The Lorentz-Invariant Deformation of the Whitham System for the Nonlinear Klein--Gordon Equation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 28-43.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a deformation of the Whitham system for the nonlinear Klein–Gordon equation. This deformation has a Lorentz-invariant form. Using the Lagrangian formalism of the original system, we obtain the first nontrivial correction to the Whitham system in the Lorentz-invariant approach.
Keywords: asymptotic method, slow modulation.
@article{FAA_2008_42_2_a4,
     author = {A. Ya. Maltsev},
     title = {The {Lorentz-Invariant} {Deformation} of the {Whitham} {System} for the {Nonlinear} {Klein--Gordon} {Equation}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {28--43},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a4/}
}
TY  - JOUR
AU  - A. Ya. Maltsev
TI  - The Lorentz-Invariant Deformation of the Whitham System for the Nonlinear Klein--Gordon Equation
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2008
SP  - 28
EP  - 43
VL  - 42
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a4/
LA  - ru
ID  - FAA_2008_42_2_a4
ER  - 
%0 Journal Article
%A A. Ya. Maltsev
%T The Lorentz-Invariant Deformation of the Whitham System for the Nonlinear Klein--Gordon Equation
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2008
%P 28-43
%V 42
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a4/
%G ru
%F FAA_2008_42_2_a4
A. Ya. Maltsev. The Lorentz-Invariant Deformation of the Whitham System for the Nonlinear Klein--Gordon Equation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 28-43. http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a4/

[1] G. Whitham, “A general approach to linear and non-linear dispersive waves using a Lagrangian”, J. Fluid Mech., 22 (1965), 273–283 | DOI | MR

[2] G. Whitham, “Non-linear dispersive waves”, Proc. Roy. Soc. London Ser. A, 283 (1965), 238–261 | DOI | MR | Zbl

[3] Dzh. Uizem, Lineinye i nelineinye volny, Mir, M., 1977

[4] J. C. Luke, “A perturbation method for nonlinear dispersive wave problems”, Proc. Roy. Soc. London Ser. A, 292:1430 (1966), 403–412 | DOI | MR | Zbl

[5] M. J. Ablowitz, D. J. Benney, “The evolution of multi-phase modes for nonlinear dispersive waves”, Stud. Appl. Math., 49 (1970), 225–238 | DOI | MR | Zbl

[6] M. J. Ablowitz, “Applications of slowly varying nonlinear dispersive wave theories”, Stud. Appl. Math., 50 (1971), 329–344 | DOI | Zbl

[7] M. J. Ablowitz, “Approximate methods for obtaining multi-phase modes in nonlinear dispersive wave problems”, Stud. Appl. Math., 51 (1972), 17–55 | DOI | MR | Zbl

[8] W. D. Hayes, “Group velocity and non-linear dispersive wave propagation”, Proc. Roy. Soc. London Ser. A, 332 (1973), 199–221 | DOI | MR | Zbl

[9] A. V. Gurevich, L. P. Pitaevskii, “Raspad nachalnogo razryva v uravnenii Kortevega–de Friza”, Pisma v ZhETF, 17:5 (1973), 268–271

[10] A. V. Gurevich, L. P. Pitaevskii, “Nestatsionarnaya struktura besstolknovitelnoi udarnoi volny”, ZhETF, 65:8 (1973), 590–604

[11] H. Flaschka, M. G. Forest, D. W. McLaughlin, “Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation”, Comm. Pure Appl. Math., 33:6 (1980), 739–784 | DOI | MR | Zbl

[12] S. Yu. Dobrokhotov, V. P. Maslov, “Konechnozonnye pochti periodicheskie resheniya v VKB-priblizheniyakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 15, VINITI, M., 1980, 3–94 | MR

[13] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR

[14] B. A. Dubrovin, S. P. Novikov, “Gamiltonov formalizm odnomernykh sistem gidrodinamicheskogo tipa i metod usredneniya Bogolyubova–Uizema”, Dokl. AN SSSR, 270:4 (1983), 781–785 | MR | Zbl

[15] P. D. Lax, C. D. Levermore, “The small dispersion limit for the Korteweg–de Vries equation I, II, III”, Comm. Pure Appl. Math., 36:3 (1983), 253–290 | DOI | MR | Zbl

[16] S. P. Novikov, “Geometriya konservativnykh sistem gidrodinamicheskogo tipa. Metod usredneniya dlya teoretiko-polevykh sistem”, UMN, 40:4 (1985), 79–89 | MR | Zbl

[17] V. V. Avilov, S. P. Novikov, “Evolyutsiya uitemovskoi zony v teorii KdF”, Dokl. AN SSSR, 294:2 (1987), 325–329 | MR

[18] A. V. Gurevich, L. P. Pitaevskii, “Usrednennoe opisanie voln v uravnenii Kortevega–de Friza–Byurgersa”, ZhETF, 93:3 (1987), 871–880 | MR

[19] V. V. Avilov, I. M. Krichever, S. P. Novikov, “Evolyutsiya uitemovskoi zony v teorii Kortevega–de Frisa”, Dokl. AN SSSR, 295:2 (1987), 345–349 | MR | Zbl

[20] I. M. Krichever, “Metod usredneniya dlya dvumernykh «integriruemykh» uravnenii”, Funkts. analiz i ego pril., 22:3 (1988), 37–52 | MR | Zbl

[21] R. Haberman, “The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg-de Vries type”, Stud. Appl. Math., 78:1 (1988), 73–90 | DOI | MR | Zbl

