Instability and stochastic analyses of a pad-on-disc frictional system in moving interactions

In this paper, we establish a new nonlinear equation which is called the two-mode Korteweg–de Vries–Burgers equation (TMKdV–BE). The new equation describes the propagation of two different wave modes simultaneously. First, we introduce a new Cole–Hopf transformation needed for the modified simplified bilinear method to find necessary conditions on the solution of TMKdV–BE. Also, a finite series in terms of tanh–coth function is presented as an alternative method to study the solution of the equation. Finally, the reliability of the obtained results is discussed in the last section.