KdV cnoidal waves are spectrally stable
Nate Bottman Bernard Deconinck
Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solutions of the Korteweg-deVries equation, the so-called cnoidal waves. In this paper, we exploit the squared-eigenfunction connection between the linear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrum of the linear stability problem for perturbations that are bounded on the real line. We find that this spectrum is confined to the imaginary axis, leading to the conclusion of spectral stability. An additional argument allows us to conclude the completeness of the associated eigenfunctions.
keywords: KdV equation stability. cnoidal waves
Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation
Vishal Vasan Bernard Deconinck
A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary-value problem is found to be ill posed.
keywords: PDEs with mixed derivatives Fokas method.
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
Deconinck Bernard Olga Trichtchenko

Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation.

keywords: Instability Hamiltonian partial differential equations Krein signature

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