Co-periodic stability of periodic waves in some Hamiltonian PDEs (Q2834290)

The authors address stability of periodic traveling wave solutions to dispersive PDEs with respect to perturbations of the same period as the wave. Stability criteria are derived and investigated in a general abstract framework with applications to three basic examples. The examples include a quasilinear version of the generalized Korteweg-de Vries equation, the Euler-Korteweg system in Eulerian coordinates, and the same system in mass Lagrangian coordinates. In particular, the authors derive a necessary condition for spectral stability and a sufficient condition for orbital stability of periodic waves. Both conditions are expressed in terms of a single function, which an action integral along the orbits of periodic waves in the phase plane. An odd value for the difference between the number of negative eigenvalues of the Hessian of the action and the number of negative eigenvalues of the energy function computed at the orbits of periodic waves implies spectral instability, while the zero value for the difference implies orbital stability. These stability criteria can be checked numerically for many dispersive PDEs. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for the three basic examples.