The nonlinear propagation of the Alfvén and magnetosonic waves in the solar corona is investigated in terms of model equations. Due to viscous effects taken into account the propagation of the Alfvén wave itself is governed by a Burgers-type equation. The Alfvén waves exhibit a tendency to drive both the slow and fast magnetosonic waves. For this process model equations are a generalization of the Zakharov equations. The propagation of the magnetosonic waves is described by linearized Boussinesq-type equations with ponderomotive terms due to the Alfvén wave. Both long and short Alfvén waves are considered. Also the limits of the slow and fast modes are investigated. An approximate shock wave solution has been found for a vertically propagating slow mode. Numerical results for the fast mode propagating perpendicular to the magnetic field show the effect of inhomogeneity and pumping on a shock as the solution of the homogeneous Burgers equation.
A new model equation governing the propagation of nonlinear pulses in optical fibres has been derived on the assumption of a saturated nonlinearity of the refractive index. This equation is a combination of the exponential nonlinear Schrödinger equation and the derivative one. It is valid for the long fibres. A modulational stability has been calculated to find out a cut-off in an angular frequency of a carrier wave. Moreover, it has been shown that the equation possesses family of stationary solutions. An initial value problem has been discussed on the basis of the implicit pseudo-spectral scheme.
Modulational instability of a plane wave of the nonlinear Schrödinger equation is discussed numerically on the basis of the pseudo-spectral method. The linear theory is verified and influence of the attenuation is considered.
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