Nonlinear Sturm Oscillation: from the interval to a star

TL;DR

This paper extends the classical Sturm oscillation property to nonlinear Schrödinger equations on intervals but reveals its failure on star graphs, highlighting the impact of nonlinearity and graph topology.

Contribution

It introduces a nonlinear version of the Sturm oscillation property and demonstrates its breakdown on star graphs, contrasting with linear cases.

Findings

Nonlinear Schrödinger solutions on intervals satisfy a Sturm oscillation property.

The nonlinear Sturm oscillation property fails on star graphs.

Conditions for the violation of the property are identified.

Abstract

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.