Drift of Spectrally Stable Shifted States on Star Graphs

Abstract

When the coefficients of the cubic terms match the coefficients in the boundary conditions at a vertex of a star graph and satisfy a certain constraint, the nonlinear Schrödinger (NLS) equation on the star graph can be transformed to the NLS equation on a real line. Such balanced star graphs have appeared in the context of reflectionless transmission of solitary waves. Steady states on such balanced star graphs can be translated along the edges with a translational parameter and are referred to as the shifted states. When the star graph has exactly one incoming edge and several outgoing edges, the steady states are spectrally stable if their monotonic tails are located on the outgoing edges. These spectrally stable states are degenerate minimizers of the action functional with the degeneracy due to the translational symmetry. Nonlinear stability of these spectrally stable states has been an open problem up to now. In this paper, we prove that these spectrally stable states are nonlinearly unstable because of the irreversible drift along the incoming edge towards the vertex of the star graph. When the shifted states reach the vertex as a result of the drift, they become saddle points of the action functional, in which case the nonlinear instability leads to their destruction. In addition to rigorous mathematical results, we use numerical simulations to illustrate the drift instability and destruction of the shifted states on the balanced star graph.

Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

“…A breakthrough result, due to Matrasulov and coworkers, is the discovery of a class of non-reflecting matching conditions that make the cubic NLS on graphs inherit the integrability from the corresponding one-dimensional system ( [40,39]) and, at least for star graphs, make possible to restore the structure and the methods typical of integrable systems, like Lax pairs and inverse scattering [21]). Other milestone results on graphs with the same non-reflecting (hence non-Kirchhoff) conditions have been obtained by Pelinovsky and collaborators [34,35,36] in a series of works where the spectral stability of special solutions (like half-solitons or shifted states) was investigated.…”

Standing waves on quantum graphs

“…Shifted states can be constructed for even number of half lines or for generalized Neumann-Kirchhoff conditions and most of them are linearly unstable [84]. Even if the shifted states are linearly stable, they are nonlinearly unstable [86].…”

“…Stability and instability of standing waves on star graphs in the case of a δ vertex have been considered in [21,72]. The case of generalized Kirchhoff boundary conditions has been studied in [84,86] when the strength of nonlinearity is suitably adapted on any edge to allow for the existence of shifted states. It is shown that these shifted translating states are orbitally unstable.…”

Standing waves of the quintic NLS equation on the tadpole graph

“…Since n(L) = 1 and z(L) = 0 by Theorem 2, the following Corollary 1 follows from Theorem 3 by the orbital stability theory of standing waves (see the recent application of this theory on star graphs in [24][25][26]).…”