×

zbMATH — the first resource for mathematics

On the canonical forms of the multi-dimensional averaged Poisson brackets. (English) Zbl 1381.35154
Summary: We consider here special Poisson brackets given by the “averaging” of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the “physical meaning” of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of “Hydrodynamic Type” in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a “pseudo-canonical” form under some special (“physical”) requirements on the initial bracket.{
©2016 American Institute of Physics}
MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ablowitz, M. J.; Benney, D. J., The evolution of multi-phase modes for nonlinear dispersive waves, Stud. Appl. Math., 49, 225-238, (1970) · Zbl 0203.41001
[2] Alekseev, V. L., On non-local Hamiltonian operators of hydrodynamic type connected with Whitham’s equations, Russ. Math. Surv., 50, 6, 1253-1255, (1995) · Zbl 0857.35108
[3] Grinberg, N. I., On Poisson brackets of hydrodynamic type with a degenerate metric, Russ. Math. Surv., 40, 4, 231-232, (1985) · Zbl 0611.58017
[4] Dobrokhotov, S. Yu.; Maslov, V. P., Finite-zone, almost periodic solutions in the WKB approximations, J. Sov. Math., 16, 1433-1487, (1980) · Zbl 0461.35010
[5] Dobrokhotov, S. Yu.; Maslov, V. P., Multi-phase asymptotics of nonlinear partial differential equations with a small parameter, Sov. Sci. Rev.-Math. Phys. Rev., 3, 221-311, (1982) · Zbl 0551.35072
[6] Dobrokhotov, S. Yu., Resonances in asymptotic solutions of the Cauchy problem for the Schrodinger equation with rapidly oscillating finite-zone potential, Math. Notes, 44, 3, 656-668, (1988) · Zbl 0667.35058
[7] Dobrokhotov, S. Yu., Resonance correction to the adiabatically perturbed finite-zone almost periodic solution of the Korteweg-de Vries equation, Math. Notes, 44, 4, 551-555, (1988) · Zbl 0667.35058
[8] Dobrokhotov, S. Yu.; Krichever, I. M., Multi-phase solutions of the Benjamin-Ono equation and their averaging, Math. Notes, 49, 6, 583-594, (1991) · Zbl 0752.35058
[9] Dubrovin, B. A.; Novikov, S. P., Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method, Sov. Math. Dokl., 27, 3, 665-669, (1983) · Zbl 0553.35011
[10] Dubrovin, B. A.; Novikov, S. P., On Poisson brackets of hydrodynamic type, Sov. Math. Dokl., 30, 651-654, (1984) · Zbl 0591.58012
[11] Dubrovin, B. A.; Novikov, S. P., Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russ. Math. Surv., 44, 6, 35-124, (1989) · Zbl 0712.58032
[12] Dubrovin, B. A.; Novikov, S. P., Hydrodynamics of soliton lattices, Sov. Sci. Rev. C. Math. Phys., 9, 4, 1-136, (1993) · Zbl 0845.58027
[13] Dubrovin, B. A., Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models, Commun. Math. Phys., 145, 1, 195-207, (1992) · Zbl 0753.58039
[14] Ferapontov, E. V., Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funct. Anal. Appl., 25, 3, 195-204, (1991) · Zbl 0742.58018
[15] Ferapontov, E. V., Dirac reduction of the Hamiltonian operator \(\delta^{i j} \frac{d}{d x}\) to a submanifold of the Euclidean space with flat normal connection, Funct. Anal. Appl., 26, 4, 298-300, (1992) · Zbl 0802.58023
[16] Ferapontov, E. V.; Odesskii, A. V.; Stoilov, N. M., Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions, J. Math. Phys., 52, 7, 073505, (2011) · Zbl 1317.37071
[17] Ferapontov, E. V.; Lorenzoni, P.; Savoldi, A., Hamiltonian operators of Dubrovin-Novikov type in 2D, Lett. Math. Phys., 105, 3, 341-377, (2015) · Zbl 1310.37027
[18] Flaschka, H.; Forest, M. G.; McLaughlin, D. W., Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Commun. Pure Appl. Math., 33, 6, 739-784, (1980) · Zbl 0454.35080
[19] Haberman, R., The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg-deVries type, Stud. Appl. Math., 78, 1, 73-90, (1988) · Zbl 0647.35079
[20] Haberman, R., Standard form and a method of averaging for strongly nonlinear oscillatory dispersive traveling waves, SIAM J. Appl. Math., 51, 6, 1489-1798, (1991) · Zbl 0752.35067
[21] Hayes, W. D., Group velocity and non-linear dispersive wave propagation, Proc. R. Soc. A, 332, 199-221, (1973) · Zbl 0271.76006
[22] Krichever, I. M., The averaging method for two-dimensional integrable equations, Funct. Anal. Appl., 22, 3, 200-213, (1988) · Zbl 0688.35088
[23] Krichever, I. M., The τ-function of the universal Whitham hierarchy, matrix models and topological field theories, Commun. Pure Appl. Math., 47, 4, 437-475, (1994) · Zbl 0811.58064
[24] Krichever, I. M.; Phong, D. H., On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Differ. Geom., 45, 349-389, (1997) · Zbl 0889.58044
[25] Luke, J. C., A perturbation method for nonlinear dispersive wave problems, Proc. R. Soc. A, 292, 1430, 403-412, (1966) · Zbl 0143.13603
[26] Maltsev, A. Ya., Whitham’s method and Dubrovin-Novikov bracket in single-phase and multiphase cases, SIGMA, 8, 103, (2012) · Zbl 1384.37081
[27] Maltsev, A. Ya., The multi-dimensional Hamiltonian structures in the Whitham method, J. Math. Phys., 54, 5, 053507, (2013) · Zbl 1306.37080
[28] Maltsev, A. Ya., On the minimal set of conservation laws and the Hamiltonian structure of the Whitham equations, J. Math. Phys., 56, 2, 023510, (2015) · Zbl 1308.76039
[29] Mokhov, O. I.; Ferapontov, E. V., Nonlocal Hamiltonian operators of hydrodynamic type related to metrics of constant curvature, Russ. Math. Surv., 45, 3, 218-219, (1990) · Zbl 0712.35080
[30] Mokhov, O. I., Poisson brackets of Dubrovin-Novikov type (DN-brackets), Funct. Anal. Appl., 22, 4, 336-338, (1988) · Zbl 0671.58006
[31] Mokhov, O. I., The classification of nonsingular multidimensional Dubrovin-Novikov brackets, Funct. Anal. Appl., 42, 1, 33-44, (2008) · Zbl 1180.37088
[32] Newell, A. C., Solitons in Mathematics and Physics, (1985), Society for Industrial and Applied Mathematics · Zbl 0565.35003
[33] Novikov, S. P., The Hamiltonian formalism and a many-valued analogue of Morse theory, Russ. Math. Surv., 37, 5, 1-56, (1982) · Zbl 0571.58011
[34] Novikov, S. P.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of solitons, The Inverse Scattering Method, (1984), Plemun: Plemun, New York · Zbl 0598.35002
[35] Novikov, S. P., The geometry of conservative systems of hydrodynamic type. The method of averaging for field-theoretical systems, Russ. Math. Surv., 40, 4, 85-98, (1985) · Zbl 0654.76004
[36] Olver, P. J., Applications of Lie Groups to Differential Equations, 107, (1986), Springer-Verlag: Springer-Verlag, Berlin, Heidelberg, New York, Tokyo · Zbl 0588.22001
[37] Tsarev, S. P., On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Sov. Math. Dokl., 31, 3, 488-491, (1985) · Zbl 0605.35075
[38] Tsarev, S. P., The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR-Izv., 37, 2, 397, (1991) · Zbl 0796.76014
[39] Whitham, G., A general approach to linear and non-linear dispersive waves using a Lagrangian, J. Fluid Mech., 22, 273-283, (1965)
[40] Whitham, G., Non-linear dispersive waves, Proc. R. Soc. A, 283, 1393, 238-261, (1965) · Zbl 0125.44202
[41] Whitham, G., Linear and Nonlinear Waves, (1974), Wiley: Wiley, New York · Zbl 0373.76001
[42] The definition of f_{1} in Refs. 26 and 27 differs by a phase shift from that used here which is included there also in the corresponding orthogonality conditions.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.