Lie point symmetry, conservation laws and exact power series solutions to the Fujimoto-Watanabe equation

References

• N.H. Asmar, Partial Differential Equations with Fourier Series and Boundary Value Problems, second edition, China Machine Press, Beijing, 2005.
   Google Scholar
• G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, Springer, New York, 1974.
   Google Scholar
• G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer- Verlag, New York, 1989.
   Google Scholar
• C.W. Cao, X.G. Ceng, and Y.T. Wu, From the special 2+1 Toda lattice to the Kadomtsev-Petviashvili equation, J. Phys. A-Math. Theor. 32 (1999), 8059-8078. doi: 10.1088/0305-4470/32/46/306
   Google Scholar
• H.H. Dong, B.Y. Guo, and B.S. Yin, Generalized Fractional Supertrace Identity For Hamiltonian Structure of NLS-MKdV Hierarchy With Self-Consistent Sources, Anal. Math. Phys. 6 (2016), 199-209. doi: 10.1007/s13324-015-0115-3
   Web of Science ®Google Scholar
• H.H. Dong, Y. Zhang, and X.E. Zhang, The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation, Commun. Nonlinear Sci. 36 (2016), 354-365. doi: 10.1016/j.cnsns.2015.12.015
   Web of Science ®Google Scholar
• H.H. Dong, Y.F. Zhang, Y.F. Zhang, and B.S. Yin, Generalized Bilinear Differential Operators, Binary Bell Polynomials, And Exact Periodic Wave Solution of Boiti-Leon-Manna-Pempinelli Equation, Abstr. Appl. Anal. 1 (2014), 1-6.
   Google Scholar
• H.H. Dong, K. Zhao, H.W. Yang, and Y.Q. Li, Generalised (2+1)-Dimensional Super Mkdv Hierarchy For Integrable Systems In Soliton Theory, E. Asian J. Appl. Math. 5(2015), 256-272. doi: 10.4208/eajam.110215.010815a
   Web of Science ®Google Scholar
• Z.Z. Dong, F. Huang, and Y. Chen, Symmetry reductions and exact solutions of the two-layer model in atmosphere, Z. Naturforsch. 66a (2011), 75-86. doi: 10.5560/ZNA.2011.66a0075
   Google Scholar
• Y. Fang, H.H. Dong, Y.J. Hou, and Y. Kong, Frobenius Integrable Decomposi­tions of Nonlinear Evolution Equations With Modified Term, Appl. Math. Comput. 226 (2014), 435-440.
   Web of Science ®Google Scholar
• C. Fu, C.N. Lu, and H.W. Yang, Time-space fractional (2 + 1) dimensional non­linear Schrodinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions, Adv. Differ. Equ-Ny. 2018 (2018), Article number 56, https://doi.org/10.1186/s13662-018-1512-3.
   Google Scholar
• B. Gao and H.X. Tian, Symmetry reductions and exact solutions to the ill-posed Boussinesq equation, Int. J. Nonlin. Mec. 72 (2015), 80-83. doi: 10.1016/j.ijnonlinmec.2015.03.004
   Web of Science ®Google Scholar
• C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Method for solving the korteweg-devries equation, Phys. Rev. Lett. 19 (1967), 1095-1097. doi: 10.1103/PhysRevLett.19.1095
   Web of Science ®Google Scholar
• M. Guo, Y. Zhang, M. Wang, Y.D. Chen, and H.W. Yang, A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism, Comput. Math. Appl. 75 (2018), 3589-3603. doi: 10.1016/j.camwa.2018.02.019
   Web of Science ®Google Scholar
• X.R. Hu, F. Huang, and Y. Chen, Symmetry reductions and exact solutions of the (2+1)-dimensional navier-stokes equations, Z. Naturforsch. 65a (2010), 504-510.
   Google Scholar
• N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007), 311-328. doi: 10.1016/j.jmaa.2006.10.078
   Web of Science ®Google Scholar
• A.G. Johnpillai, A.H. Kara, and A. Biswas, Symmetry reduction,exact group- invariant solutions and conservation laws of the Benjamin-Bona-Mahoney equation, Appl. Math. Lett. 26 (2013), 376-381. doi: 10.1016/j.aml.2012.10.012
   Web of Science ®Google Scholar
• Y.S. Li, W.X. Ma, and J.E Zhang, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A. 275 (2000), 60-66. doi: 10.1016/S0375-9601(00)00583-1
   Web of Science ®Google Scholar
• S. Lie, Uber die integration durch bestimmte Integrale von einer Klasse linearer partieller Differential gleichungen, Arch. Math. 6 (1881), 328-368.
