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Articles

Repeated application of the recursion operator for a new hierarchy of negative-order integrable KdV equations

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Pages 300-307 | Received 04 Jan 2018, Accepted 25 Jul 2018, Published online: 10 Aug 2018

References

  • Baldwin D, Hereman W. A symbolic algorithm for computing recursion operators of nonlinear partial differential equations. Int J Comput Math. 2010;87(5):1094–1119. doi: 10.1080/00207160903111592
  • Fokas AS. Symmetries and integrability. Stud Appl Math. 1987;77:253–299. doi: 10.1002/sapm1987773253
  • Sanders JA, Wang JP. Integrable systems and their recursion operators. Nonlinear Anal. 2001;47:5213–5240. doi: 10.1016/S0362-546X(01)00630-7
  • Olver PJ. Evolution equations possessing infinitely many symmetries. J Math Phys. 1977;18(6):1212–1215. doi: 10.1063/1.523393
  • Magri F. Lectures notes in physics. Berlin: Springer; 1980.
  • Zhang D, Ji J, Zhao S. Soliton scattering with amplitude changes of a negative order AKNS equation. Phys D. 2009;238:2361–2367. doi: 10.1016/j.physd.2009.09.018
  • Verosky JM. Negative powers of Olver recursion operators. J Math Phys. 1991;32(7):1733–1736. doi: 10.1063/1.529234
  • Qiao Z, Fan E. Negative-order Korteweg-de Vries equation. Phys Rev E. 2012;86:016601. doi: 10.1103/PhysRevE.86.016601
  • Weiss J, Tabor M, Carnevale G. The Painlevé property of partial differential equations. J Math Phys A. 1983;24:522–526. doi: 10.1063/1.525721
  • Hirota R. The direct method in soliton theory. Cambridge: Cambridge University Press; 2004.
  • Khalique CM. Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities. Pramana. 2013;80:413–427. doi: 10.1007/s12043-012-0489-9
  • Kara AH, Khalique CM. Nonlinear evolution-type equations and their exact solutions using inverse variational methods. J Phys A Math Gen. 2005;38:4629–4636. doi: 10.1088/0305-4470/38/21/008
  • Leblond H, Mihalache D. Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys Rep. 2013;523:61–126. doi: 10.1016/j.physrep.2012.10.006
  • Leblond H, Mihalache D. Few-optical-cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models. Phys Rev A. 2009;79:063835.
  • Khoury SA. New ansätz for obtaining wave solutions of the generalized Camassa-Holm equation. Chaos Solitons Fractals. 2005;25(3):705–710. doi: 10.1016/j.chaos.2004.11.083
  • Khoury SA. Exact solutions for a class of nonlinear evolution equations: a unified ansätze approach. Chaos Solitons Fractals. 2008;36(5):1181–1188. doi: 10.1016/j.chaos.2006.09.066
  • Selima ES, Yao X, Wazwaz AM. Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convictive fluid via Painlevé analysis. Phys Rev E. 2017;95:062211. doi: 10.1103/PhysRevE.95.062211
  • Wazwaz AM. Partial differential equations and solitary waves theorem. Berlin: Springer; 2009.
  • Wazwaz AM. Multiple soliton solutions for a (2+1)-dimensional integrable KdV6 equation. Commun Nonlinear Sci Numer Simul. 2010;15:1466–1472. doi: 10.1016/j.cnsns.2009.06.024
  • Wazwaz AM. A study on the (2+1)-dimensional KdV4 equation derived by using the KdV recursion operator. Math Methods Appl Sci. 2012;36(13):1360–1367.
  • Wazwaz AM. A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions. Waves Random Complex Media. 2017; In Press. doi:10.1080/17455030.2017.1367440
  • Wazwaz AM. The generalized Kaup-Boussinesq equation for water wave: multiple soliton solutions. Waves Random Complex Media. 2015;25(4):473–481. doi: 10.1080/17455030.2015.1016474
  • Wazwaz AM. Multiple soliton solutions for two integrable couplings of the modified Korteweg-de Vries equation. Proc Rom Acad A. 2013;14(3):219–225.
  • Wazwaz AM. Two forms of (3+1)-dimensional B-type Kadomtsev–Petviashvili equation: multiple soliton solutions. Phys Scr. 2012;86:035007.
  • Wazwaz AM. Abundant solutions of various physical features for the (2+1)- dimensional modified KdV-Calogero–Bogoyavlenskii–Schiff equations. Nonlinear Dyn. 2017;89:1727–1732. doi: 10.1007/s11071-017-3547-5
  • Wazwaz AM. Negative-order KdV equations in (3+1)-dimensions by using the KdV recursion operator. Waves Random Complex Media. 2017;27(4):768–778. doi: 10.1080/17455030.2017.1317115
  • Wazwaz AM. Negative-order KdV equation and negative-order KP equation: multiple soliton solutions. Proc Natl Acad Sci Indian Sect A Phys Sci. 2017;87(2):291–296. doi: 10.1007/s40010-017-0349-6
  • Wazwaz AM. A new integrable nonlocal modified KdV equation: abundant solutions with distinct physical structures. J Ocean Eng Sci. 2017;2:1–4. doi: 10.1016/j.joes.2016.11.001
  • Yu F. Nonautonomous rogue waves and ‘catch’ dynamics for the combined Hirota-LPD equation with variable coefficients. Commun Nonlinear Sci Numer Simulat. 2016;34:142–153. doi: 10.1016/j.cnsns.2015.10.018
  • Yu FJ. Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz–Musslimani equation with PT-symmetric potential. Chaos. 2017;27:023108.
  • Yu FJ. Localized analytical solutions and numerically stabilities of generalized Gross-Pitaevskii (GP(p, q)) equation with specific external potentials. Appl Math Lett. 2018;85:1–7. doi: 10.1016/j.aml.2018.05.003

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