arXiv subjects

Dinh-Thi Nguyen

Publications and source records attributed to Dinh-Thi Nguyen.

16 recordsLinked to original sources

arXiv2026

Homogeneous attractive Bose-Einstein condensates with repulsive three-body interactions: the one-dimensional case

We investigate the behavior of a homogeneous one-dimensional Bose gas composed of $N$ identical bosons experiencing attractive two-body and repulsive three-body interactions. Utilizing intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the many-body Hamiltonian. Particular attention is given to the impact of the repulsive three-body term on ground state energy estimates and mass concentration properties

math-ph↗

arXiv2026

Homogeneous attractive Bose-Einstein condensates with repulsive three-body interactions: the two-dimensional case

We consider a homogeneous Bose gas composed of $N$ identical bosons occupying an infinite two-dimensional space. This gas exhibits both an attractive two-body interaction and a repulsive three-body interaction. Via the intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schroedinger functional as the mean-field limit of the model. Our investigation focuses on the system's behavior in relation to the two-body interaction.

math-ph↗

arXiv2026

Local Density Approximation and Other Limit Regimes for a Homogeneous Bose Gas with Repulsive Three-Body Interactions in Low-Dimensional Space

In the whole space of low spatial dimensions, namely $d \leq 2$, we study the minimizers of an energy functional with an attractive cubic nonlinearity and a repulsive quintic nonlinearity, which describes a quantum Bose gas with a two-body attraction and a three-body repulsion. We prove the existence of minimizers at fixed effective statistics parameter. In the limit of a large effective statistics parameter of the mass-critical nonlinearity (with respect to the kinetic energy), we derive an effective Thomas-Fermi-like model for the homogeneous Bose gas. We also consider other limit regimes depending on the mass-critical nonlinearity.

math-ph↗

arXiv2026

Vortex patterns of a two-dimensional Bose-Einstein condensate at the almost critical rotation speed

We study vortex patterns of a two-dimensional Bose-Einstein condensate rotating close to the centrifugal limit, treating the two signs of the contact interaction with the method each requires: for repulsion, a GPU-accelerated variational minimization with exact projection onto the Lowest Landau Level (LLL); for attraction, imaginary-time evolution of the full real-space Gross-Pitaevskii (GP) equation. For repulsive interactions, our approach reproduces Abrikosov vortex lattices, achieving quantitative alignment with Thomas-Fermi theory and recovering the Abrikosov constant $e^{\rm Ab}(1)\approx1.1596$, in close analogy with the vortex ordering of type-II superconductors. In the attractive regime, the rotating ground state carries no vortex lattice at any rotation frequency: the cloud contracts and collapses as $G\to -G_*$, where $G_*=\|Q\|_{L^2}^2\simeq11.70$, essentially independently of rotation. The only stationary vortex states we find are giant vortices, trapped vortex (Townes) solitons whose charge-$m$ sector collapses as $G\to -\|Q_m\|_{L^2}^2$ ($\|Q_m\|_{L^2}^2\simeq11.70,\,48.3,\,89.8,\,132.4$ for $m=0,1,2,3$), fixed by an equivariant Gagliardo-Nirenberg inequality and confirmed numerically up to the critical region. These findings provide a numerical benchmark for vortex formation and collapse in rotating two-dimensional quantum gases.

cond-mat.quant-gas↗

arXiv2026

Nonlinear Landau levels in the almost-bosonic anyon gas

We consider the quantitative description of a many-particle gas of interacting abelian anyons in the plane, confined in a trapping potential. If the anyons are modeled as bosons with a magnetic flux attachment, and if the total magnetic flux is small compared to the number of particles, then an average-field description becomes appropriate for the low-energy collective state of the gas. Namely, by means of a Hartree-Jastrow ansatz, we derive a two-parameter Chern-Simons-Schrödinger energy functional which extends the well-known Gross-Pitaevskii / nonlinear Schrödinger density functional theory to the magnetic (anyonic) self-interaction. One parameter determines the total number of self-generated magnetic flux units in the system, and the other the effective strength of spin-orbit self-interaction. This latter interaction can be either attractive/focusing or repulsive/defocusing, and depends both on the intrinsic spin-orbit interaction and the relative length scale of the flux profile of the anyons. Densities and energies of ground and excited states are studied analytically and numerically for a wide range of the parameters and align well with a sequence of exact nonlinear Landau levels describing Jackiw-Pi self-dual solitons. With increasing flux, counter-rotating vortices are formed, enhancing the stability of the gas against collapse. Apart from clarifying the relations between various different anyon models that have appeared in the literature, our analysis sheds new light on the many-anyon spectral problem, and also exemplifies a novel supersymmetry-breaking phenomenon.

cond-mat.quant-gas↗

arXiv2025

A generalized Liouville equation and magnetic stability

This work considers two related families of nonlinear and nonlocal problems in the plane $\mathbb{R}^2$. The first main result derives the general integrable solution to a generalized Liouville equation using the Wronskian of two coprime complex polynomials. The second main result concerns an application to a generalized Ladyzhenskaya-Gagliardo-Nirenberg interpolation inequality, with a single real parameter $β$ interpreted as the strength of a magnetic self-interaction. The optimal constant of the inequality and the corresponding minimizers of the quotient are studied and it is proved that for $β\ge 2$, for which the constant equals $2πβ$, such minimizers only exist at quantized $β\in 2\mathbb{N}$ corresponding to nonlinear generalizations of Landau levels with densities solving the generalized Liouville equation. This latter problem originates from the study of self-dual vortex solitons in the abelian Chern-Simons-Higgs theory and from the average-field-Pauli effective theory of anyons, i.e. quantum particles with statistics intermediate to bosons and fermions. An immediate application is given to Keller-Lieb-Thirring stability bounds for a gas of such anyons which self-interact magnetically (vector nonlocal repulsion) as well as electrostatically (scalar local/point attraction), thus generalizing the stability theory of the 2D cubic nonlinear Schrödinger equation.

