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 Schriidinger equation on an interval satisfy a nonlinear version of the Sturm oscillation property. However, we show that unlike for the linear theory, the nonlinear version of Sturm oscillation breaks down already for a star graph. We point out conditions under which this violation can be assured.