Group classification and exact solutions of a higher-order Boussinesq equation

Hasanoğlu Y., Özemir C.

Nonlinear Dynamics, vol.104, pp.2599-2611, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 104
  • Publication Date: 2021
  • Doi Number: 10.1007/s11071-021-06382-7
  • Journal Name: Nonlinear Dynamics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.2599-2611
  • Keywords: Higher-order Boussinesq equation, Lie symmetries, Exact solutions, Elliptic solutions
  • Istanbul Technical University Affiliated: Yes


© 2021, The Author(s), under exclusive licence to Springer Nature B.V.We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for a one-dimensional lattice model considering higher order effects, and later it was re-derived in the context of nonlinear nonlocal elasticity. In the sense of one-dimensional wave propagation in solids, the nonlinearity function of the equation is connected with the stress-strain relation of the physical model. Considering a general nonlinearity, we determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. We find that the maximal dimension of the symmetry algebra is four, which is realized when the nonlinearity assumes some special canonical form. After that we perform reductions to ordinary differential equations. In case of a quadratic nonlinearity, we provide several exact solutions, some of which are in terms of elliptic functions.