Symmetry Analysis and Reductions Through Conservation Laws of a Generalized Bogoyavlensky-Konopelchenko Equation in $(2+1)$-Dimensions
DOI:
https://doi.org/10.5890/JVTSD.2021.09.005Abstract
In this paper we obtain Lie symmetries and travelling wave solutions for a generalized Bogoyavlensky-Konopelchenko equation in $(2+1)$-dimensions. Moreover, we determine some low-order conservation laws which are invariant under the translation symmetry; consequently they are inherited by the reduced differential equations.References
[1] Ray, S.S. (2017), On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in wave propagation, Computers and Mathematics with Applications, 74(6), 1158-1165.
[2] Li, Q., Chaolu, T., and Wanga, Y.H. (2019), Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation., Computers and Mathematics with Applications, 77, 2077-2085
[3] Bruzon, M.S. and Gandarias, M.L. (2003), Symmetry Reductions for a Dissipation-Modified KdV Equation, Applied Mathematics Letters, 16, 155-159.
[4] Bruzon, M.S., de la Rosa, R., and Tracina, R. (2018), Exact solutions via equivalence transformations of variable-coefficient fifth-order KdV equations, Applied Mathematics and Computation, 325, 239-245.
[5] Bruzon, M.S., Marquez, A.P., Garrido, T.M., Recio, E., and de la Rosa, R. (2019), Conservation laws for a generalized seventh order KdV equation, Journal of Computational and Applied Mathematics, 354, 682-688.
[6] de la Rosa, R., Recio, E., Garrido, T.M., and Bruzon, M.S. (2019), Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients, International Journal of Computer Mathematics, 97(1-2), 330-340.
[7] Anco, S. and Bluman, G. (1997), Direct constrution of conservation laws from field equations, Phys. Rev. Lett., 78, 2869-2873.
[8] Anco, S.C. and Bluman, G. (2002), Direct constrution method for conservation laws of partial differential equations Part I: Examples of conservation law classifications, Eur. J. Appl. Math., 13, 545-566.
[9] Anco, S.C. and Bluman, G. (2002), Direct construction method for conservation laws of partial differential equations Part II: General treatment, Eur. J. Appl. Math., 13, 567-585.
[10] Anco, S.C. (2016), Symmetry properties of conservation laws, Internat, J. Modern Phys. B, 30, 1640003.
[11] Anco, S.C. (2017), Generalization of Noethers theorem in modern form to non-variational partial differential equations, In: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 119-182, Fields Institute Communications, Volume 79.
[12] Ibragimov, N.H. (2007), Quasi self-adjoint differential equations, Arch. ALGA, 4, 55-60.
[13] Ibragimov, N.H. (2007), A new conservation theorem, Journal of Mathematical Analysis and Applications, 333, 311-328.
[14] Bruzon, M.S., Gandarias, M.L., and Ibragimov, N.H. (2009), Self-adjoint sub-classes of generalized thin film equations, J. Math. Anal. Appl., 357, 307-313.
[15] Anco, S.C., Gandarias, M.L., and Recio, E. (2018), 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), 1393-1411.
[16] Anco, S.C. and Gandarias, M.L. Multi-reduction theory for PDEs with conservation laws. Comunications in Nonlinear Science and Numerical Simulations (Accepted).
[17] Anco, S.C. and Kara, A. (2018), Symmetry invariance of conservation laws, Euro. J. Appl. Math., 29(1), 78-117.
[18] Bluman, G. and Anco, S.C. (2006), New conservation laws obtained directly from symmetry action on known conservation laws, J. Math. Anal. Appl., 322, 233-250.
[19] Bokhari, A.H., Dweik, A.Y., Zaman, F.D., Kara, A.H., and Mahomed, F.M. (2010), Generalization of the double reduction theory, Nonlinear Analysis: Real World Applications, 11(5), 3763-3769.
[20] Gandarias, M.L. and Rosa, M. (2016), On double reductions from symmetries and conservation laws for a damped Boussinesq equation, Chaos, Solitons, Fractals, 89, 560-565.
[21] Gandarias, M.L. and Bruzon, M.S. (2017), Conservation laws for a Boussinesq equation, Appl. Math. and Nonlin. Sci., 2(2), 465-472.
[22] Halder, A.K., Paliathanasis, A., and Leach, P.G.L. (2020), Similarity solutions and Conservation laws for the Bogoyavlensky-Konopelchenko Equation by Lie point symmetries arXiv:2003.10131v1.
[23] Ibragimov, N.H., Torrisi, M., and Tracina, R. (2010), Quasi self-adjoint nonlinear wave equations, J. Phys. A: Math. Theor., 43, 442001.
[24] Ibragimov, N.H. (2006), Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl., 318, 742-757.
[25] Ibragimov, N.H. (2006), The answer to the question put to me by LV Ovsiannikov 33 years ago, Arch. ALGA, 3, 53-80.
[26] Ibragimov, N.H. (2011), Nonlinear self-adjointness in constructing conservation laws, Arch. ALGA, 8, 59-63.
[27] Ray, S.S. (2018), Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky-Konopelchenko equation by geometric approach, Modern Physics Letters B, 32(11), 1850127.
[28] San, S., Akbulut, A., Unsal, O., and Ta, F. (2017), Conservation laws and double reduction of(2+1) dimensional Calogero-Bogoyavlenskii-Schiff equation, Math. Meth. Appl. Sci., 40, 1703-1710.
[29] Sjoberg, A. (2007), Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput., 184, 608-616.
[30] Sjoberg, A. (2009), On double reduction from symmetries and conservation laws, Nonlinear Analysis: Real World Applications, 10, 3472-3477.
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