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January  2008, 20(1): 139-158. doi: 10.3934/dcds.2008.20.139

Relative Morse indices, Fredholm indices, and group velocities

Björn Sandstede 1, and  Arnd Scheel 2,

1. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom
2. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  December 2006 Revised  May 2007 Published  October 2007

We discuss Fredholm properties of the linearization of partial differential equations on cylindrical domains about travelling and modulated waves. We show that the Fredholm index of each such linearization is given by a relative Morse index which depends only on the asymptotic coefficients. Several strategies are outlined that help to compute relative Morse indices using crossing numbers of spatial eigenvalues, and the results are applied to prove the existence of small viscous shock waves in hyperbolic conservation laws upon adding small localized time-periodic source terms.
Keywords: Morse index, spatial dynamics, group velocity., Fredholm operator.
Mathematics Subject Classification: Primary: 35K57, 35B32, 35B3.
Citation: Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139
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