Mikhail Lebedev
Publications:
Alfimov G. L., Lebedev M. E.
Complete Description of Bounded Solutions for a DuffingType Equation with a Periodic Piecewise Constant Coefficient
2023, Vol. 19, no. 4, pp. 473506
Abstract
We consider the equation $u_{xx}^{}u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $u(x)<\xi$, can be put in onetoone correspondence with biinfinite sequences of numbers $n\in \{N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the mapoverperiod (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.
