Hyperbolic travelling fronts

Source abstract

Since the work of R. A. Fisher [ 2 ] and Kolmogorov, Petrovskij and Piskunov [ 7 ] (see [ 6 ] for further references) the problem of travelling fronts in reaction–diffusion equations has been extensively studied. For the equation with F (0) = F (1) = 0 a travelling front is a solution where the function of one variable φ is decreasing and satisfies φ(−∞) = 1, φ(+∞) = 0. The function φ describes the shape of the front and the constant c is the speed of propagation. There are two main types of the problem. In the all-positive case, where the function F satisfies there is a half-line [ c 0 , ∞), c 0 >0, of speeds. For each c ∈[ c 0 , ∞) there is, up to translation, a unique travelling front. Fronts for different c can be distinguished by the rate of decay towards +∞. In the threshold case, where F has the property, for some λ∈(0, 1), there is a unique speed c 0 with a travelling front, which is unique up to translation. In this case the sign of c 0 is determined by the sign of the integral