Subsonic phase transition waves in bistable lattice models with small spinodal region (Q2870582)

The authors study phase transition waves in atomic chains NEWLINE\[NEWLINE \ddot u_j(t) = \Phi'(u_{j + 1}(t) - u_j(t)) - \Phi'(u_j(t) - u_{j - 1}(t)) NEWLINE\]NEWLINE with \(\Phi(x) = \Phi_{\delta}(x)\), that is a perturbation of the bi-quadratic potential: \(\Phi_{\delta}(x) = \frac{1}{2}r^2 - \Psi_{\delta}(x)\). Existence and uniqueness of transition waves, satisfying some additional conditions, are proved for sufficiently small \(\delta > 0\). The proof is based on a fixed point theorem. The authors show that the perturbation of the wave is a fixed point of a nonlinear and nonlocal operator, which is contractive on a small ball in a suitable Banach space. In conclusion, the kinetic relations are discussed.