Taiwanese Journal of Mathematics

Quasi-periodic Waves and Solitary Waves to a Generalized KdV-Caudrey-Dodd-Gibbon Equation from Fluid Dynamics

Jian-Min Tu, Shou-Fu Tian, Mei-Juan Xu, and Tian-Tian Zhang

Full-text: Open access

Abstract

In this paper, a generalized KdV-Caudrey-Dodd-Gibbon (KdV-CDG) equation is investigated, which describes certain situations in the fluid mechanics, ocean dynamics and plasma physics. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study its Hirota's bilinear form and $N$-soliton solution, respectively. Furthermore, based on the Riemann theta function, the one-quasi- and two-quasi-periodic wave solutions are also constructed. Finally, an asymptotic relation of the quasi-periodic wave solutions are strictly analyzed to reveal the relations between quasi-periodic wave solutions and soliton solutions.

Article information

Source
Taiwanese J. Math., Volume 20, Number 4 (2016), 823-848.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874493

Digital Object Identifier
doi:10.11650/tjm.20.2016.6850

Mathematical Reviews number (MathSciNet)
MR3535676

Zentralblatt MATH identifier
1357.35256

Subjects
Primary: 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35C99: None of the above, but in this section 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10] 74J35: Solitary waves

Keywords
generalized KdV-Caudrey-Dodd-Gibbon equation Hirota's bilinear method Riemann theta function soliton wave solution quasi-periodic wave solution

Citation

Tu, Jian-Min; Tian, Shou-Fu; Xu, Mei-Juan; Zhang, Tian-Tian. Quasi-periodic Waves and Solitary Waves to a Generalized KdV-Caudrey-Dodd-Gibbon Equation from Fluid Dynamics. Taiwanese J. Math. 20 (2016), no. 4, 823--848. doi:10.11650/tjm.20.2016.6850. https://projecteuclid.org/euclid.twjm/1498874493


