arXiv subjects

Francisco Agostinho

Publications and source records attributed to Francisco Agostinho.

3 recordsLinked to original sources

arXiv2025

Stability transitions of NLS action ground-states on metric graphs

We study the orbital stability of action ground-states of the nonlinear Schrödinger equation over two particular cases of metric graphs, the $\mathcal{T}$ and the tadpole graphs. We show the existence of stability transitions near the $L^2$-critical exponent, a new dynamical feature of the nonlinear Schrödinger equation. More precisely, as the frequency $λ$ increases, the action ground-state transitions from stable to unstable and then back to stable (or vice-versa). This result is complemented with the stability analysis of ground-states in the asymptotic cases of low/high frequency and weak/strong nonlinear interaction. Finally, we present a numerical simulation of the stability of action ground-states depending on the nonlinearity and the frequency parameter, which validates the aforementioned theoretical results.

math.AP↗

arXiv2025

A comprehensive study of bound-states for the nonlinear Schrödinger equation on single-knot metric graphs

We study the existence and qualitative properties of action ground-states (that is, bound-states with minimal action) {of the nonlinear Schrödinger equation} over single-knot metric graphs -- which are made of half-lines, loops and pendants, all connected at a single vertex. First, we prove existence of action ground-state for generic single-knot graphs, even in the absence of an associated variational problem. Second, for regular single-knot graphs of length $\ell$, we perform a complete analysis of positive monotone bound-states. Furthermore, we characterize all positive bound-states when $\ell$ is small and prove some symmetry-breaking results for large $\ell$. Finally, we apply the results to some particular graphs to illustrate the complex relation between action ground-states and the topological {and metric} features of the underlying metric graph. The proofs are nonvariational, using a careful phase-plane analysis, the study of sections of period functions, asymptotic estimates and blowup arguments. We show, in particular, how nonvariational techniques are complementary to variational ones in order to deeply understand bound-states of the nonlinear Schrödinger equation on metric graphs.

math.AP↗

arXiv2023

Classification and stability of positive solutions to the NLS equation on the $\mathcal{T}$-metric graph

Given $λ>0$ and $p>2$, we present a complete classification of the positive $H^1$-solutions of the equation $-u''+λu=|u|^{p-2}u$ on the $\mathcal{T}$-metric graph (consisting of two unbounded edges and a terminal edge of length $\ell>0$, all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states. Moreover, for $p\sim 6^-$, the notions of action and energy ground states do not coincide and energy ground states are not unique. In the $L^2$-supercritical case $p>6$, we prove that, for $λ\sim 0^+$ and $λ\sim +\infty$, action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation $i\partial_tu + \partial^2_{xx} u + |u|^{p-2}u=0$. Finally, we provide numerical evidence of the uniqueness of energy ground states for $p\sim 2^+$ and of the existence of both stable and unstable action ground states for $p\sim6$.

math.AP↗