223.
  45. Schmidt W.M., Diophantine approximation, Lecture Notes in Mathematics, Vol. 785, Springer-Verlag, Berlin - Heidelberg - New York, 1980.
  46. Tsarev S.P., On Poisson bracket and one-dimensional systems of hydrodynamic type, Soviet Math. Dokl. 31 (1985), 488-491.
  47. 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 finite-zone potential, Funct. Anal. Appl. 22 (1988), 137-139.
  48. Whitham G.B., A general approach to linear and non-linear dispersive waves using a Lagrangian, J. Fluid Mech. 22 (1965), 273-283.
  49. Whitham G.B., Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience, New York, 1974.
  50. Whitham G.B., Non-linear dispersive waves, Proc. R. Soc. Lond. Ser. A 283 (1965), 238-261.

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