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On the Integrability of (2+1)-Dimensional Quasilinear Systems

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Abstract

A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of ‘sufficiently many’ n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Moscow: Nauka, 1979

  2. Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Classification of integrable equations on quad-graphs. The consistency approach. nlin.SI/0202024

  3. Blaszak, M., Szablikowski, B.M.: Classical R-matrix theory of dispersionless systems: II. (2+1)-dimension theory. nlin.SI/0211018

  4. Boyer, C.P., Finley, J.D.III: Killing vectors in self-dual, Euclidean Einstein spaces. J. Math. Phys. 23, 1126–1130 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burnat, M.: The method of Riemann invariants for multi-dimensional nonelliptic system. Bull. Acad. Polon. Sci. Sr. Sci. Tech. 17, 1019–1026 (1969)

    MATH  Google Scholar 

  6. Burnat, M.: The method of Riemann invariants and its applications to the theory of plasticity. I, II. Arch. Mech. (Arch. Mech. Stos.) 23, 817–838 (1971); ibid 24, 3–26 (1972)

    Google Scholar 

  7. Burnat, M.: The method of characteristics and Riemann’s invariants for multidimensional hyperbolic systems. (Russian) Sibirsk. Mat. Z. 11, 279–309 (1970)

    MATH  Google Scholar 

  8. Canic, S., Keyfitz, B.L., Kim, E.H.: Mixed Hyperbolic-Elliptic Systems in Self-Similar Flows. Boletim da Sociedade Brasileira de Matematica 32, 1–23 (2002)

    Google Scholar 

  9. Dafermos, C.: Hyperbolic conservation laws in continuum physics. Berlin-Heidelberg-New York: Springer-Verlag, 2000

  10. Dinu, L.: Some remarks concerning the Riemann invariance, Burnat-Peradzyński and Martin approaches. Rev. Roumaine Math. Pures Appl. 35, 203–234 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Dubrovin, B.A., Novikov, S.P.: Hydrodynamics of weakly deformed soliton lattices: differential geometry and Hamiltonian theory. Russ. Math. Surv. 44 (6), 35–124 (1989)

    MATH  Google Scholar 

  12. Dubrovin, B.A.: Geometry of 2D topological field theories. Lect. Notes in Math. 1620, Berlin-Heidelberg New York: Springer-Verlag, 1996, pp. 120–348

  13. Ferapontov, E.V.: Integrable systems in projective differential geometry. Kyushu J. Math. 54 (1), 183–215 (2000)

    MATH  Google Scholar 

  14. Ferapontov, E.V., Korotkin, D.A., Shramchenko, V.A.: Boyer-Finley equation and systems of hydrodynamic type. Class. Quantum Grav 19, L1–L6 (2002)

    Google Scholar 

  15. Ferapontov, E.V., Pavlov, M.V.: Hydrodynamic reductions of the heavenly equation. Class. Quantum Grav. 20, 1–13 (2003)

    Article  Google Scholar 

  16. Gibbons, J.: Collisionless Boltzmann equations and integrable moment equations. Physica. 3D, 503–511 (1981)

    Article  MathSciNet  Google Scholar 

  17. Gibbons, J., Kodama, Y.: A method for solving the dispersionless KP hierarchy and its exact solutions. II. Phys. Lett. A 135(3), 167–170 (1989)

    Google Scholar 

  18. Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211, 19–24 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gibbons, J., Tsarev, S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258, 263–271 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Godunov, S.K.: An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR 139, 521–523 (1961)

    MATH  Google Scholar 

  21. Grundland, A., Zelazny, R.: Simple waves in quasilinear hyperbolic systems. I, II. Riemann invariants for the problem of simple wave interactions. J. Math. Phys. 24 (9), 2305–2328 (1983)

    Google Scholar 

  22. Guil, F., Manas, M., Martinez, Alonso, L.: On the Whitham Hierarchies: Reductions and Hodograph Solutions. nlin.SI/0209051

  23. Hereman, W., Banerjee, P.P., Korpel, A., Assanto, G., Van Immerzule, A., Meerpoel, A.: Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method. J. Phys. A: Math. Gen. 19, 607–628 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hietarinta, J.: Introduction to the Hirota bilinear method. In: Integrability of nonlinear systems (Pondicherry, 1996), Lecture Notes in Phys., 495, Berlin: Springer, 1997, pp. 95–103

