On solitary waves in case of amplitude-dependent nonlinearity

The aim of the present paper is to study soliton emergence, modelled by a Boussinesq-type equation with nonstandard nonlinear terms. Such a model has been proposed to describe mechanical waves in cylindrical biomembranes. While the governing equation is of the Boussinesq-type, the nonlinearity is of the nonstandard f(u)·uxx-type instead of the conventional f(ux)·uxx-type and the dispersion can be either normal or anomalous. It is shown that the ratio between the two nonlinear parameters can have significant impact on the solution behaviour and it is shown how the dispersion related parameters affect the evolution of solutions including the demonstration of cases where smaller amplitude waves travel faster than the larger amplitude waves.