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We discuss the spread of a piece of news in a population. This is modeled bySIR model of epidemic spread. The model can be reduced to a nonlineardifferential equation for the number of people affected by the news ofinterest. The differential equation has an exponential nonlinearity and it canbe approximated by a sequence of nonlinear differential equations withpolynomial nonlinearities. Exact solutions to these equations can be obtainedby the Simple Equations Method (SEsM). Some of these exact solutions can beused to model a class of waves associated with the spread of the news in apopulation. The presence of exact solutions allow to study in detail thedependence of the amplitude and the time horizon of the news waves on the waveparameters such as the size of the population, initial number of spreaders ofthe piece of the news, transmission rate and recovery rate. This allows forrecommendations about the change of wave parameters in order to achieve a largeamplitude or appropriate time horizon of the news wave. We discuss 5 types ofnews waves on the basis of the values of the transmission rate and recoveryrate - the types A,B,C,D and E of news waves. In addition, we discuss thepossibility of building wavetrains by news waves. There are three possiblekinds of wavetrains with respect of the amplitude of the wave: increasingwavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrainis especially interesting as it is connected to an increasing amplitude of thenews wave with respect to the amplitude of the previous wave of the wavetrain.It can find applications in advertising, propaganda, etc.

On the mathematical theory of news waves