The weakly nonlinear dynamics of fast magnetosonic waves ducted by a region of density enhancement in a cold plasma is considered. A self-consistent 2D quasi-hyperbolic equation with quadratic and cubic nonlinearities is derived for perturbations of the transverse component of the plasma velocity. For a wave package with a narrow spectrum, the governing equation is reduced to the nonlinear Schr?dinger equation, whose structure is independent of the detailed density profile provided that it has a maximum (to be a potential well for the waves). The derived equation allows us to take into account the effect of the shape and steepness of the density profile on the wave dynamics.
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