On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$

Abstract

The paper is devoted to the equation $u_{xx}+Q(x)u+P(x)u^3=0$. The equations of such kind have been used to describe stationary modes in the models of Bose–Einstein condensate. It is known that under some conditions for $P(x)$ and $Q(x)$, the “most part” of solutions for such equations are singular, i.e. tend to infinity at some point of the real axis. In some situations this fact allows us to apply the methods of symbolic dynamics to describe non-singular solutions of this equation and to construct comprehensive classification of these solutions. In the paper we present (i) necessary conditions for existence of singular solutions as well as conditions for their absence; (ii) the results of numerical study of the case when $Q(x)$ is a constant and $P(x)$ is an alternate periodic function. Basing on these results, we formulate a conjecture that all the non-singular solutions of the equation can be coded by bi-infinite sequences of symbols of a countable alphabet.