Monotonous property of non-oscillations of the damped Duffing's equation (Q813662)

This paper deals with the monotony property of the bounded nonoscillations for the damped Duffing equation \[ \ddot x+\delta\dot x-\mu x+ x^3= 0, \] where \(\delta\) is the damping coefficient. The author is interested in establishing analytic results. It is shown that under suitable assumptions on \(\delta\) and \(\mu\), the only two nontrivial bounded exact solutions are monotone: one is monotone decreasing and the other one is monotone increasing. An explicit exact solution is obtained by a direct method. This approach is used to obtain travelling waves solutions of reaction-diffusion equations.