Invariance, conservation laws and reductions of some classes of “high” order partial differential equations

References

• Ablowitz, M.J., & Clarkson, P.A. 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge, Cambridge University Press.
   Google Scholar
• Abramowitz, M. & Stegun, I. 1972. Handbook of Mathematical Functions. New York, Dover.
   Google Scholar
• Anco, S.C. & Bluman, G. 2002. Direct construction method for conservation laws of partial differential equations part ii: General treatment. European Journal of Applied Mathematics 13: 567–585.
   Web of Science ®Google Scholar
• Anco, S.C. & Gandarias, M.L. 2020. Symmetry multi-reduction method for partial differential equations with conservation laws. Communications in Nonlinear Science and Numerical Simulation 91: 105349.
   Web of Science ®Google Scholar
• Anco, S.C. & Kara, A.H. 2018. Symmetry-invariant conservation laws of partial differential equations. European Journal of Applied Mathematics 29: 78–117.
   Web of Science ®Google Scholar
• Billingham, J. & King, A.C. 2000. Wave Motion. Cambridge Texts in Applied Mathematics. 24 vols. Cambridge, Cambridge University Press.
   Google Scholar
• Biswas, A., Coleman, N., Kara, A.H., Khan, S., Moraru, L., Moldovanu, S., Iticescu, C. & Yıldırım, Y. 2022. Shallow water waves and conservation laws with dispersion triplet. Applied Sciences 12: 3647.
   Google Scholar
• Bluman, G.W., Cheviakov, A.F. & Anco, S.C. 2010. Applications of Symmetry Methods to Partial Differential Equations. 168 vols. New York, Springer-Verlag.
   Google Scholar
• Bluman, G.W. & Kumei, S.. 1989. Symmetries and Differential Equations Berlin, Springer-Verlag.
   Google Scholar
• Gardner, L. & Gardner, G. 1992. Solitary waves of the equal width wave equation. Journal of Computational Physics 101: 218–223.
   Web of Science ®Google Scholar
• Gardner, L., Gardner, G., Ayoub, F. & Amein, N. 1997. Simulations of the ew undular bore. Communications in Numerical Methods in Engineering 13: 583–592.
   Web of Science ®Google Scholar
• Gu, C. 1990. Soliton Theory and its Application. Hangzhou, China, Zhejiang Science and Technology.
   Google Scholar
• Han, Z., Zhang, Y.-F. & Zhao, Z.-L. 2013. Double reduction and exact solutions of Zakharov–Kuznetsov modified equal width equation with power law nonlinearity via conservation laws. Communications in Theoretical Physics 60: 699.
   Web of Science ®Google Scholar
• Hereman, W. 2006. Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions. International journal of quantum chemistry 106: 278–299.
   Web of Science ®Google Scholar
• Hirota, R. 1971. Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Physical Review Letters 27: 1192.
   Web of Science ®Google Scholar
• Ibragimov, N.H. 1995. CRC Handbook of Lie Group Analysis of Differential Equations. 3 vols. Boca Raton, CRC press.
   Google Scholar
• Ibragimov, N.K. 1999. Elementary Lie Group Analysis and Ordinary Differential Equations. 197 vols. New York, Wiley New York.
   Google Scholar
• Ibragimov, N.H. 2007. A new conservation theorem. Journal of Mathematical Analysis and Applications333: 311–328.
   Web of Science ®Google Scholar
• Ibragimov, N.H. 2009. A Practical Course in Differential Equations and Mathematical Modelling. Beijing, Higher Education Press.
   Google Scholar
• Ibragimov, N., Kara, A. & Mahomed, F. 1998. Lie–Bäcklund and Noether symmetries with applications. Nonlinear Dynamics 15: 115–136.
   Web of Science ®Google Scholar
• Kara, A. & Mahomed, F. 2006. Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dynamics 45: 367–383.
   Web of Science ®Google Scholar
• Khalique, M.C., Mahomed, F.M. & Muatjetjeja, B. 2008. Lagrangian formulation of a generalized Lane-Emden equation and double reduction. Journal of Nonlinear Mathematical Physics 15: 152–161.
   Web of Science ®Google Scholar
• Kudryashov, N. 2004. Analytical theory of nonlinear differential equations. Moskow-Igevsk, Institute of computer investigations 360.
   Google Scholar
• Kudryashov, N.A. 2005. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals 24: 1217–1231.
   Web of Science ®Google Scholar
• Kudryashov, N.A. 2012. One method for finding exact solutions of nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation 17: 2248–2253.
   Web of Science ®Google Scholar
• Lie, S. 1895. Zur allgemeinen theorie der partiellen differentialgleichungen beliebeger ordnung. Leipz. Berich 47: 53–128.
   Google Scholar
• Liu, S., Fu, Z., Liu, S. & Zhao, Q. 2001. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A 289: 69–74.
   Web of Science ®Google Scholar
• Morrison, P., Meiss, J. & Carey, J. 1984. Scattering of regularized-long-waves. Physica D 11: 324–336.
   Web of Science ®Google Scholar
• Noether, E. 1918. Invariante variations probleme, nachr. ges. wiss. Gottingen Math.-Phys. Kl. 1918: 235–257.
   Google Scholar
• Olver, P.J. 1986. Applications of Lie Groups to Differential Equations. New York, Springer-Verlag.
   Google Scholar
• Sjöberg, A. 2007. Double reduction of pdes from the association of symmetries with conservation laws with applications. Applied Mathematics and Computation 184: 608–616.
   Web of Science ®Google Scholar
• Sjöberg, A. 2009. On double reductions from symmetries and conservation laws. Nonlinear Analysis: Real World Applications 10: 3472–3477.
   Web of Science ®Google Scholar
• Stephani, H. 1989. Differential Equations: Their Solution Using Symmetries. Cambridge, Cambridge University Press.
   Google Scholar
• Steudel, H. 1962. Über die zuordnung zwischen lnvarianzeigenschaften und erhaltungssätzen. Zeitschrift für Naturforschung A 17: 129–132.
   Web of Science ®Google Scholar
• Wang, M., Li, X. & Zhang, J. 2008. The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A 372: 417–423.
   Web of Science ®Google Scholar
• Wazwaz, A.-M. 2018. Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions. Nonlinear Dynamics 94: 2655–2663.
   Web of Science ®Google Scholar