Dynamics of unstable solutions for the wave equation with sources

Two new accurate solutions of the wave equation with sources are obtained. The dynamics of unstable states described by these solutions is studied. Analytical forms are given for the partial derivatives of the required function with respect to the spatial coordinate and time on the plane of independent variables “the required function–time”. This structure of the solution allows us to consider nonstationary analogs of self-similar kinks describing the transition between two equilibrium states of the “medium–source” system. For the classical wave equation, a nonlinear rheonomic source is used, the behavior of which affects the properties of the relaxing kink. The conditions under which the speed of movement of the formed self-similar transfer wave is subsonic or supersonic are determined. An important role of the velocity of the inflection point of a nonselfsimilar kink has been analyzed; the threshold value of the velocity is calculated, which separates the subsonic and supersonic regimes. An unstable version of the presented solution gives a strong discontinuity of the required function with an unlimited increase in time. The stopping of the kink inflection point is an indicator of a strong rupture. An estimate of the value of the moment in time preceding the beginning of the return motion of the inflection point is indicated.
A solution to a spatially nonlocal fourth-order wave equation with two additively entering sources is given. One source depends on the desired function in a linear homogeneous way; the second one depends on the modulus of the gradient of the desired function the same way. The solution is an analog of an overthrow wave in an interval with non-stationary boundaries. At each finite moment of time this solution is continuous, and for an infinite time there is a loss of smoothness of the solution, we have the so-called “slow explosion”. In the unstable solution, the isolines of the sought-for function on the concave section (the lower part of the kink) move towards the convex section, which is adjacent to the upper boundary of the kink. In the stable version, the kink degenerates into a homogeneous state. It has been analyzed that for a nonselfsimilar process, the inversion of the sign of the gradient source gives an inversion of the stability conditions for the kink and antikink. An unstable kink/ antikink corresponds to a gradient sink/source.