Static and Dynamic Patterns in Three-Dimensional Equilibrium and Nonequilibrium Systems
Time-dependent solutions to a class of nonlinear partial differential equations in three spatial dimensions are obtained analytically. As examples, explicit solutions to the nonlinear diffusion equation, the nonlinear Klein — Gordon equation with dissipation and coupled reaction-diffusion equations are presented. The solutions obtained have a form of vortex rings, rotating and travelling spirals and localized solutions with Cn symmetry. Pattern evolution in diffusion limited and rate limited situations in systems undergoing phase transitions is discussed. In diffusion limited cases spiral solutions with dislocations can be formed in two-dimensional systems. The solutions in three spatial dimensions are discussed in connection with observed dislocations and three-dimensional helical crystal aggregates and spiral and other patterns observed in chemical systems. For thermodynamic systems undergoing phase transitions, localized nonhomogeneous structures for which the amplitude decays like r−1/2 at the critical point and like r−1/4 at the tricritical point are obtained.
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