305
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon
Pages 330-340 | Received 30 Jul 2018, Accepted 16 Mar 2019, Published online: 04 Apr 2019

References

  • M.J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979), pp. 691–715. doi: 10.1017/S0022112079000835
  • S.C. Anco, M. Mohiuddin, and T. Wolf, Traveling waves and conservation laws for complex mKdV-type equations, Appl. Math. Comput. 219 (2012), pp. 679–698.
  • S.C. Anco, M. Rosa, and M.L. Gandarias, Conservation laws and symmetries of time-dependent generalized KdV equations, Discret. Contin. Dyn. Syst. Ser. S 11(4) (2018), pp. 607–615.
  • S.C. Anco, M.L. Gandarias, and E. Recio, Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions, Theor. Math. Phys. 197(1) (2018), pp. 1393–1411. doi: 10.1134/S004057791810001X
  • T.B. Benjamin, J.L. Bona, and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A 272(1220) (1972), pp. 47–78. doi: 10.1098/rsta.1972.0032
  • G.W. Bluman, A.F. Cheviakov, and S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York, 2010.
  • M. Boiti, J.P. Leon, M. Manna, and F. Pempinelli, On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions, Inverse Probl. 2(3) (1986), pp. 271–279. doi: 10.1088/0266-5611/2/3/005
  • J.L. Bona and R.L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys. 118 (1988), pp. 15–29. doi: 10.1007/BF01218475
  • J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, J. Math. Pures Appl. 2 (1872), pp. 55–108.
  • J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires presents par divers savants à lAcadémie des Sciences de l'Institut de France 23(1) (1877), pp. 1–680.
  • M.S. Bruzón, R. de la Rosa, and R. Tracinà, Exact solutions via equivalence transformations of a variable-coefficient fifth-order KdV equations, Appl. Math. Comput. 325 (2018), pp. 239–245.
  • R. de la Rosa, E. Recio, T.M. Garrido, and M.S. Bruzón, Lie symmetry analysis of the (2+1)-dimensional KdV equation with variable coefficients, to appear in Proceedings of the 18th International Conference on Mathematical Methods in Science and Engineering, J. Vigo-Aguiar, ed., CMMSE, Almería, 2018.
  • J. Escorcia and E. Suazo, Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrödinger equation, Appl. Math. Comput. 301 (2017), pp. 155–176.
  • L.R.T. Gardner and G.A. Gardner, Solitary waves of the equal width wave equation, J. Comput. Phys. 101 (1992), pp. 218–223. doi: 10.1016/0021-9991(92)90054-3
  • R.K. Gazizov, N.H. Ibragimov, and S.Yu. Lukashchuk, Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations, Commun. Nonlinear Sci. Numer. Simul. 23 (2015), pp. 153–163. doi: 10.1016/j.cnsns.2014.11.010
  • F. Gesztesy, H. Holden, E. Saab, and B. Simon, Explicit construction of solutions of the modified Kadomtsev-Petviashvili equation, J. Funct. Anal. 98(1) (1991), pp. 211–228. doi: 10.1016/0022-1236(91)90096-N
  • C.A. Gómez Sierra, On a KdV equation with higher-order nonlinearity: Traveling wave solutions, J. Comput. Appl. Math. 235 (2011), pp. 5330–5332. doi: 10.1016/j.cam.2011.05.028
  • A.G. Johnpillai and C.M. Khalique, Group analysis of KdV equation with time dependent coefficients, Appl. Math. Comput. 216 (2010), pp. 3761–3771.
  • B.B. Kadomtsev and V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl. 15 (1970), pp. 539–541.
  • B.G. Konopelchenko and V.G. Dubrovsky, Inverse spectral transform for the modified Kadomtsev-Petviashvili equation, Stud. Appl. Math. 86(3) (1992), pp. 219–268. doi: 10.1002/sapm1992863219
  • D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39(5) (1895), pp. 422–443. doi: 10.1080/14786449508620739
  • F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differ. Equ. 106 (1993), pp. 257–293. doi: 10.1006/jdeq.1993.1108
  • H. Liu, B. Sang, X. Xin, and X. Liu, CK transformations, symmetries, exact solutions and conservation laws of the generalized variable-coefficient KdV types of equations, J. Comput. Appl. Math. 345(1) (2019), pp. 127–134. doi: 10.1016/j.cam.2018.06.019
  • S.Y. Lou and X.B. Hu, Infinitely many lax pairs and symmetry constraints of the KP equation, J. Math. Phys. 38(12) (1997), pp. 6401–6427. doi: 10.1063/1.532219
  • S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its, and V.B. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A 63 (1977), pp. 205–206. doi: 10.1016/0375-9601(77)90875-1
