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New types of stable nonlinear whistler waveguides
International audienceThe stationary self-focusing of whistler waves with frequencies near half of the electron-cyclotron frequency in the ionospheric plasma is considered in the framework of a two-dimensional generalized nonlinear Schrödinger equation including fourth-order dispersion effects and nonlinearity saturation. New types of soliton-like (with zero topological charge) and vortex-like nonlinear waveguides are found, and their stability confirmed both analytically and numerically
On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion
This paper deals with the Cauchy problem associated to the nonlinear
fourth-order Schr\"odinger equation with isotropic and anisotropic mixed
dispersion. This model is given by the equation where represents either the operator (isotropic
dispersion) or (anisotropic
dispersion), and are given real parameters. We
obtain local and global well-posedness results in spaces of initial data with
low regularity, such as weak- spaces. Our analysis also includes the
biharmonic and anisotropic biharmonic equation for which, the
existence of self-similar solutions is obtained as consequence of his scaling
invariance. In a second part, we investigate the vanishing second order
dispersion limit in the framework of weak- spaces. We also analyze the
convergence of the solutions for the nonlinear fourth-order Schr\"odinger
equation , as goes to zero, in -norm, to the solutions of the
corresponding biharmonic equation
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