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Motivated by an exact mapping between equilibrium properties of a one-dimensional chain of quantum Ising spins in a transverse field (the transverse field Ising (TFI) model) and a two-dimensional classical array of particles in double-well potentials (the “ 4 model”) with weak inter-chain coupling, we explore connections between the driven variants of the two systems. We argue that coupling between the fundamen- tal topological solitary waves in the form of kinks between neighboring chains in the classical  4 system is the analog of the competing effect of the transverse field on spin flips in the quantum TFI model. As an example application, we mimic simplified measurement protocols in a closed quantum model system by studying the classical  phi 4 model subjected to periodic perturbations. This reveals memory/loss of mem- ory and coherence/decoherence regimes, whose quantum analogs are essential in annealing phenomena. In particular, we examine regimes where the topological excitations control the thermal equilibration following perturbations. This paves the way for further explorations of the analogy between lower-dimensional linear quantum and higher-dimensional classical nonlinear systems.