Periodic solutions of a singular equation with indefinite weight. (English) Zbl 1232.34064

The authors study the existence and uniqueness of \(T\)-periodic solutions for the equation \[ x''= \frac{a(t)}{x^3}, \] where \(a\) is a \(T\)-periodic function given by \[ a(t) = a_+ \;\;\text{if} \;0 \leq t < t_+, \;\;a(t) = -a_- \;\;\text{if} \;t_+ \leq t < T \] with \(a_+,a_- > 0.\) These problems arise in different physical situations such as in the stabilization of matter-wave breathers in Bose-Einstein condensates, in the propagation of guided waves in optical fibers and in the electromagnetic trapping of a neutral atom near a charged wire. If the parameters \(a_+, a_-\) are fixed, and \(T := t_+ + t_-,\) an interesting question is how to control the switching times \(t_-,t_+\) in order to get periodic states with a particular amplitude. This question is studied in the paper as well as the stability properties (in the linear sense) of the \(T\)-periodic solutions.


34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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