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The Lorentz-invariant deformation of the Whitham system for the nonlinear Klein-Gordon equation

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Abstract

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.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Whitham, Linear and Nonlinear Waves, Wiley, New York-London-Sydney, 1974.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  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,” 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.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

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

  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.

  46. A. Ya. Maltsev, “Whitham systems and deformations,” J. Math. Phys., 47:7 (2006).

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

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  1. L. D. Landau Institute for Theoretical Physics, Russia

    A. Ya. Maltsev

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  1. A. Ya. Maltsev
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Correspondence to A. Ya. Maltsev.

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 2, pp. 28–43, 2008

Original Russian Text Copyright © by A. Ya. Maltsev

The work was partially supported by the grant of President of Russian Federation (MD-8906.2006.2) and Russian Science Support Foundation.

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Maltsev, A.Y. The Lorentz-invariant deformation of the Whitham system for the nonlinear Klein-Gordon equation. Funct Anal Its Appl 42, 103–115 (2008). https://doi.org/10.1007/s10688-008-0016-4

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  • DOI: https://doi.org/10.1007/s10688-008-0016-4

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