arXiv subjects

Simone Dovetta

Publications and source records attributed to Simone Dovetta.

At least 19 recordsLinked to original sources

arXiv2026

Uniqueness and non-uniqueness of least energy normalized solutions for nonlinear Schr\"odinger equations on compact metric graphs

We investigate uniqueness and non-uniqueness of least energy normalized and least energy nodal normalized solutions for nonlinear Schr\"odinger equations on compact metric graphs. We first prove existence of least energy nodal normalized solutions in the $L^2$-subcritical regime and, at the critical exponent, below a graph-dependent threshold. We then establish a conditional non-uniqueness result for slightly $L^2$-subcritical powers and identify broad classes of graphs for which the required condition either holds or fails. In particular, we prove uniqueness of least energy nodal normalized solutions on the interval for every mass and every $p\in(2,6)$. Finally, using ODE and phase-plane techniques, we show that least energy normalized solutions on the interval are unique for $p$ sufficiently close to $2$. Overall, the results reveal a strong dependence of the uniqueness picture on the nonlinearity power, the topology, and the metric of the graph, and a structural difference between the constant sign and the sign-changing settings.

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arXiv2026

Existence of optimizers for the sharp stability constant in the logarithmic Sobolev inequality

We study the sharp stability constant in the logarithmic Sobolev inequality, defined as the infimum of the logarithmic Sobolev deficit divided by the squared $L^2$-distance from the manifold of Gaussian optimizers. For every $N\geq 1$, first we prove that this infimum is attained in the Euclidean formulation of the inequality. Then, we show that every such optimizer decays exponentially fast at infinity, and therefore gives rise to an optimizer in the Gaussian formulation, where the same sharp constant appears. The proof is based on an extension of the general strategy introduced by Bianchi and Egnell for the Sobolev inequality and recently developed by K\"onig to prove the existence of optimizers for the corresponding sharp stability constant. The main difference in the logarithmic Sobolev setting is the analysis of minimizing sequences approaching the manifold of optimizers, since the entropy term $u^2\log u^2$ does not allow for a direct global second-order expansion. This is overcome by a perturbative estimate near the Gaussian manifold, which provides the compactness threshold needed to rule out the loss of compactness.

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arXiv2025

Dirac solitons in one-dimensional nonlinear Schr\"odinger equations

In this paper we study a family of one-dimensional stationary cubic nonlinear Schr\"odinger (NLS) equations with periodic potentials and linear part displaying Dirac points in the dispersion relation. By introducing a suitable periodic perturbation, one can open a spectral gap around the Dirac-point energy. This allows to construct standing waves of the NLS equation whose leading-order profile is a modulation of Bloch waves by means of the components of a spinor solving an appropriate cubic nonlinear Dirac (NLD) equation. We refer to these solutions as Dirac solitons. Our analysis thus provides a rigorous justification for the use of the NLD equation as an effective model for the original NLS equation.

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arXiv2025

Doubly nonlinear Schr\"odinger normalized ground states on 2D grids: existence results and singular limits

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schr\"odinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\R^2)$ to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.

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arXiv2025

Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions

We investigate normalized solutions for doubly nonlinear Schr\"odinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $\delta$-type at the origin. We provide a complete characterization of existence and uniqueness for normalized solutions and for energy ground states for every value of the nonlinearity powers. We show that the interplay between a defocusing standard and a focusing point nonlinearity gives rise to new phenomena with respect to those observed with single nonlinearities, standard combined nonlinearities, and combined focusing standard and pointwise nonlinearities.

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arXiv2025

Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions

We provide a non-uniqueness result for normalized ground states of nonlinear Schr\"odinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the $L^2$-critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.

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arXiv2024

An action approach to nodal and least energy normalized solutions for nonlinear Schr\"odinger equations

We develop a new approach to the investigation of normalized solutions for nonlinear Schr\"odinger equations based on the analysis of the masses of ground states of the corresponding action functional. Our first result is a complete characterization of the masses of action ground states, obtained via a Darboux-type property for the derivative of the action ground state level. We then exploit this result to tackle normalized solutions with a twofold perspective. First, we prove existence of normalized nodal solutions for every mass in the $L^2$-subcritical regime, and for a whole interval of masses in the $L^2$-critical and supercritical cases. Then, we show when least energy normalized solutions/least energy normalized nodal solutions are action ground states/nodal action ground states.