Export citation

References

  • M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes Series 149, Cambridge University Press, Cambridge, 1991.
  • E. T. Bell, Exponential polynomials, Ann. of Math. (2) 35 (1934), no. 2, 258–277.
  • E. Belokolos, A. Bobenko, V. Enol'skij, A. Its and V. Matveev, Algebro-Geometrical Approach to Nonlinear Integrable Equations, Springer 1994.
  • G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences 81, Springer-Verlag, New York, 1989.
  • K. W. Chow, A class of exact, periodic solutions of nonlinear envelope equations, J. Math. Phys. 36 (1995), no. 8, 4125–4137.
  • E. Fan, Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik-Novikov-Veselov equation, J. Phys. A 42 (2009), no. 9, 095206, 11 pp.
  • E. Fan and Y. C. Hon, Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii's breaking soliton equation in $(2+1)$-dimensions, Phys. Rev. E (3) 78 (2008), no. 3, 036607, 13 pp.
  • H. M. Farkas and I. Kra, Riemann Surfaces, Second edition, Graduate Texts in Mathematics 71, Springer-Verlag, New York, 1992.
  • C. Gilson, F. Lambert, J. Nimmo and R. Willox, On the combinatorics of the Hirota $D$-operators, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1945, 223–234.
  • R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics 155, Cambridge University Press, Cambridge, 2004.
  • X.-B. Hu, C.-X. Li, J. J. C. Nimmo and G.-F. Yu, An integrable symmetric $(2+1)$-dimensional Lotka-Volterra equation and a family of its solutions, J. Phys. A 38 (2005), no. 1, 195–204.
  • F. Lambert, I. Loris and J. Springael, Classical Darboux transformations and the KP hierarchy, Inverse Problems 17 (2001), no. 4, 1067–1074.
  • T.-C. Lin and J. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearity 19 (2006), no. 12, 2755–2773.
  • ––––, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phys. D 220 (2006), no. 2, 99–115.
  • T.-C. Lin and P. Zhang, Incompressible and compressible limits of coupled systems of nonlinear Schrödinger equations, Comm. Math. Phys. 266 (2006), no. 2, 547–569.
  • C. Liu and Z. Dai, Exact soliton solutions for the fifth-order Sawada-Kotera equation, Appl. Math. Comput. 206 (2008), no. 1, 272–275.
  • S.-Y. Lou, Extended painlevé expansion, nonstandard truncation and special reductions of nonlinear evolution equations, Z. Naturforsch. A 53 (1998), no. 5, 251–258.
  • W.-X. Ma, Bilinear equations and resonant solutions characterized by Bell polynomials, Rep. Math. Phys. 72 (2013), no. 1, 41–56.
  • W.-X. Ma and Y. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc. 357 (2005), no. 5, 1753–1778.
  • W.-X. Ma, R. Zhou and L. Gao, Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in $(2+1)$ dimensions, Modern Phys. Lett. A 24 (2009), no. 21, 1677–1688.
  • V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer Seriers in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  • Q. Miao, Y. Wang, Y. Chen and Y. Yang, PDEBellII: A Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations, Comput. Phys. Commun. 185 (2014), no. 1, 357–367.
  • D. Mumford, Tata Lectures on Theta II: Jacobian theta functions and differential equations, Progress in Mathematics 43, Birkhäuser, Boston, Boston, MA, 1984.
  • A. Nakamura, A direct method of calculating periodic wave solutions to nonlinear evolution equations II: Exact one- and two-periodic wave solution of the coupled bilinear equations, J. Phys. Soc. Japan 48 (1980), no. 4, 1365–1370.
  • Ameina S. Nuseir, New exact solutions to the modified Fornberg-Whitham equation, Taiwanese J. Math. 16 (2012), no. 6, 2083–2091.
  • C. Rogers and S. Carillo, On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Phys. Scripta 36 (1987), no. 6, 865–869.
  • 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), no. 5, 1355–1367.
  • S.-F. Tian and H.-Q. Zhang, Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations, J. Math. Anal. Appl. 371 (2010), no. 2, 585–608.
  • S. F. Tian and H. Q. Zhang, A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 173–186.
  • ––––, On the integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation, J. Phys. A 45 (2012), no. 5, 055203, 29 pp.
  • ––––, Riemann theta functions periodic wave solutions and rational characteristics for the $(1+1)$-dimensional and $(2+1)$-dimensional Ito equation, Chaos Solitons Fractals 47 (2013), 27–41.
  • ––––, On the integrability of a generalized variable-coefficient forced Korteweg-de Vries equation in fluids, Stud. Appl. Math. 132 (2014), no. 3, 212–246.
  • S. Tian, Y. Zhang, B. Feng and H. Zhang, On the Lie algebras, generalized symmetries and Darboux transformations of the fifth-order evolution equations in shallow water, Chin. Ann. Math. Ser. B 36 (2015), no. 4, 543–560.
  • S.-F. Tian, T.-T. Zhang, P.-L. Ma and X.-Y. Zhang, Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach, J. Nonlinear Math. Phys. 22 (2015), no. 2, 180–193.
  • M. Wadati, H. Sanuki and K. Konno, Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progr. Theoret. Phys. 53 (1975), no. 2, 419–436.
  • Y.-H. Wang and Y. Chen, Binary Bell polynomials, bilinear approach to exact periodic wave solutions of $(2+1)$-dimensional nonlinear evolution equations, Commun. Theor. Phys. (Beijing) 56 (2011), no. 4, 672–678.
  • A. M. Wazwaz, Multiple soliton solutions for the $(2+1)$-dimensional Sawada-Kotera and Caudrey-Dodd-Gibbon (CDG) equations, Math. Methods Appl. Sci. 34 (2011), no. 13, 1580–1586.
  • J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. 24 (1983), no. 3, 522–526.
  • D.-J. Zhang, J. Ji and S.-L. Zhao, Soliton scattering with amplitude changes of a negative order AKNS equation, Phys. D 238 (2009), no. 23-24, 2361–2367.