Nonlinear Sturm oscillation: From the interval to a star

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

“…n, (m), the spectral curves never cross (see Figure 3.1) and we obtain the first nonlinear generalization of Sturm's oscillation theorem as a corollary (see also Theorem 2.4 in [16]). Corollary 3.3.…”

“…The NLS Equation (1.1) on an interval of length with Dirichlet boundary conditions has a one-parameter family of real-valued solutions with n nodal domains, for each n ∈ N. The relation between the spectral parameter µ = k 2 and the deformation parameter m is dictated by Equation (3.4) and may be explicitly written as k (m) (or its implicitly defined inverse m (±) n, (k)) as spectral curves. As k(±) n+1, (m) > k (±)n, (m), the spectral curves never cross (seeFigure 3.1) and we obtain the first nonlinear generalization of Sturm's oscillation theorem as a corollary (see also Theorem 2.4 in[16]Spectral curves k (±) n, (m) for the repulsive (a) and attractive case (b). The n-th curve is obtained from the curve for n = 1 by rescaling k…”

“…We refer to [14,15] where a detailed overview of the recent literature and some applications is given and just summarise here the relevant work related to the nodal counting. In a previous work some of us have shown that Sturm's oscillation theorem is generically broken for nonlinear quantum stars, apart from the special case of an interval [16]. This is not unexpected as the set of solutions is known to have a far more complex structure.…”

“…All statements in this proposition follow straight-forwardly from the known properties of elliptic integrals and elliptic functions and we thus omit the proof here. Furthermore, some of the statements in the proposition are mentioned explicitly in [16,14,17] and others follow easily from the definitions as given in the Appendix A.…”