In this paper, specific functional operators, referred to as two-mode operators, are utilized to extend the \((1+1)\) -dimensional Chaffee-Infante model into a generalized second-order evolutionary partial differential equation in the time coordinate. This extended model, termed the two-mode Chaffee-Infante model, characterizes the motion of two synchronized symmetric waves propagating under the influence of three embedded parameters: dispersion, nonlinearity, and phase velocity. Two effective methods, the extended tanh(coth)-expansion method and the sine(cosine)-function method, are employed to derive several traveling solutions for the proposed model. Additionally, the impact of other parameters on the propagation behavior of the TMCI is explored. It is believed that the findings in this paper will offer valuable insights into the study of nonlinear models of second-order in the time-coordinate.
