In this paper, we primarily investigate Lie symmetry analysis and exact solutions for the coupled integrable dispersionless equations. First of all, based on the Lie symmetry analysis, an optimal system of one dimensional subalgebras is constructed. Furthermore, similarity reductions and group invariant solutions are given. Next, exact solutions of the reduced equations have been derived by the method of power series. Finally, by means of Ibragimov’s method, conservation laws are obtained.

In this paper, based on the classical symmetry method, the group-invariant solutions of the evolution equation of a hyperbolic curve flow are investigated. The optimal system of the obtained symmetries is found, and the reduced equations and exact solutions of the evolution equation are discussed. Then explicit solutions are obtained by the power series method. In addition, the convergence of the power series solutions is proved. Theobjective shapes of the solutions of the evolution equation are performed.

In this article, we take into account the (1 + 1)-dimensional dispersive long-wave equation which is reduced from the Wu–Zhang equation. Firstly, the Bäcklund transformation and the non-local symmetry are successfully acquired from the truncated Painlevé expansion. At the same time, the non-local symmetry is transformed to Lie point symmetry by a suitable prolonged system. Then some solitary solutions are derived without a hitch via new Bäcklund transformation which originates from Lie point symmetry. Secondly, by utilising the consistent Riccati expansion method and the consistent tanh-expansion method, we find the interaction solutions between soliton and cnoidal wave by using the Jacobi elliptic function. Lastly, the conservation laws which are related to symmetries of the equation are successfully obtained by Ibragimov’s method.