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Generalized conditional symmetries and exact solutions of non-integrable equations

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Abstract

We introduce the concept of a generalized conditional symmetry. This concept provides an algorithm for constructing physically important exact solutions of non-integrable equations. Examples include 2-shock and 2-soliton solutions. The existence of such exact solutions for non-integrable equations can be traced back to the relation of these equations with integrable ones. In this sense these exact solutions are remnants of integrability.

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References

  1. M. J. Ablowitz and A. Zeppetella, Bull. Math. Biol. 41 (1979) 835.

    Google Scholar 

  2. R. Fitzhugh, Biophys. J. 1 (1961) 445; J. S Nagumo, S. Arimoto, and S. Yoshizawa, Proc. IRE 50 (1962) 2061; A. Kolmogorov, I. Petrovsky, and N. Piscounov, Bull. Univ. Etat. Moscou (Ser. Int.) A1 (1937) 1; A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38 (1969) 239.

    Google Scholar 

  3. J. D. Cole, Quart. Appl. Math. 9 (1951) 225; E. Hopf, Comm. Pure Appl. Math. 3 (1950) 201.

    Google Scholar 

  4. T. Kawahara and M. Tanaka, Phys. Lett. 97A (1983) 311.

    Google Scholar 

  5. J. Satsuma, Exact solutions of Burgers equation with reaction terms, preprint, 1987.

  6. R. Conte, Phys. Lett. 134A (1988) 100; F. Cariello and M. Tabor, Physica 39D (1989) 77; P. G. Estévez and P. R. Gordoa, J. Phys. 23A (1990) 4831; Z. Chen and B. Guo, IMA J. Appl. Math. 48 (1992) 645.

    Google Scholar 

  7. M. C. Nucci and P. A. Clarkson, Phys. Lett. 164A (1992) 49–56.

    Google Scholar 

  8. G. W. Bluman and J. D. Cole, J. Math. Mech. 18 (1969) 1025.

    Google Scholar 

  9. P. J. Olver and P. Rosenau, Phys. Lett. 114A (1986) 107; SIAM J. Appl. Math 47 (1987) 263.

    Google Scholar 

  10. P. A. Clarkson and P. Winternitz, Physica D 49 (1991) 257.

    Google Scholar 

  11. N. A. Kudryashov, Phys. Lett. 169A (1992) 237.

    Google Scholar 

  12. I. M. Gel'fand and L. A. Dikii, Uspeki Matem. Nauk 30 (1975) 67.

    Google Scholar 

  13. R. L. Anderson and N. H. Ibragimov,Lie-Bäcklund Transformations in Applications, SIAM Stud. Appl. Math. No. 1, Philadelphia, 1979.

  14. A. S. Fokas and I. M. Gel'fand, Bi-hamiltonian structures and integrability, inImportant Developments in Soliton Theory, ed. by A. S. Fokas and V. E. Zakharov, Springer-Verlag, 1993.

  15. L. V. Ovsiannikov,Group Analysis of Differential Equations, Academic Press, New York, 1982.

    Google Scholar 

  16. E. L. Ince,Ordinary Differential Equations, Longmans and Green, London-New York 1926.

    Google Scholar 

  17. A. S. Fokas, J. Math. Phys. 21 (1980) 1318; Stud. Appl. Math. 77 (1987) 253.

    Google Scholar 

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Authors
  1. A. S. Fokas
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  2. Q. M. Liu
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Department of Mathematics and Computer Science and Institute for Nonlinear Studies, Clarkson University, Potsdam, New York 13699-5815. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 263–277, May, 1994.

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Fokas, A.S., Liu, Q.M. Generalized conditional symmetries and exact solutions of non-integrable equations. Theor Math Phys 99, 571–582 (1994). https://doi.org/10.1007/BF01016141

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  • DOI: https://doi.org/10.1007/BF01016141

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