Abstract
The Korteweg–de Vries equation with an additional term of Landau damping is numerically and analytically investigated. It is shown that the equation has a shock-like solution for an initial ramp signal. The temporal evolution of waveforms with various magnitudes of the Landau damping is studied for several values of the initial amplitude. Dependences of widths and velocities of the leading part on initial conditions are shown. It is found that a steepening is suppressed due to the Landau damping even when its coefficient is two orders less than those of nonlinear and dispersive terms. There is a critical relation for such a steepening to take place for a fixed height of the initial ramp. An analytical estimate of the magnitude of temporal Landau damping is given for a linear sinusoidal wave.
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