Global structure of travelling pulse solutions in some reaction-diffusion systems

Global structure of travelling pulse solutions of the problem epsilon tau u(t) = epsilon(2)u(xx) + f(u, v), v(t) = v(xx) + g(u, v) is considered. epsilon(> 0) is a sufficiently small parameter and tau is a positive one. First, it is shown that there exist two types of destabilization of standing pulse solutions when tau decreases. One is the appearance of travelling pulse solutions via the static bifurcation and the other is that of in-phase breathers via the Hopf bifurcation. Second, when f and g are piecewise linear nonlinearities, which type of destabilization occurs first with decreasing tau is discussed. Moreover we can trace the branches of these bifurcated solutions globally. Travelling breathers, which bifurcate from the branch of travelling pulse solutions via the Hopf bifurcation, play an important role.