SUPPLEMENTAL MATERIAL for : Unidirectional Transition Waves in Bistable Lattices

where L is the lattice parameter and p < −1. We can therefore deduce the following relations: 1. The mass density should be a constant: m/L = constant ⇒ m = O(L). 2. The energy density of the nonlinear spring should be constant: − A p+1 (un+1 − un + L) p+1 /L = − A p+1 (L + L) p+1 /L = constant ⇒ A = O(L−p) where ∼ O(1). 3. The energy density of the bistable function should be constant: βφ(un)/L = constant ⇒ β = O(L). 4. The dissipation potential density should be a constant: 12αu 2 n,t/L = constant ⇒ α = O(L). Assuming a traveling wave solution of the form, un(t) = u(nL− vt) = u(ξ) gives mvuξξ +A (u(ξ + L)− u(ξ) + L) −A (u(ξ)− u(ξ − L) + L) − vαuξ + βφ′(u) = 0. (2) Using appropriate Taylor expansions for u(ξ + L) and u(ξ − L) results in mvuξξ +AL p ( 1 + uξ + L 2 uξξ + ... )p −AL (1 + uξ − L2 uξξ + ...)p − vαuξ + βφ′(u) = 0. (3)