Travelling Waves in Continuum/discrete Bistable/monostable Dynamics

Traveling waves play a key role in mathematical models since they characterize observable fundamental phenomena in nature. A typical traveling wave corresponds to a state which is time independent (in some cases time periodic) in a moving coordinates. In a one space dimensional setting, it depends only on z := x − ct where c is the wave speed and x and t are respectively space and time variables. Consider an autonomous dynamical system u̇ = A[u] where u : t ∈ [0,∞) → u(t) = u(t; ·) ∈ C(R) and A : φ(·) ∈ C(R) → A[φ](·) ∈ C(R) is an operator. Here autonomous means that A is translation invariant; namely, for any h ∈ R and φ in the definition domain of A, with φ(·) := φ(·+ h), A[φ](·) = A[φ](·+ h). (1.1) The following are examples of translation invariant operators: A1[φ](x) := φxx(x) + f(φ(x)), A2[φ](x) := φ(x+ 1) + φ(x− 1)− 2φ(x) + f(φ(x)), A3[φ](x) := J ∗ φ(x)− b φ(x) + f(φ(x)), where f : R→ R is a given smooth function and J ∗ φ(x) := ∫