  • P.J. Morrison, J.D. Meiss, and J.R. Cary, Scattering of regularized-long-wave solitary waves, Physica D 11 (1984), pp. 324–336. doi: 10.1016/0167-2789(84)90014-9
  • P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
  • M.S. Osman and A.M. Wazwaz, An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients, Appl. Math. Comput. 321 (2018), pp. 282–289.
  • L.V. Ovsyannikov, Group analysis of differential equations, Academic, New York, 1982.
  • D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966), pp. 321–330. doi: 10.1017/S0022112066001678
  • T. Rehman, G. Gambino, and S.R. Choudhury, Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), pp. 1746–1769. doi: 10.1016/j.cnsns.2013.10.029
  • A.K. Roy Chowdhury, Lie Algebraic Methods in Integrable Systems, Chapman and Hall/CRC, Boca Raton, FL, 2000.
  • S. Saha Ray, Invariant analysis and conservation laws for the time fractional (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis, Comput. Math. Appl. 76 (2018), pp. 2110–2118. doi: 10.1016/j.camwa.2018.08.008
  • S. Saha Ray and S. Sahoo, Invariant analysis and conservation laws of (2+1) dimensional time-fractional ZK-BBM equation in gravity water waves, Comput. Math. Appl. 75(7) (2018), pp. 2271–2279. doi: 10.1016/j.camwa.2017.12.001
  • S. Sahoo and S. Saha Ray, Invariant analysis with conservation laws for the time fractional Drinfeld-Sokolov-Satsuma-Hirota equations, Chaos Solitons Fract. 104 (2017), pp. 725–733. doi: 10.1016/j.chaos.2017.09.031
  • S. Sahoo and S. Saha Ray, Lie symmetry analysis and exact solutions of (3+1) dimensional Yu-Toda-Sasa-Fukuyama equation in mathematical physics, Comput. Math. Appl. 73 (2017), pp. 253–260. doi: 10.1016/j.camwa.2016.11.016
  • J. Satsuma, N-Soliton solution of the two-dimensional Korteweg-de Vries equation, J. Phys. Soc. Jpn. 40 (1976), pp. 286–290. doi: 10.1143/JPSJ.40.286
  • J. Satsuma and M.J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys. 20 (1979), pp. 1496–1503. doi: 10.1063/1.524208
  • M. Senthilkumaran, D. Pandiaraja, and B.M. Vaganan, New exact explicit solutions of the generalized KdV equations, Appl. Math. Comput. 202 (2008), pp. 693–699.
  • M. Torrisi and R. Tracinà, Exact solutions of a reaction-diffusion system for Proteus mirabilis bacterial colonies, Nonlinear Anal. Real World Appl. 12 (2011), pp. 1865–1874. doi: 10.1016/j.nonrwa.2010.12.004
  • R. Tracinà, On the nonlinear self-adjointness of the Zakharov-Kuznetsov equation, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), pp. 377–382. doi: 10.1016/j.cnsns.2013.06.014
  • R. Tracinà, I.L. Freire, and M. Torrisi, Nonlinear self-adjointness of a class of third order nonlinear dispersive equations, Commun. Nonlinear Sci. Numer. Simul. 32 (2016), pp. 225–233. doi: 10.1016/j.cnsns.2015.08.016
  • H. Triki, T.R. Taha, and A.M. Wazwaz, Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients, Math. Comput. Simul. 80 (2010), pp. 1867–1873. doi: 10.1016/j.matcom.2010.02.001
  • O. Vaneeva, O. Kuriksha, and C. Sophocleous, Enhanced group classification of Gardner equations with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), pp. 1243–1251. doi: 10.1016/j.cnsns.2014.09.016
  • V. Veerakumar and M. Daniel, Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton, Math. Comput. Simul. 62 (2003), pp. 163–169. doi: 10.1016/S0378-4754(02)00176-3
  • A.M. Wazwaz, The variational iteration method for rational solutions for KdV, K(2,2), burgers, and cubic Boussinesq equations, J. Comput. Appl. Math. 207 (2007), pp. 18–23. doi: 10.1016/j.cam.2006.07.010
  • Y. Yildirim and E. Yaşar, A (2+1)-dimensional breaking soliton equation: Solutions and conservation laws, Chaos Solitons Fract. 107 (2018), pp. 146–155. doi: 10.1016/j.chaos.2017.12.016
  • V.E. Zakharov and E.A. Kuznetsov, Three-dimensional solitons, Sov. Phys. 39(2) (1974), pp. 285–286.
  • Y. Zhang, J. Liu, and G. Wei, Lax pair, auto-Bäcklund transformation and conservation law for a generalized variable-coefficient KdV equation with external-force term, Appl. Math. Lett. 45 (2015), pp. 58–63. doi: 10.1016/j.aml.2015.01.007
  • Y. Zhang, J. Li, and Y.N. Lv, The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vries equation, Ann. Phys. 323 (2008), pp. 3059–3064. doi: 10.1016/j.aop.2008.04.012

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.