Nonlinear Sturm Oscillation: from the interval to a star

The Sturm oscillation property, i.e. that the $n$-th eigenfunction of a Sturm-Liouville operator on an interval has $n-1$ zeros (nodes), has been well studied. This result is known to hold when the interval is replaced by a metric (quantum) tree graph. We prove that the solutions of the real stationary nonlinear Schr"odinger equation on an interval satisfy a nonlinear version of the Sturm oscillation property. However, we show that unlike the linear theory, the nonlinear version of the Sturm oscillation breaks down already for a star graph. We point out conditions under which this violation can be assured.