In this work, we consider varying aspects of the stability of periodic traveling wave solutions to nonlinear dispersive equations. In particular, we are interested in deriving universal geometric criterion for the stability of particular third order nonlinear dispersive PDE's. We begin by studying the spectral stability of such solutions to the generalized Korteweg-de Vries (gKdV) equation. Using the integrable structure of the ODE governing the traveling wave solutions of the gKdV, we are able to determine the role of the null-directions of the linearized operator in the stability of the traveling wave to perturbations of long-wavelength by conducting what amounts to a rigorous Whitham theory calculation. By then considering the characteristic polynomial of the monodromy map (the periodic Evans function) in a neighborhood of the origin in the spectral plane, we derive two separate instability indices. The first is a modulational stability index which, assuming a particular non-degeneracy condition holds, determines a rigorous normal form of the spectrum in the neighborhood of the origin, and yields necessary and sufficient criterion for the traveling wave to be modulationally stable. The second is an orientation index which counts modulo 2 the total number of periodic eigenvalues of the linearized operator with the positive real axis. This is essentially a generalization of the solitary wave stability index. Both of these indices are expressible in terms of a map between a parameter space which parameterizes the periodic traveling waves of the gKdV to the conserved quantities of the governing PDE. Moreover, we show how our general methods can be used to derive transverse-modulational instability indices, i.e., in analyzing the stability of such solutions to long-wavelength transverse perturbations in higher dimensional equations.
We also study the nonlinear stability of periodic traveling wave solutions of the gKdV within the class of solutions having the same period. In particular, by conducting a detailed analysis of the Hamiltonian system satisfied by the traveling wave profile, we prove that in many cases the periodic spectral instability index mentioned above determines the orbital stability of the underlying traveling wave. However, the signs of two other indices play a role in our analysis, neither of which are present when one considers exponentially decaying solutions: this stands in stark contrast to the solitary wave theory, where such a solution is nonlinearly stable if and only if it is spectrally stable (assuming a particular non-degeneracy condition holds).
Finally, we show how our results extend to other classes of dispersive equations. In particular, we derive modulational and finite-wavelength instability indices for the generalized Benjamin-Bona-Mahony (gBBM) equation, as well as the generalized Camassa-Holm equation. Moreover, we prove a transverse instability result for the gBBM equation analogous to that for the gKdV.
|Advisor:||Bronski, Jared C.|
|School:||University of Illinois at Urbana-Champaign|
|School Location:||United States -- Illinois|
|Source:||DAI-B 70/06, Dissertation Abstracts International|
|Keywords:||Generalized Korteweg-de Vries equation, Nonlinear dispersive equations, Periodic waves|
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