2015/05/14 by Bernard Deconinck, Deconinck, Bernard, Olga Trichtchenko +1
Physics and Astronomy · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1505.03688
Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition\nfor the presence of high-frequency ( i.e., not modulational) instabilities of\nsmall-amplitude periodic solutions of Hamiltonian partial differential\nequations is presented, entirely in terms of the Hamiltonian of the linearized\nproblem. With the exception of a Krein signature calculation, the theory is\ncompletely phrased in terms of the dispersion relation of the linear problem.\nThe general theory changes as the Poisson structure of the Hamiltonian partial\ndifferential equation is changed. Two important cases of such Poisson\nstructures are worked out in full generality. An example not fitting these two\nimportant cases is presented as well, using a candidate Boussinesq-Whitham\nequation.\n