Symmetric homoclinic solutions to the periodic orbits in the Michelson system
In the paper, the Michelson system \[ x'''+x'+\frac{1}{2}x^2=c^2 \tag{1} \] for the parameter value \(c=1\) is studied. Equation (1) arises as the equation for steady state or traveling wave solutions of the Kuramoto-Sivashinsky partial differential equation. In \textit{W.~C.~Troy} [J. Differ. Equations 82, No. 2, 269--313 (1989; Zbl 0693.34053)], it has been proven that equation (1) possesses two odd periodic solutions \(S_1\) and \(S_2\). The paper presents the following results. Theorem 1. Equation (1) with the parameter value \(c=1\) possesses infinitely many heteroclinic solutions connecting the periodic orbits \(S_1\) and \(S_2\) in both directions. Theorem 2. Equation (1) with the parameter value \(c=1\) possesses infinitely many homoclinic solutions both to the periodic orbits \(S_1\) and \(S_2\). Moreover, both families of such homoclinic solutions contain a countable set of odd homoclinic solutions.
Kuramoto-Sivashinsky partial differential equation
scientific article; zbMATH DE number 5077088