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Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions, strong symmetry operators and hereditary operators, we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs. For the first order Hopf equation, there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions. The general solution of the first order Hopf equation is obtained via hodograph transformation. For the second order Hopf equation, the Hopf-diffusion equation, there are five sets of infinitely many symmetries. Especially, there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation. Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.