Maltsev, A. Ya. The Lorentz-invariant deformation of the Whitham system for the nonlinear Klein-Gordon equation. (English) Zbl 1171.37029 Funct. Anal. Appl. 42, No. 2, 103-115 (2008); translation from Funkts. Anal. Prilozh. 42, No. 2, 28-43 (2008). A Lorentz invariant deformation of the Whitham system associated to the nonlinear Klein-Gordon equation is described. Reviewer: Tomasz Brzeziński (Swansea) Cited in 1 Document MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:Klein-Gordon equation; Whitham system; slow modulation PDFBibTeX XMLCite \textit{A. Ya. Maltsev}, Funct. Anal. Appl. 42, No. 2, 103--115 (2008; Zbl 1171.37029); translation from Funkts. Anal. Prilozh. 42, No. 2, 28--43 (2008) Full Text: DOI arXiv References: [1] G. Whitham, ”A general approach to linear and non-linear dispersive waves using a Lagrangian,” J. Fluid Mech., 22 (1965), 273–283. · Zbl 0125.44202 [2] G. Whitham, ”Non-linear dispersive waves,” Proc. Roy. Soc. London Ser. A, 283 (1965), 238–261. · Zbl 0125.44202 [3] G. Whitham, Linear and Nonlinear Waves, Wiley, New York-London-Sydney, 1974. · Zbl 0373.76001 [4] J. C. Luke, ”A perturbation method for nonlinear dispersive wave problems,” Proc. Roy. Soc. London Ser. A, 292:1430 (1966), 403–412. · Zbl 0143.13603 [5] M. J. Ablowitz and D. J. Benney, ”The evolution of multi-phase modes for nonlinear dispersive waves,” Stud. Appl. Math., 49 (1970), 225–238. · Zbl 0203.41001 [6] M. J. Ablowitz, ”Applications of slowly varying nonlinear dispersive wave theories,” Stud. Appl. Math., 50 (1971), 329–344. · Zbl 0229.76016 [7] M. J. Ablowitz, ”Approximate methods for obtaining multi-phase modes in nonlinear dispersive wave problems,” Stud. Appl. Math., 51 (1972), 17–55. · Zbl 0261.76010 [8] W. D. Hayes, ”Group velocity and non-linear dispersive wave propagation,” Proc. Roy. Soc. London Ser. A, 332 (1973), 199–221. · Zbl 0271.76006 [9] A. V. Gurevich and L. P. Pitaevskii, ”Decay of initial discontinuity in the Korteweg-de Vries equation,” Pis’ma v ZhETF, 17:5 (1973), 268–271; English transl.: JETP Lett., 17 (1973), 193–195. [10] A. V. Gurevich and L. P. Pitaevskii, ”Nonstationary structure of collisionless shock waves,” ZhETF, 65:8 (1973), 590–604; English transl.: Sov. Phys. JETP, 38 (1974), 291–297. [11] H. Flaschka, M. G. Forest, and 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. · Zbl 0454.35080 [12] S. Yu. Dobrokhotov and V. P. Maslov, ”Finite-gap almost periodic solutions in the WKB approximation,” in: Itogi Nauki i Tekhniki. Current Problems in Mathematics [in Russian], vol. 15, VINITI, Moscow, 1980, 3–94; English transl.: J. Soviet Math., 15 (1980), 1433–1487. [13] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. the Inverse Scattering Method, Plenum, New York, 1984. · Zbl 0598.35002 [14] B. A. Dubrovin and S. P. Novikov, ”Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR, 270:4 (1983), 781–785; English transl.: Soviet Math. Dokl., 27:3 (1983), 665–669. · Zbl 0553.35011 [15] P. D. Lax and 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, 36:5 (1983), 571–593, 36:6 (1983), 809–830. · Zbl 0532.35067 [16] S. P. Novikov, ”The geometry of conservative systems of hydrodynamic type. The method of averaging for field-theoretical systems,” Uspekhi Mat. Nauk, 40:4 (1985), 79–89; English transl.: Russian Math. Surveys, 40:4 (1985), 85–98. · Zbl 0654.76004 [17] V. V. Avilov and S. P. Novikov, ”Evolution of the Whitham zone in KdV theory,” Dokl. Akad. Nauk SSSR, 294:2 (1987), 325–329; English transl.: Soviet Phys. Dokl., 32 (1987), 366–368. · Zbl 0655.65132 [18] A. V. Gurevich and L. P. Pitaevskii, ”Averaged description of waves in the Korteweg-de Vries-Burgers equation,” ZhETF, 93:3 (1987), 871–880; English transl.: Soviet Phys. JETP, 66 (1987), 490–495. [19] V. V. Avilov, I. M. Krichever, and S. P. Novikov, ”Evolution of the Whitham zone in the Korteweg-de Vries theory,” Dokl. Akad. Nauk SSSR, 295:2 (1987), 345–349; English transl.: Soviet Phys. Dokl., 32 (1987), 564–566. · Zbl 0655.65132 [20] I. M. Krichever, ”Method of averaging for two-dimensional ’integrable’ equations,” Funkts. Anal. Prilozhen., 22:3 (1988), 37–52; English transl.: Functional Anal. Appl., 22 (1988), 200–213. · Zbl 0688.35088 [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. · Zbl 