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arXiv2024

Non-uniqueness of normalized ground states for nonlinear Schr\"odinger equations on metric graphs

We establish general non-uniqueness results for normalized ground states of nonlinear Schr\"odinger equations with power nonlinearity on metric graphs. Basically, we show that, whenever in the $L^2$-subcritical regime a graph hosts ground states at every mass, for nonlinearity powers close to the $L^2$-critical exponent $p=6$ there is at least one value of the mass for which ground states are non-unique. As a consequence, we also show that, for all such graphs and nonlinearities, there exist action ground states that are not normalized ground states.

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arXiv2024

Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs

We consider the existence of normalized solutions to nonlinear Schr\"odinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive solutions on two classes of graphs: periodic graphs, and noncompact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an $L^2$ supercritical nonlinearity extended on the whole graph and unraveling the role of topology in the existence of solutions.

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arXiv2023

Normalized ground states for Schr\"odinger equations on metric graphs with nonlinear point defects

We investigate the existence of normalized ground states for Schr\"odinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear $\delta$-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every $L^2$-subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on $\mathbb{Z}$-periodic graphs and on a prototypical $\mathbb{Z}^2$-periodic graph, the two-dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.

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arXiv2023

Singular limit of periodic metric grids

We investigate the asymptotic behaviour of nonlinear Schrödinger ground states on $d$-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\mathbb{R}^d)$ to the corresponding ground states on $\mathbb{R}^d$. As an application of such convergence results, qualitative properties of ground states and multiplicity results for fixed mass critical points of the energy on grids are derived. Moreover, we compare optimal constants in $d$-dimensional Gagliardo-Nirenberg inequalities on $\mathbb{R}^d$ and on grids. For $L^2$-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on $\mathbb{R}^d$ but, contrary to $\mathbb{R}^d$, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo-Nirenberg inequalities on grids.

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arXiv2023

On the continuum limit of epidemiological models on graphs: convergence and approximation results

We focus on an epidemiological model (the archetypical SIR system) defined on graphs and study the asymptotic behavior of the solutions as the number of vertices in the graph diverges. By relying on the theory of so called graphons we provide a characterization of the limit and establish convergence results. We also provide approximation results for both deterministic and random discretizations.

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arXiv2023

Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs

We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to $3$. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than $1$. The proof is based on a suitable Ljapunov-Schmidt reduction.

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arXiv2023

Constant sign and sign changing NLS ground states on noncompact metric graphs

We investigate existence and nonexistence of action ground states and nodal action ground states for the nonlinear Schrödinger equation on noncompact metric graphs with rather general boundary conditions. We first obtain abstract sufficient conditions for existence, typical of problems with lack of compactness, in terms of ``levels at infinity'' for the action functional associated with the problems. Then we analyze in detail two relevant classes of graphs. For noncompact graphs with finitely many edges, we detect purely topological sharp conditions preventing the existence of ground states or of nodal ground states. We also investigate analogous conditions of metrical nature. The negative results are complemented by several sufficient conditions to ensure existence, either of topological or metrical nature, or a combination of the two. For graphs with infinitely many edges, all bounded, we focus on periodic graphs and infinite trees. In these cases, our results completely describe the phenomenology. Furthermore, we study nodal domains and nodal sets of nodal ground states and we show that the situation on graphs can be totally different from that on domains of $\mathbb{R}^N$.

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arXiv2023

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

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arXiv2022

Doubly nonlinear Schrödinger ground states on metric graphs

We investigate the existence of ground states at prescribed mass on general metric graphs with half-lines for focusing doubly nonlinear Schrödinger equations involving both a standard power nonlinearity and delta nonlinearities located at the vertices. The problem is proved to be sensitive both to the topology and to the metric of the graph and to exhibit a phenomenology richer than in the case of the sole standard nonlinearity considered in [Adami et al. '15, Adami et al. '16]. On the one hand, we provide a complete topological characterization of the problem, identifying various topological features responsible for existence/non-existence of doubly nonlinear ground states in specific mass regimes. On the other hand, we describe the role of the metric in determining the exact interplay between these different topological properties.

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arXiv2021

Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover

We discuss the model robustness of the infinite two-dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so-called dimensional crossover identified in [Adami et al. 2019]. Such a phenomenon has two evidences: the coexistence of the one and the two-dimensional Sobolev inequalities and the appearence of a continuum of $L^2$-critical exponents for the ground states at fixed mass of the nonlinear Schrödinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not.

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arXiv2021

Action versus energy ground states in nonlinear Schrödinger equations

We investigate the relations between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold. Our first general result is that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre-Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We prove that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers.

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