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Articles

A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

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Pages 555-568 | Received 09 Apr 2019, Accepted 04 May 2019, Published online: 09 Jul 2019

References

  • R.M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear trans- formation, J. Mathematical Phys. 9 (1968) 1202–1204. doi: 10.1063/1.1664700
  • A.V. Mikhaĭlov, A.B. Shabat and R.I. Yamilov, A symmetric approach to the classification of nonlinear equations. Complete lists of integrable systems, Uspekhi Mat. Nauk 42 (4(256)) (1987) 3–53.
  • J. Schiff, The nonlinear Schrödinger equation and conserved quantities in the deformed parafermion and sl(2, r)/u(1) coset models, arXiv preprint hep-th/9210029 (1992).
  • R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. doi: 10.1103/PhysRevLett.71.1661
  • R. Camassa, D.D. Holm and J.M. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics 31 (1994) 1–33. doi: 10.1016/S0065-2156(08)70254-0
  • A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), (World Sci. Publ., River Edge, NJ, 1999) 23–37.
  • A. Degasperis, D.D. Holm and A.N.I. Hone, A new integrable equation with peakon solutions, Teoret. Mat. Fiz. 133 (2) (2002) 170–183. doi: 10.4213/tmf388
  • V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A 42 (34) (2009) 342002, 14 pages. doi: 10.1088/1751-8113/42/34/342002
  • A.N.W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A 41 (37) (2008) 372002, 10 pages. doi: 10.1088/1751-8113/41/37/372002
  • A.G. Rasin and J. Schiff, Unfamiliar aspects of Bäcklund transformations and an associated Degasperis-Procesi equation, Teoret. Mat. Fiz. 196 (3) (2018) 449–464. doi: 10.4213/tmf9477
  • Y. Matsuno, Smooth multisoliton solutions and their peakon limit of Novikov’s Camassa-Holm type equation with cubic nonlinearity, J. Phys. A 46 (36) (2013) 365203, 27 pages. doi: 10.1088/1751-8113/46/36/365203
  • L. Wu, C. Li and N. Li, Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy, Appl. Math. Lett. 87 (2019) 134–140. doi: 10.1016/j.aml.2018.07.036
  • M. Musette and R. Conte, Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations, J. Math. Phys. 32 (6) (1991) 1450–1457. doi: 10.1063/1.529302
  • R. Hirota and J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan 40 (2) (1976) 611–612. doi: 10.1143/JPSJ.40.611
  • K. Sawada and T. Kotera, A method for finding N-soliton solutions of the K.d.V. equation and K.d.V.- like equation, Progr. Theoret. Phys. 51 (1974) 1355–1367. doi: 10.1143/PTP.51.1355
  • J. Kang, X. Liu, P.J. Olver and C. Qu, Liouville correspondences between integrable hierarchies, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017) Paper No. 035, 26 pages.
  • D.J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx + 6Qψx + 6Rψ = λψ , Stud. Appl. Math. 62 (3) (1980) 189–216. doi: 10.1002/sapm1980623189
  • M.C. Nucci, Painlevé property and pseudopotentials for nonlinear evolution equations, J. Phys. A 22 (15) (1989) 2897–2913. doi: 10.1088/0305-4470/22/15/009
  • A. Shabat, V. Adler, V. Marikhin and V. Sokolov, Encyclopedia of integrable systems, LD Landau Institute for Theoretical Physics (2010) p. 303.
  • Y. Shang, Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation, Appl. Math. Comput. 187 (2) (2007) 1286–1297.
  • A.G. Rasin and J. Schiff, Bäcklund transformations for the Boussinesq equation and merging solitons, J. Phys. A 50 (32) (2017) 325202, 21. doi: 10.1088/1751-8121/aa7af7
  • M. Fisher and J. Schiff, The Camassa Holm equation: conserved quantities and the initial value problem, Phys. Lett. A 259 (5) (1999) 371–376. doi: 10.1016/S0375-9601(99)00466-1
  • P.J. Olver, Applications of Lie groups to Differential Equations, 2nd edn. (Springer-Verlag, New York, 1993).
  • A.G. Rasin and J. Schiff, The Gardner method for symmetries, J. Phys. A 46 (15) (2013) 155202, 15 pages. doi: 10.1088/1751-8113/46/15/155202
  • A.G. Rasin and J. Schiff, Bäcklund transformations for the Camassa-Holm equation, J. Nonlinear Sci. 27 (1) (2017) 45–69. doi: 10.1007/s00332-016-9325-6
  • V. Golovko, P. Kersten, I. Krasilshchik and A. Verbovetsky, On integrability of the Camassa-Holm equation and its invariants: a geometrical approach, Acta Appl. Math. 101 (1-3) (2008) 59–83. doi: 10.1007/s10440-008-9200-z
  • V.E. Adler and A.B. Shabat, Toward a theory of integrable hyperbolic equations of third order, J. Phys. A 45 (39) (2012) 395207, 17 pages. doi: 10.1088/1751-8113/45/39/395207
  • J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, Journal of Mathematical Physics 24 (3) (1983) 522–526. doi: 10.1063/1.525721
  • J. Weiss, The singular manifold method, in Painlevé Transcendents, (Springer, 1992) 225–247.
  • C.M. Cosgrove and G. Scoufis, Painlevé classification of a class of differential equations of the second order and second degree, Studies in Applied Mathematics 88 (1) (1993) 25–87. doi: 10.1002/sapm199388125
  • J. Hietarinta, Introduction to the hirota bilinear method, in Integrability of nonlinear systems, (Springer, 1997) 95–103.
  • J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (8) (1987) 1732–1742. doi: 10.1063/1.527815
  • Y. Matsuno, Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves, J. Math. Phys. 31 (12) (1990) 2904–2916. doi: 10.1063/1.528943

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