Traveling waves in a profile of phase field: exact analytical solutions of a hyperbolic Allen–Cahn equation

To obtain solutions of the hyperbolic Allen–Cahn equation, the first integral method, which follows from well-known Hilbert Null-theorem, is used. Exact analytical solutions are obtained in a form of traveling waves, which define complete class of the hyperbolic Allen–Cahn equation. It is shown that two subclasses of solutions exist within this complete class. The first subclass exhibits continual solutions and the second subclass is represented by solutions with singularity at the origin of coordinate system. Such non-uniqueness of solutions stands a question about stable attractor, i. e., about the traveling wave to which non-stationary solutions may attract. The obtained solutions include earlier solutions for the parabolic Allen–Cahn equation in a form of finite number of $\tanh$-functions.