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Traveling waves of bistable dynamics on a lattice. (English) Zbl 1050.37041
Summary: We prove the existence of stationary or traveling waves in a lattice dynamical system arising in the theory of binary phase transitions. The system allows infinite-range couplings with positive and negative weights. The allowance for negative coupling coefficients precludes the possibility of a maximum principle. Instead, a weakened type of ellipticity is stipulated that is used with spectral theory in a perturbative fixed point argument to construct a traveling wave when the nonlinearity is unbalanced and the coupling is sufficiently strong. When the nonlinearity is balanced, a variational technique is used to obtain stationary waves, which are then analyzed in more detail for strong couplings. From a physical perspective these models are important since long-range and indefinite interactions occur in nature and can lead to pattern formation. Our results provide conditions under which patterned states tend to be swept away by traveling waves even when the interaction is of excitatory-inhibitory type. The results also have implications for the numerical analysis of spatially discretized reaction-diffusion equations, where it is important to know whether solutions to the discretized equations converge to solutions to the continuum equation as the mesh size tends to zero.
37L60Lattice dynamics (infinite-dimensional dissipative systems)
34K20Stability theory of functional-differential equations
34K30Functional-differential equations in abstract spaces
34K60Qualitative investigation and simulation of models
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B26Phase transitions (general)
35K99Parabolic equations and systems