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The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions.

We establish the existence of traveling wave solution for a reaction-diffusion predator-prey system with Holling type-IV functional response. For simplicity, only one space dimension will be involved, the traveling solution equivalent to the heteroclinic orbits inR3. The methods used to prove the result are the shooting argument and the invariant manifold theory.

With the help of the bifurcation theory of dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions and bifurcations of the (2 + 1)-dimensional dissipative long wave equation. The study of travelling wave solutions for long wave equation derives a planar Hamiltonian system. Based on phase portraits, we obtain exact explicit expressions of some bounded traveling wave solutions and some important singular traveling wave solutions, under different parametric conditions.