   Google Scholar
• C.S. Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181 (2010), 317-324. doi: 10.1016/j.cpc.2009.10.006
   Web of Science ®Google Scholar
• H.Z. Liu, J.B. Li, and L. Liu, Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations, J. Math. Anal. Appl. 368 (2010), 551-558. doi: 10.1016/j.jmaa.2010.03.026
   Web of Science ®Google Scholar
• H.Z. Liu, J.B. Li, and Q.X. Zhang, Lie symmetry analysis and exact explicit solutions for general Burgers equation, J. Comput. Appl. Math. 228 (2009), 1-9. doi: 10.1016/j.cam.2008.06.009
   Web of Science ®Google Scholar
• Y. Liu, H.H. Dong, and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys. 2 (2018), 1-17.
   Google Scholar
• C.N. Lu, C. Fu, and H.W. Yang, Time-fractional generalized Boussinesq Equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions, Appl. Math. Comput. 327 (2018), 104-116.
   Web of Science ®Google Scholar
• W.X. Ma and A. Abdeljabbar, A bilinear Backlund transformation of a (3+1)- dimensional generalized KP equation, Appl. Math. Lett. 25 (2012), 1500-1504. doi: 10.1016/j.aml.2012.01.003
   Web of Science ®Google Scholar
• W.X. Ma, Y. Zhang, Y.N. Tang, and J.Y. Tu, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012), 7174-7183.
   Web of Science ®Google Scholar
• P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Springer, Berlin, 1986.
   Google Scholar
• L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
   Google Scholar
• C. Rogers and W.K. Schief, Backlund and Darboux transformations geometry and modern applications in soliton theory, Cambridge texts in applied mathematics, Cambridge University Press, Cambridge, 2002.
   Google Scholar
• W. Rudin, Principles of Mathematical Analysis, third edition, China Machine Press, Beijing, 2004.
   Google Scholar
• L.J. Shi and Z.S. Wen, Bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation, Commun. Theor. Phys. 69 (2010), 631-636. doi: 10.1088/0253-6102/69/6/631
   Google Scholar
• L.J. Shi and Z.S. Wen, Dynamical behaviors of traveling wave solutions to a Fujimoto-Watanabe equation, Chinese Phys. B. 27 (2018), 090201. doi: 10.1088/1674-1056/27/9/090201
   Web of Science ®Google Scholar
• M.S. Tao and H.H. Dong, Algebro-geometric solutions for a discrete integrable equation, Discrete Dyn. Nat. Soc. 5 (2017), 1-9. doi: 10.1155/2017/5258375
   Google Scholar
• X.Z. Wang, H.H. Dong, and Y.X. Li, Some Reductions From A Lax Integrable System and Their Hamiltonian Structures, Appl. Math. Comput. 218 (2012), 10032­10039.
   Web of Science ®Google Scholar
• J. Weiss, M. Tabor, and G. Carnevale, The painlev property for partial differ­ential equations, J. Math. Phys. 24 (1983), 522-526. doi: 10.1063/1.525721
   Web of Science ®Google Scholar
• H.W. Yang, X. Chen, M. Guo, and Y.D. Chen, A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property, Nonlinear Dynam. 4 (2017), 1-14.
   Google Scholar
• H.W. Yang, Z.H. Xu, D.Z. Yang, X.R. Feng, B.S. Yin, and H.H. Dong, ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect, Adv. Differ. Equ-Ny. 2016 (2016), Article number 167, https://doi.org/10.1186/s13662-016-0901-8.
   Google Scholar
• X.G. Zhang and Y.F. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett. 25 (2012), 555-560. doi: 10.1016/j.aml.2011.09.058
   Web of Science ®Google Scholar
• X.G. Zhang, L.S. Liu, Y.H. Wu, and Y.N. Lu, The iterative solutions of nonlinear fractional differential equations, Appl. Math. Comput. 219 (2013), 4680-4691.
   Web of Science ®Google Scholar
• Q.L. Zhao, X.Y. Li, and F.S. Liu, Two Integrable Lattice Hierarchies and Their Respective Darboux Transformations, Appl. Math. Comput. 219 (2013), 5693-5705.
   Web of Science ®Google Scholar
• J.Y. Zhu and X.G. Geng, Algebro-geometric constructions of the 2+1 dimensional differential-difference equation, Phys. Lett. A. 368 (2007), 464-469. doi: 10.1016/j.physleta.2007.04.041
   Web of Science ®Google Scholar