Mathew A. Johnson

NONLINEAR STABILITY OF PERIODIC TRAVELING WAVE SOLUTIONS OF THE GENERALIZED KORTEWEG-DE VRIES EQUATION

In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg–de Vries equation

The Modulational Instability for a Generalized Korteweg-de Vries Equation

u_t=u_{xxx}+f(u)_x

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation

. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solutions. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case when

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

f(u)=u^2

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

(the Korteweg–de Vries equation) and in neighborhoods of the homoclinic and equilibrium solutions if

Nonlinear Stability of Viscous Roll Waves

f(u)=u^{p+1}

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

for some

Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

p\geq1

Stability of Small Periodic Waves in Fractional KdV-Type Equations

.

Convergence of Hill's Method for Nonselfadjoint Operators

We study the spectral stability of a family of periodic standing wave solutions to the generalized Korteweg–de Vries in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evans function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first, an orientation index, counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second, a modulational instability index, provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Hǎrǎguş and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator—in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests a geometric interpretation attached to the vanishing of the orientation index previously mentioned.