Searcharxiv⌕ Search

arXiv subjects

Alex H. Ardila

Publications and source records attributed to Alex H. Ardila.

At least 19 recordsLinked to original sources
arXiv2026

Blow-up or grow-up for the focusing 3D cubic NLS with a repulsive inverse-power potential at the mass--energy threshold

We consider the focusing cubic nonlinear Schrodinger equation with a repulsive inverse-power potential $V(x)=a|x|^{-μ}$, where $a>0$ and $1<μ\leq 2$. At the mass-energy threshold, Miao, Murphy, and Zheng, as well as Ardila, Hamano, and Ikeda, established sharp scattering results in the positive virial region. In this paper, we continue to study the dynamics in the negative virial region and show that, in each time direction, solutions either blow up in finite time or grow up.

math.AP↗
arXiv2026

Threshold dynamics for the 4$d$ mass-energy double critical NLS

We consider the 4$d$ mass-energy double critical NLS \[ (i\partial_t+Δ)u = -|u|^2 u + |u| u. \] In Luo (2024) and Cheng--Miao--Zhao (2016), the authors established a scattering/blowup dichotomy for solutions satisfying the energy constraint $E(u_0)< E^c(W)$, where $W$ is the energy-critical NLS ground state and $E^c$ is the energy for the underlying cubic NLS. We prove that the scattering/blowup dichotomy persists even at the energy threshold $E(u_0)=E^c(W)$.

math.AP↗
arXiv2025

Dynamics of the Energy-Critical Nonlinear Schrödinger System in ${\mathbb R}^{4}$

In this paper, we investigate the dynamics of radial solutions at threshold energy for a 3-component Schrödinger system with cubic nonlinearity in four dimensions. The main difference from the cases previously addressed in the literature is that, in our system, the kernel of the imaginary part $L_I$ of the linearized operator $-i{\mathcal L}=L_{R}+iL_{I}$ has dimension 2. To overcome this difficulty, we carry out a detailed study of the coercivity properties of these operators. We also introduce a new modulation parameter associated with the additional eigenfunction in the kernel of the operator $L_{I}$, which enables us to perform the modulation analysis and establish the uniqueness of exponentially decaying solutions to the linearized equation.

math.AP↗
arXiv2025

Threshold solutions for the energy-critical NLS system with quadratic interaction

In this paper, we study the Cauchy problem for a quadratic nonlinear Schrödinger system in dimension six. In~\cite{GaoMengXuZheng}, the authors classified the behavior of solutions under the energy constraint $E(u) < E(Q)$, where $Q$ denotes the ground state. In this work, we classify the dynamics of radial solutions at the threshold energy $E(u) = E(Q)$.

math.AP↗
arXiv2023

Threshold dynamics for the 3$d$ radial NLS with combined nonlinearity

We consider the nonlinear Schrödinger equation with focusing quintic and defocusing cubic nonlinearity in three space dimensions: \[ (i\partial_t+Δ)u = |u|^2 u - |u|^4 u. \] In [18, 23], the authors classified the dynamics of solutions under the energy constraint $E(u)< E^c(W)$, where $W$ is the quintic NLS ground state and $E^c$ is the quintic NLS energy. In this work we classify the dynamics of $H^1$ solutions at the threshold $E(u)=E^c(W)$.

math.AP↗
arXiv2022

Threshold solutions for the 3d cubic-quintic NLS

We study the cubic-quintic NLS in three space dimensions. It is known that scattering holds for solutions with mass-energy in a region corresponding to positive virial, the boundary of which is delineated both by ground state solitons and by certain rescalings thereof. We classify the possible behaviors of solutions on the part of the boundary attained solely by solitons. In particular, we show that non-soliton solutions either scatter in both time directions or coincide (modulo symmetries) with a special solution, which scatters in one time direction and converges exponentially to the soliton in the other.

math.AP↗
arXiv2021

The cubic-quintic nonlinear Schrödinger equation with inverse-square potential

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of \cite{R.Killip, T. Oh, O. Pocovnicu, and M. Visan} in the setting of the standard cubic-quintic NLS.

math.AP↗
arXiv2021

Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_γ(u_{0})[M(u_{0})]^σ=E_{0}(Q)[M(Q)]^σ$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_γ$ is the energy, $M$ is the mass and $σ=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy threshold.

math.AP↗
arXiv2021

Blow-up and scattering for the 1D NLS with point nonlinearity above the mass-energy threshold

In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig-Merle's compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.

math.AP↗
arXiv2021

Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with $χ^3$ nonlinear response

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

math.AP↗
arXiv2021

Instability of ground states for the NLS equation with potential on the star graph

We study the nonlinear Schrödinger equation with an arbitrary real potential $V(x)\in (L^1+L^\infty)(Γ)$ on a star graph $Γ$. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $-γ<0$. We show the existence of ground states $φ_ω(x)$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on $V(x)$. Moreover, for $V(x)=-\dfracβ{x^α}$, in the supercritical case, we prove that the standing waves $e^{iωt}φ_ω(x)$ are orbitally unstable in $H^{1}(Γ)$ when $ω$ is large enough. Analogous result holds for an arbitrary $γ\in\mathbb{R}$ when the standing waves have symmetric profile.

math.AP↗
arXiv2021

Global well-posedness, blow-up and stability of standing waves for supercritical NLS with rotation

We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are $L^{2}$-supercritical.

math.AP↗
arXiv2020

Scattering of the energy-critical NLS with dipolar interaction

In this paper, we investigate the global well-posedness and $H^{1}$ scattering theory for a 3d energy-critical Schrödinger equation under the influence of magnetic dipole interaction $λ_{1}|u|^{2}u+λ_{2}(K\ast|u|^{2})u$, where $K$ is the dipole-dipole interaction kernel. Our proof of global well-posedness result is based on the argument of Zhang [23]. Moreover, adopting the induction of energy technique of Killip-Oh-Pocovnicu-Visan [20], we obtain a condition for scattering.

math.AP↗
arXiv2019

Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation

We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.

math.AP↗
arXiv2019

Logarithmic Bose-Einstein condensates with harmonic potential

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schrödinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.

math.AP↗
arXiv2018

Orbital stability of standing waves for supercritical NLS with potential on graphs

In this paper we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is $L^{2}$-critical and supercritical.

math.AP↗
arXiv2017

Logarithmic NLS equation on star graphs: existence and stability of standing waves

In this paper we consider the logarithmic Schrödinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the existence of several families of standing waves. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we show that the ground states are orbitally stable via a variational approach.

math.AP↗