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January & February  2004, 10(1&2): 1-19. doi: 10.3934/dcds.2004.10.1

Preservation of spatial patterns by a hyperbolic equation

A. V. Babin 1,

1. 

Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States

Received  December 2001 Revised  November 2002 Published  October 2003

Oscillations in a nonlinear, strongly spatially inhomogenious media are described by scalar semilinear hyperbolic equations with coefficients strongly depending on spatial variables. The spatial patterns of solutions may be very complex, we describe them in terms of binary lattice functions and show that the patterns are preserved by the dynamics. We prove that even when solutions of the equations are not unique, the large-scale spatial patterns of solutions are preserved. We consider arbitrary large spatial domains and show that the number of distinct invariant domains in the function space depends exponentially on the volume of the domain.
Keywords: Complex spatial patterns, invariant domains, nonunique solutions, hyperbolic equation, homotopy invariants, multiple scales..
Mathematics Subject Classification: 35L70, 35B05, 58.
Citation: A. V. Babin. Preservation of spatial patterns by a hyperbolic equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 1-19. doi: 10.3934/dcds.2004.10.1
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