Abstract
We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.
Highlights
Sturm’s oscillation theorem [1] is a classic example for how solutions of linear self-adjoint differential eigenvalue problems Dφ( x ) = λφ( x ) are ordered and classified by the number of nodal points
With this more detailed description we show that a much larger set of nodal structures is possible in nonlinear quantum star graphs compared to the linear case, as is stated
N=1 of positive real numbers, increasingly ordered, such that φn = χm,k n is a solution of the nonlinear Schrödinger (NLS) equation on the interval [0, `] with spectral parameters μn = k2n and n is the number of nodal domains
Summary
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. Our main result here is that the nonlinear case of a metric star allows for solutions with any given number of nodal domains on each edge.
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