2017/06/18 by L. Miguel Rodrigues, Rodrigues, L. Miguel · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Gyrotron and Vacuum Electronics Research
paper · pdf · doi:10.48550/arxiv.1706.05752
We provide a detailed study of the dynamics obtained by linearizing the\nKorteweg-de Vries equation about one of its periodic traveling waves, a cnoidal\nwave. In a suitable sense, linearly analogous to space-modulated stability, we\nprove global-in-time bounded stability in any Sobolev space, and asymptotic\nstability of dispersive type. Furthermore, we provide both a leading-order\ndescription of the dynamics in terms of slow modulation of local parameters and\nasymptotic modulation systems and effective initial data for the evolution of\nthose parameters. This requires a global-in-time study of the dynamics\ngenerated by a non normal operator with non constant coefficients. On the road\nwe also prove estimates on oscillatory integrals particularly suitable to\nderive large-time asymptotic systems that could be of some general interest.\n