0647.35079 [22] B. A. Dubrovin and S. P. Novikov, ”Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,” Uspekhi Mat. Nauk, 44:6 (1989), 29–98; English transl.: Russian Math. Surveys, 44:6 (1989), 35–124. · Zbl 0712.58032 [23] B. A. Dubrovin and S. P. Novikov, ”Hydrodynamics of soliton lattices,” Sov. Sci. Rev., sect. C, Math. Phys., 9:4 (1993), 1–136. · Zbl 0845.58027 [24] S. P. Tsarev, ”On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 282:3 (1985), 534–537; English transl.: Soviet Math. Dokl., 31:3 (1985), 488–491. · Zbl 0605.35075 [25] O. I. Mokhov and E. V. Ferapontov, ”Nonlocal Hamiltonian operators of hydrodynamic type related to metrics of constant curvature,” Uspekhi Matem. Nauk, 45:3 (1990), 191–192; English transl.: Russian Math. Surveys, 45:3 (1990), 218–219. · Zbl 0712.35080 [26] E. V. Ferapontov, ”Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funkts. Anal. Prilozhen., 25:3 (1991), 37–49; English transl.: Functional Anal. Appl., 25:3 (1991), 195–204. · Zbl 0742.58018 [27] E. V. Ferapontov, ”Dirac reduction of the Hamiltonian operator \(\delta ^{IJ} \frac{d}{{dx}}\) to a submanifold of the Euclidean space with flat normal connection,” Funkts. Anal. Prilozhen., 26:4 (1992), 83–86; English transl.: Functional Anal. Appl., 26:4 (1992), 298–300. · Zbl 0802.58023 [28] E. V. Ferapontov, ”Nonlocal matrix Hamiltonian operators, differential geometry, and applications,” Teor. Mat. Fiz., 91:3 (1992), 452–462; English transl.: Theor. Math. Phys., 91:3 (1992), 642–649. · Zbl 0760.58013 [29] E. V. Ferapontov, ”Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications,” in: Amer. Math. Soc. Transl. (2), vol. 170, Amer. Math. Soc., Providence, RI, 1995, 33–58. · Zbl 0845.58029 [30] M. V. Pavlov, ”Elliptic coordinates and multi-Hamiltonian structures of systems of hydrodynamic type,” Dokl. Ross. Akad. Nauk, Ser. Mat., 339:1 (1994), 21–23; English transl.: Russian Acad. Sci. Dokl. (Math.), 50:3 (1995), 374–377. · Zbl 0872.58025 [31] A. Ya. Maltsev and S. P. Novikov, ”On the local systems Hamiltonian in the weakly nonlocal Poisson brackets,” Phys. D, 156:1–2 (2001), 53–80. · Zbl 0991.37041 [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. · Zbl 1046.70015 [33] B. A. Dubrovin, ”Integrable systems in topological field theory,” Nucl. Phys. B, 379:3 (1992), 627–689. [34] B. A. Dubrovin, Integrable Systems and Classification of 2-dimensional Topological Field Theories, http://arxiv.org/abs/hep-th/9209040 . · Zbl 0824.58029 [35] B. A. Dubrovin, Geometry of 2d Topological Field Theories, http://arxiv.org/abs/hep-th/9407018 . · Zbl 0841.58065 [36] B. A. Dubrovin, ”Flat pencils of metrics and Frobenius manifolds,” in: 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 . · Zbl 0963.53054 [37] B. A. Dubrovin and Y. Zhang, ”Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation,” Comm. Math. Phys., 198 (1998), 311–361. · Zbl 0923.58060 [38] B. A. Dubrovin, Geometry and Analytic Theory of Frobenius Manifolds, http://arxiv.org/abs/math.AG/9807034 . · Zbl 0916.32018 [39] B. A. Dubrovin and 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. · Zbl 1010.37041 [41] B. A. Dubrovin and Y. Zhang, Virasoro Symmetries of the Extended Toda Hierarchy, http://arxiv.org/abs/math.DG/0308152 . · Zbl 1071.37054 [42] S.-Q. Liu and Y. Zhang, Deformations of Semisimple Bihamiltonian Structures of Hydrodynamic Type, http://arxiv.org/abs/math.DG/0405146 . · Zbl 1079.37058 [43] S.-Q. Liu and Y. Zhang, On the Quasitriviality of Deformations of Bihamiltonian Structures of Hydrodynamic Type, http://arxiv.org/abs/math.DG/0406626 . [44] B. Dubrovin, S.-Q. Liu, and Y. Zhang, On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, http://arxiv.org/abs/math.DG/0410027 . [45] B. Dubrovin, Y. Zhang, and D. Zuo, Extended Affine Weyl Groups and Frobenius Manifolds-II, http://arxiv.org/abs/math.DG/0502365 . · Zbl 0964.32020 [46] A. Ya. Maltsev, ”Whitham systems and deformations,” J. Math. Phys., 47:7 (2006). · Zbl 1112.37052 [47] A. Ya. Maltsev, The Deformations of Whitham Systems and Lagrangian Formalism, http://arxiv.org/abs/nlin.SI/0601050 . · Zbl 1112.37052 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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