We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit a bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations are quite different: indeed the KdV averaged bi- Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.
Nous construisons les équations modulées (équations de Whitham) pour l\rq équation de Camassa-Holm (CH). Nous démontrons que ces équations modulées sont hyperboliques et bi- hamiltoniennes. En particulier, il existe une transformation réciproque telle qu'aux équations modulées du premier flot négatif de l\rq équation de Korteweg-de Vries (KdV) correspondent aux équations modulées de CH. Cette transformation réciproque est engendrée par le Casimir du deuxième crochet de Poisson associé au flot moyenné de KdV. Nous démontrons que la géométrie des structures bi-hamiltoniennes des équations modulées de KdV et CH sont très différentes : en effet, la structure de Poisson moyennée de KdV est liée à une variété semi-simple de Frobenius et non celle de CH.
Keywords: Camassa-Holm equation, Korteweg de Vries hierarchy, modulation equations, Whitham equations, reciprocal transformations, Hamiltonian structures
Mot clés : équation de Camassa-Holm, équation de Korteweg de Vries, équations de modulation, équation de Whitham, transformations réciproques, structures Hamiltoniennes
@article{AIF_2005__55_6_1803_0, author = {Abenda, Simonetta and Grava, Tamara}, title = {Modulation of the {Camassa-Holm} equation and reciprocal transformations}, journal = {Annales de l'Institut Fourier}, pages = {1803--1834}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {6}, year = {2005}, doi = {10.5802/aif.2142}, mrnumber = {2187936}, zbl = {02230058}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2142/} }
TY - JOUR AU - Abenda, Simonetta AU - Grava, Tamara TI - Modulation of the Camassa-Holm equation and reciprocal transformations JO - Annales de l'Institut Fourier PY - 2005 SP - 1803 EP - 1834 VL - 55 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2142/ DO - 10.5802/aif.2142 LA - en ID - AIF_2005__55_6_1803_0 ER -
%0 Journal Article %A Abenda, Simonetta %A Grava, Tamara %T Modulation of the Camassa-Holm equation and reciprocal transformations %J Annales de l'Institut Fourier %D 2005 %P 1803-1834 %V 55 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2142/ %R 10.5802/aif.2142 %G en %F AIF_2005__55_6_1803_0
Abenda, Simonetta; Grava, Tamara. Modulation of the Camassa-Holm equation and reciprocal transformations. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1803-1834. doi : 10.5802/aif.2142. http://archive.numdam.org/articles/10.5802/aif.2142/
[1] On the weak Kowalevski-Painlevé property for hyperelliptically separable systems, Acta Appl. Math., Volume 60 (2000) no. 2, pp. 137-178 | DOI | MR | Zbl
[2] The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math., Volume 53 (1974), pp. 249-315 | MR | Zbl
[3] Completely integrable systems, Euclidean Lie algebras and curves, Adv. in Math., Volume 38 (1980), pp. 318-379 | MR | Zbl
[4] Wave solutions of evolution equations and Hamiltonian flow on nonlinear subvarieties of generalized Jacobians, J. Phys. A, Volume 33 (2000), pp. 8409-8425 | DOI | MR | Zbl
[5] The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type, Comm. Math. Phys., Volume 221 (2001) no. 1, pp. 197-227 | DOI | MR | Zbl
[6] Multipeakons and the classical moment problem, Adv. Math., Volume 154 (2000) no. 2, pp. 229-257 | DOI | MR | Zbl
[7] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993), pp. 1661-1664 | DOI | MR | Zbl
[8] Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa-Holm equation, Bull. Sci. Math., Volume 122 (1998) no. 7, pp. 487-494 | DOI | MR | Zbl
[9] On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 457 (2001) no. 2008, pp. 953-970 | DOI | MR | Zbl