[22] B. A. Dubrovin, S. P. Novikov, “Gidrodinamika slabo deformirovannykh solitonnykh reshetok. Differentsialnaya geometriya i gamiltonova teoriya”, UMN, 44:6 (1989), 29–98 | MR | Zbl

[23] B. A. Dubrovin, S. P. Novikov, “Hydrodynamics of soliton lattices”, Sov. Sci. Rev., sect. C, Math. Phys., 9:4 (1993), 1–136 | MR | Zbl

[24] S. P. Tsarev, “O skobkakh Puassona i odnomernykh gamiltonovykh sistemakh gidrodinamicheskogo tipa”, Dokl. AN SSSR, 282:3 (1985), 534–537 | MR | Zbl

[25] O. I. Mokhov, E. V. Ferapontov, “Nelokalnye gamiltonovy operatory gidrodinamicheskogo tipa, svyazannye s metrikami postoyannoi krivizny”, UMN, 45:3 (1990), 191–192 | MR | Zbl

[26] E. V. Ferapontov, “Differentsialnaya geometriya nelokalnykh gamiltonovykh operatorov gidrodinamicheskogo tipa”, Funkts. analiz i ego pril., 25:3 (1991), 37–49 | MR | Zbl

[27] E. V. Ferapontov, “Ogranichenie po Diraku gamiltonova operatora $\delta^{IJ}\frac{d}{dx}$ na poverkhnost evklidova prostranstva s ploskoi normalnoi svyaznostyu”, Funkts. analiz i ego pril., 26:4 (1992), 83–86 | MR | Zbl

[28] E. V. Ferapontov, “Nelokalnye matrichnye gamiltonovy operatory. Differentsialnaya geometriya i prilozheniya”, Teoret. matem. fiz., 91:3 (1992), 452–462 | MR | Zbl

[29] E. V. Ferapontov, “Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications”, Amer. Math. Soc. Transl. (2), 170, Amer. Math. Soc., Providence, RI, 1995, 33–58 | MR | Zbl

[30] M. V. Pavlov, “Ellipticheskie koordinaty i multigamiltonovy struktury sistem gidrodinamicheskogo tipa”, Dokl. RAN, ser. matem., 339:1 (1994), 21–23 | MR | Zbl

[31] A. Ya. Maltsev, S. P. Novikov, “On the local systems hamiltonian in the weakly nonlocal Poisson brackets”, Phys. D, 156:1–2 (2001), 53–80 | DOI | MR | Zbl

[32] A. Ya. Maltsev, “The averaging of non-local Hamiltonian structures in Whitham's method”, Intern. J. Math. Math. Sci., 30:7 (2002), 399–434 | DOI | MR | Zbl

[33] B. A. Dubrovin, “Integrable systems in topological field theory”, Nucl. Phys. B, 379:3 (1992), 627–689 | DOI | MR

[34] B. A. Dubrovin, Integrable Systems and Classification of 2-dimensional Topological Field Theories, http://arxiv.org/abs/hep-th/9209040 | MR

[35] B. A. Dubrovin, Geometry of 2d Topological Field Theories, http://arxiv.org/abs/hep-th/9407018

[36] B. A. Dubrovin, “Flat pencils of metrics and Frobenius manifolds”, Proc. of 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”, World Sci. Publ., River Edge, NJ, 1998, 47–72 ; http://arxiv.org/abs/math.DG/9803106 | MR | Zbl

[37] B. A. Dubrovin, Y. Zhang, “Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation”, Comm. Math. Phys., 198 (1998), 311–361 | DOI | MR | Zbl

[38] B. A. Dubrovin, Geometry and Analytic Theory of Frobenius Manifolds, http://arxiv.org/abs/math.AG/9807034 | MR

[39] B. A. Dubrovin, Y. Zhang, Normal Forms of Hierarchies of Integrable PDEs, Frobenius Manifolds and Gromov–Witten Invariants, http://arxiv.org/abs/math.DG/0108160

[40] P. Lorenzoni, “Deformations of bihamiltonian structures of hydrodynamic type”, J. Geom. Phys., 44:2–3 (2002), 331–371 | DOI | MR

[41] B. A. Dubrovin, Y. Zhang, Virasoro Symmetries of the Extended Toda Hierarchy, http://arxiv.org/abs/math.DG/0308152 | MR

[42] S.-Q. Liu, Y. Zhang, Deformations of Semisimple Bihamiltonian Structures of Hydrodynamic Type, http://arxiv.org/abs/math.DG/0405146 | MR

[43] S.-Q. Liu, Y. Zhang, On the Quasitriviality of Deformations of Bihamiltonian Structures of Hydrodynamic Type, http://arxiv.org/abs/math.DG/0406626 | MR

[44] B. Dubrovin, S.-Q. Liu, Y. Zhang, On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, http://arxiv.org/abs/math.DG/0410027

[45] B. Dubrovin, Y. Zhang, D. Zuo, Extended Affine Weyl Groups and Frobenius Manifolds–II, http://arxiv.org/abs/math.DG/0502365

[46] A. Ya. Maltsev, “Whitham systems and deformations”, J. Math. Phys., 47:7 (2006), 073505, 18 pp. | DOI | MR | Zbl

[47] A. Ya. Maltsev, The Deformations of Whitham Systems and Lagrangian Formalism, http://arxiv.org/abs/nlin.SI/0601050 | MR