math.AP↗

arXiv2024

Thomas-Fermi profile of a fast rotating Bose-Einstein condensate

We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

math-ph↗

arXiv2024

2D attractive almost-bosonic anyon gases

In two-dimensional space, we consider a system of $N$ anyons interacts via a short range attractive two-body interaction. In the stable regime, we derive the average-field Pauli functional as the mean-field limit of many-body quantum mechanics. Furthermore, we investigate the collapse phenomenon in the collapse regime where the strength of attractions tends to a critical value (defined by the cubic NLS equation) while simultaneously considering the weak field regime where the strength of the self-generated magnetic field tends to zero.

math-ph↗

arXiv2023

On one-dimensional Bose gases with two- and (critical) attractive three-body interactions

We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $a N^{α-1} U(N^α(x-y))$ and an attractive three-body interaction potential of the form $-b N^{2β-2} W(N^β(x-y,x-z))$, where $a\in\mathbb{R}$, $b,α>0$, $0<β<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3π^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $β$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$.

math-ph↗

arXiv2023

Stabilization against collapse of 2D attractive Bose-Einstein condensates with repulsive, three-body interactions

We consider a trapped Bose gas of $N$ identical bosons in two dimensional space with both an attractive, two-body, scaled interaction and a repulsive, three-body, scaled interaction respectively of the form $-aN^{2α-1} U(N^α\cdot)$ and $bN^{4β-2} W(N^β\cdot, N^β\cdot))$, where $a,b,α,β>0$ and $\int_{\mathbb R^2}U(x) {\mathop{}\mathrm{d}} x = 1 = \iint_{\mathbb R^{4}} W(x,y) {\mathop{}\mathrm{d}} x {\mathop{}\mathrm{d}} y$. We derive rigorously the cubic--quintic nonlinear Schrödinger semiclassical theory as the mean-field limit of the model and we investigate the behavior of the system in the double-limit $a = a_N \to a_*$ and $b = b_N \searrow 0$. Moreover, we also consider the homogeneous problem where the trapping potential is removed.

math-ph↗

arXiv2022

Blow-up of 2D attractive Bose-Einstein condensates at the crittical rotational speed

We study the ground states of a 2D focusing non-linear Schrödinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo--Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular the blow-up profile remains independent of the rotation frequency.

math.AP↗

arXiv2020

Blow-Up Profile of 2D Focusing Mixture Bose Gases

We study the collapse of a many-body system which is used to model two-component Bose-Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for $N$ identical bosons in $\mathbb R^2$, interacting with potentials rescaled in the mean-field manner $-N^{2β-1}w^{(σ)}(N^βx)$ with $\int_{\mathbb R^{2}}w^{(σ)}(x){\rm d}x=1$. Assuming that $0<β<1/2$, we first show that the leading order of the quantum energy is captured correctly by the Gross-Pitaevskii energy. Secondly, we investigate the blow-up behavior of the quantum energy as well as the ground states when $N\to\infty$ and either the total interaction strength of intra-species and inter-species or the strengths of intra-species interactions of each component approaches sufficiently slowly a critical value, which is the critical strength for the focusing Gross-Pitaevskii functional. We prove that the many-body ground states fully condensate on the (unique) Gagliardo-Nirenberg solution.

math-ph↗

arXiv2019

Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory

We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.

math-ph↗

arXiv2019

Blow-up Profile of Neutron Stars in the Chandrasekhar theory

We study the Chandrasekhar variational model for neutron stars, with or without an external potential. We prove the existence of minimizers when the attractive interaction strength $τ$ is strictly smaller than the Chandrasekhar limit $τ_c$ and investigate the blow-up phenomenon in the limit $τ\uparrow τ_{c}$. We show that the blow-up profile of the minimizer(s) is given by the Lane-Emden solution.

math.AP↗

arXiv2019

Many-Body Blow-Up Profile of Boson Stars with External Potentials

We consider a 3D quantum system of $N$ identical bosons in a trapping potential $|x|^p$, with $p\geq0$, interacting via a Newton potential with an attractive interaction strength $a_{N}$. For a fixed large $N$ and the coupling constant $a_{N}$ smaller than a critical value $a_{*}$ (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when $a_{N}$ approaches $a_{*}$ sufficiently slowly in the limit $N\to\infty$. The blow-up profile is given by the Gagliardo-Nirenberg solutions.

math-ph↗

arXiv2017

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional $$\mathcal{E}_{a}(u):=\|(-Δ+m^{2})^{1/4}u\|_{L^{2}}^{2}-\frac{a}{2}\int_{\mathbb{R}^{3}}(\left|\cdot\right|^{-1}\star |u|^{2})(x)|u(x)|^{2}{\rm d}x+\int_{\mathbb{R}^{3}}V(x)|u(x)|^{2}{\rm d}x$$ under the mass constraint $\int_{\mathbb{R}^3}|u(x)|^2{\rm d}x=1$. Here $m>0$ is the mass of particles and $V\geq 0$ is an external potential. We prove that minimizers exist if and only if $a$ satisfies $0\leq a 0$.

math-ph↗