  25. Hietarinta, J.: Equations that pass Hirota’s three-soliton condition and other tests of integrability. In: Nonlinear evolution equations and dynamical systems (Kolymbari, 1989), Res. Rep. Phys., Berlin: Springer, 1990, pp. 46–50

  26. Hietarinta, J.: A search for nonlinear equations passing Hirota’s three-soliton condition. J. Math. Phys. 28, 1732–1742, 2094–2101, 2586–2592 (1987); 29, 628–635 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khusnutdinova, K.R.: Exact solutions describing interaction of solitary waves of non-integrable equations. In: Nonlinearity and Geometry, Warsaw: PWN, 1998, pp. 319–333

  28. Krichever, I.M.: The averaging method for two-dimensional ‘‘integrable’’ equations. Funct. Anal. Appl. 22(3), 200–213 (1988)

    MATH  Google Scholar 

  29. Krichever, I.M.: Spectral theory of two-dimensional periodic operators and its applications. Russ. Math. Surv. 44(2), 145–225 (1989)

    MATH  Google Scholar 

  30. Majda, A.: Compressible fluid flows and systems of conservation laws in several space variables. Appl. Math. Sci., NY: Springer-Verlag, 53, 1984

  31. Manas, M., Martinez Alonso, L., Medina, E.: Reductions and hodograph solutions of the dispersionless KP hierarchy. J. Phys. A: Math. Gen. 35, 401–417 (2002)

    MathSciNet  Google Scholar 

  32. Manas, M., Martinez, Alonso, L.: A hodograph transformation which applies to the heavenly equation. nlin.SI/0209050

  33. Mikhailov, A.V., Shabat, A.B., Sokolov, V.V.: The symmetry approach to classification of integrable equations. In: What is integrability? Springer Ser. Nonlinear Dynam., Berlin: Springer, 1991, pp. 115–184

  34. Mikhailov, A.V., Yamilov, R.I.: Towards classification of (2+1)-dimensional integrable equations. Integrability conditions. I. J. Phys. A 31(31), 6707–6715 (1998)

    MATH  Google Scholar 

  35. Pavlov, M.V.: Integrable hydrodynamic chains. nlin.SI/0301010.

  36. Pavlov, M.V.: Classification of the integrable Egorov hydrodynamic chains. Submitted to Theor. and Math. Phys. 2003

  37. Peradzyński, Z.: Riemann invariants for the nonplanar k-waves. Bull. Acad. Polon. Sci. Sr. Sci. Tech. 19, 717–724 (1971)

    Google Scholar 

  38. Peradzyński, Z.: Nonlinear plane k-waves and Riemann invariants. Bull. Acad. Polon. Sci. Sr. Sci. Tech. 19, 625–632 (1971)

    Google Scholar 

  39. Shabat, A.B., Martinez Alonso, L.: On the prolongation of a hierarchy of hydrodynamic chains. To appear in Theor. Math. Phys., 2003

  40. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: The method of differential constraints and its applications in gas dynamics. Novosibirsk: Nauka, 1984

  41. Tsarev, S.P.: Geometry of hamiltonian systems of hydrodynamic type. Generalized hodograph method. Izvestija AN USSR Math. 54(5), 1048–1068 (1990)

    MATH  Google Scholar 

  42. Yu, L.: Reductions of dispersionless integrable hierarchies. PhD Thesis, London: Imperial College, 2001

  43. Zakharov, E.V.: Dispersionless limit of integrable systems in 2+1 dimensions. In: Singular Limits of Dispersive Waves, Ercolani, N.M. et al. (eds.), NY: Plenum Press, 1994, pp. 165–174

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom

    E.V. Ferapontov & K.R. Khusnutdinova

  2. Institute of Mechanics, Ufa Branch of the Russian Academy of Sciences, Karl Marx Str. 6, Ufa, 450000, Russia

    K.R. Khusnutdinova

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  1. E.V. Ferapontov
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  2. K.R. Khusnutdinova
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Correspondence to E.V. Ferapontov.

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Communicated by G.W. Gibbons

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Ferapontov, E., Khusnutdinova, K. On the Integrability of (2+1)-Dimensional Quasilinear Systems. Commun. Math. Phys. 248, 187–206 (2004). https://doi.org/10.1007/s00220-004-1079-6

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