[10] A shallow water equation on the circle, Comm. Pure Appl. Math., Volume 52 (1999), pp. 949-982 | DOI | MR | Zbl
[11] Multiphase asymptotics of non-linear partial differential equations with a small parameter, Soviet Sci. Rev. Math. Phys. Rev., Volume 3 (1982), pp. 221-311 | MR | Zbl
[12] Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys, Volume 44 (1989), pp. 35-124 | DOI | MR | Zbl
[13] Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory. Surveys in differential geometry: integral [integrable] systems, Surv. Differ. Geom., IV (1998), pp. 213-238 | Zbl
[14] Geometry of D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) (Lecture Notes in Math.), Volume 1620 (1996), pp. 120-348 | Zbl
[15] Korteweg-de Vries and other asymptotically equivalent equations for shallow water waves. In memoriam Prof. Philip Gerald Drazin (1934-2002), Fluid Dynam. Res., Volume 33 no. 1-2, pp. 73-95 | MR | Zbl
[16] Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Russian Math. Surveys, Volume 45 (1990) no. 3, pp. 218-219 | DOI | MR | Zbl
[17] Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications (Amer. Math. Soc. Transl. Ser. 2), Volume 170 (1995), pp. 33-58 | Zbl
[18] Reciprocal tranformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for nonlinearly degenerate systems, J. Math. Phys., Volume 44 (2003), pp. 1150-1172 | DOI | MR | Zbl
[19] Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equations, Comm. Pure Appl. Math., Volume 33 (1980), pp. 739-784 | DOI | MR | Zbl
[20] Bäcklund transformations for hereditary symmetries, Nonlinear Anal., Volume 5 (1981), pp. 423-432 | DOI | MR | Zbl
[21] Some tricks from the symmetry-toolbox for nonlinear-equations: generalizations of the Camassa-Holm equation, Physica D, Volume 95 (1996), pp. 229-243 | DOI | MR | Zbl
[22] Real-Valued Algebro-Geometric Solutions of the Camassa-Holm hierarchy, 24 pp. (Preprint, http://xxx.lanl.gov/nlin.SI/0208021) | MR
[23] Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann., Volume 329 (2004) no. 2, pp. 335-364 | DOI | MR | Zbl
[24] The averaging method for two-dimensional integrable equations, Funct. Anal. Appl., Volume 22 (1988) no. 3, pp. 200-213 | MR | Zbl
[25] Group velocity and nonlinear dispersive wave propagation, Proc. Royal Soc. London Ser. A, Volume 332 (1973), pp. 199-221 | DOI | MR | Zbl
[26] The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math., Volume 36 (1983) no. 6, pp. 809-829 | DOI | MR | Zbl
[27] The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math., Volume 56 (2003) no. 7, pp. 998-1015 | MR | Zbl
[28] On the local systems Hamiltonian in the weakly non-local Poisson brackets, Phys. D, Volume 156 (2001), pp. 53-80 | DOI | MR | Zbl
[29] Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type, 64 pp. (Preprint, http://xxx.lanl.gov/nlin.SI/0405060) | MR | Zbl
[30]
(private communication.)[31] On Whitham's averaging method, Funct. Anal. Appl., Volume 29 (1995) no. 1, pp. 6-19 | DOI | MR | Zbl
[32] Tri-Hamiltonian structures of Egorov systems of hydrodynamic type (Russian), Funktsional. Anal. i Prilozhen., Volume 37 (2003) no. 1, pp. 32-45 | DOI | MR | Zbl
[33] Oscillations of the zero dispersion limit of the Korteweg-de Vries equation, Comm. Pure Appl. Math., Volume 46 (1993), pp. 1093-1129 | DOI | MR | Zbl
[34] The initial value problem for the Whitham averaged system, Comm. Math. Phys., Volume 166 (1994), pp. 79-115 | DOI | MR | Zbl
[35] Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Dokl. Akad. Nauk. SSSR, Volume 282 (1985), pp. 534-537 | MR | Zbl
[36] Integrable systems and symmetric product of curves, Math. Z., Volume 227 (1998) no. 1, pp. 93-127 | DOI | MR | Zbl
[37] A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid. Mech., Volume 22 (1965), pp. 273-283 | DOI | MR
Cited by Sources: