Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Preservation of spatial patterns by a hyperbolic equation

Pages: 1 - 19, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.1

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A. V. Babin - Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States (email)

Abstract: 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, hyperbolic equation, invariant domains,homotopy invariants, nonunique solutions, multiple scales.
Mathematics Subject Classification:  35L70, 35B05, 58E.

Received: December 2001;      Revised: November 2002;      Available Online: October 2003.