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Direct method of finding exact solutions of nonlinear evolution equations

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Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 515))

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References

  1. R. HIROTA, Exact solution of the modified Korteweg-deVries equation for multiple collisions of solitons, J. Phys. Soc. Japan 33 (1972), 1456–1458.

    Article  Google Scholar 

  2. M. WADATI, The modified Korteweg-deVries equation, J. Phys. Soc. Japan 34 (1973), 1289–1296.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. R. GRAVES-MORRIS, Ed., Pade Approximants and Their Applications, Academic Press, New York, N.Y., 1973.

    MATH  Google Scholar 

  4. G. B. WHITHAM, Linear and Nonlinear Waves, John Wiley and Sons, New York, N.Y., 1974.

    MATH  Google Scholar 

  5. R. HIROTA, A new form of Bäcklund transformation and its relation to the inverse scattering problem, Progr. Theoret Phys. 52 (1974), 1498–1512.

    Article  MATH  Google Scholar 

  6. T. TANIUTI AND N. YAJIMA, Perturbation method for a nonlinear wave modulation. I., J. Mathematical Phys. 10 (1969), 1369–1372.

    Article  MathSciNet  Google Scholar 

  7. V. E. ZAKHAROV AND A. B. SHABAT, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62–69.

    MathSciNet  Google Scholar 

  8. R. HIROTA, Exact envelope-soliton solutions of a nonlinear wave equation, J. Mathematical Phys. 14 (1973), 805–809.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. HASEGAWA AND F. TAPPERT, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion, Appl. Phys. Lett. 23 (1973), 171–172.

    Article  Google Scholar 

  10. V. E. ZAKHAROV AND A. B. SHABAT, Interaction between solitons in a stable medium, Soviet Physics JETP 37 (1973), 823–828.

    Google Scholar 

  11. A. YOSHIKAWA AND M. YAMAGUTI, On some further properties to a certain semilinear system of partial differential equations, Publ. Res. Inst. Math. Sci., Kyoto Univ. 9 (1974), 577–595.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. HASEGAWA, Propagation of wave intensity shocks in nonlinear interaction of waves and particles, Phys. Lett. 47A (1974), 165–166.

    Article  Google Scholar 

  13. H. HASHIMOTO, Exact solution of a certain semi-linear system of partial differential equations related to a migrating predation problem. Proc. Japan Acad. 50 (1974), 623–627.

    Article  MathSciNet  Google Scholar 

  14. E. HOPF, The partial differential equation ut+uux=μuxx, Comm. Pure. Appl. Math. 3 (1950), 201–230.

    Article  MathSciNet  Google Scholar 

  15. U. d’ANCONA, The Struggle for Existence, E. J. Brill, Leiden, Netherlands, 1954.

    Google Scholar 

  16. K. DAIKOKU AND Y. MIZUSHIMA, New instability concept in avalanche diode oscillation, Japan. J. Appl. Phys. 13 (1974), 989–994.

    Article  Google Scholar 

  17. K. DAIKOKU, Y. MIZUSHIMA, AND T. TAMAMA, Computer experiments on new lattice solitons propagating in Volterra’s system, Japan. J. Appl. Phys. 14 (1975), 367–376.

    Article  Google Scholar 

  18. R. HIROTA AND K. SUZUKI, Theoretical and experimental studies of lattice solitons in nonlinear lumped networks, Proc. IEEE 61 (1973), 1483–1491.

    Article  Google Scholar 

  19. M. TODA, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967), 501–506.

    Article  Google Scholar 

  20. V. E. ZAKHAROV AND S. V. MANAKOV, Resonant interaction of wave packets in nonlinear media, Soviet Physics JETP Lett. 18 (1973), 243–245.

    Google Scholar 

  21. K. NOZAKI AND T. TANIUTI, Propagation of solitary pulses in interactions of plasma waves, J. Phys. Soc. Japan 34 (1973), 796–800.

    Article  Google Scholar 

  22. V. S. DRYUMA, Analytic solution of the two-dimensional Korteweg-deVries (KdV) equation, Soviet Physics JETP Lett. 19 (1974), 387–388.

    Google Scholar 

  23. J. SATSUMA, private communication.

    Google Scholar 

  24. R. HIROTA, Exact solution of the Korteweg-deVries equation for multiple collisions of solitions, Phys. Rev. Lett. 27 (1971), 1192–1194.

    Article  MATH  Google Scholar 

  25. T. A. FULTON, R. C. DYNES, AND P. W. ANDERSON, The flux shuttle—A Josephson junction shift register employing single flux quanta, Proc. IEEE 61 (1973), 28–35.

    Article  Google Scholar 

  26. R. HIROTA, Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. Phys. Soc. Japan 35 (1973), 1566.

    Article  Google Scholar 

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Robert M. Miura

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© 1976 Springer-Verlag

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Hirota, R. (1976). Direct method of finding exact solutions of nonlinear evolution equations. In: Miura, R.M. (eds) Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. Lecture Notes in Mathematics, vol 515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081162

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  • DOI: https://doi.org/10.1007/BFb0081162

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07687-2

  • Online ISBN: 978-3-540-38220-1

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