An index theorem for the stability of periodic traveling waves of KdV type

There has been a large amount of work aimed at understanding the stability of nonlinear dispersive equations that support solitary wave solutions[20, 7, 3, 6, 19, 29, 28, 30, 35, 36]. Much of this work relies on understanding detailed properties of the spectrum of the operator obtained by linearizing the flow around the solitary wave. These spectral properties, in turn, have important implications for the long-time behavior of solutions to the corresponding partial differential equation[10, 5, 13, 17, 18, 25, 24, 26, 27, 14, 32, 34, 33]. In this paper we consider periodic solutions to equations of KortewegDevries type. While the stability theory for periodic waves has received much some attention[1, 8, 2, 9, 15, 16, 21, 12] the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equation and the conserved quantities of the partial differential equation. Department of Mathematics, University of Illinois, 1409 W. Green St. Urbana, IL 61801 USA Department of Mathematics, Indiana University,831 East 3rd St, Bloomington, IN 47405 USA Department of Mathematics and Statistics, Calvin College, 1740 Knollcrest Circle SE, Grand Rapids, MI 49546 USA 1