Non-autonomous for Modified Fifth-Order Korteweg-de Vries Equation with Variable Coefficients, Breather, and Soliton

In this paper, an investigation of a damped variable-coefficient fifth-order modified Korteweg–de Vries equation by using Hirota bilinear approach for N-soliton solution and breather. The characteristic line method is used to examine the propagation and collision of soliton. With variable coefficients (t), (t) and (t) are the dispersive, dissipative and line-damping coefficients, respectively, and discussed the effect of a variable coefficient in graphically and analytically. The amplitude of soliton is affected by \(\gamma (t)\) . . In breather α (t) and β(t) affect the central path. The soliton’s amplitude is inverse variation to line damping. For breather, the amplitude of solitons decreases as the positive coefficient of the (t) increases. The α(t) and β(t) are not only influenced by the position of the breather but also the period of the breather.