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