arXiv subjects

Thanh Trung Le

Publications and source records attributed to Thanh Trung Le.

5 recordsLinked to original sources

arXiv2026

Spectral Algorithms for 3-Wave Kinetic and $C_{12}$ Quantum Boltzmann Equations with General Resonance Manifolds in $\mathbb{R}^d$

Following recent developments in numerical schemes for 3-wave kinetic equations [2, 7, 42, 44, 43], we develop spectral algorithms for multidimensional 3-wave kinetic equations and $C_{12}$ quantum Boltzmann equations with general polynomial dispersion relations. The principal numerical difficulty arises from the resonance constraint, supported on a nonlinear manifold in wave-vector space. We approximate the Dirac distribution by a truncated Fourier representation and derive two spectral discretizations of the collision operator. The first is a direct spectral method with complexity $\mathcal{O}\big(L(2N)^{3d}\big)$, while the second exploits multidimensional FFTs to reduce the complexity to $\mathcal{O}\big(L(2N)^{2d}\log(2N)\big)$. Numerical tests show excellent agreement between the two methods, with the fast algorithm providing substantial computational savings. To suppress unresolved high-frequency modes, we combine the classical $2/3$-rule with exponential spectral filtering. Simulations in two and three dimensions capture the gain--loss dynamics of the $C_{12}$ quantum Boltzmann equation for both rapidly and algebraically decaying initial data. For the 3-wave kinetic equation, the computations exhibit strong oscillations and rapid spectral broadening, providing numerical evidence of an apparent energy cascade toward high frequencies. The results also show that the dispersion relation and spatial dimension strongly influence the transient resonant dynamics.

math.NA↗

arXiv2025

Robust Sparse Subspace Tracking from Corrupted Data Observations

Subspace tracking is a fundamental problem in signal processing, where the goal is to estimate and track the underlying subspace that spans a sequence of data streams over time. In high-dimensional settings, data samples are often corrupted by non-Gaussian noises and may exhibit sparsity. This paper explores the alpha divergence for sparse subspace estimation and tracking, offering robustness to data corruption. The proposed method outperforms the state-of-the-art robust subspace tracking methods while achieving a low computational complexity and memory storage. Several experiments are conducted to demonstrate its effectiveness in robust subspace tracking and direction-of-arrival (DOA) estimation.

eess.SP↗

arXiv2023

Numerical challenges for the understanding of localised solutions with different symmetries in non-local hyperbolic systems

We consider a one-dimensional nonlocal hyperbolic model introduced to describe the formation and movement of self-organizing collectives of animals in homogeneous 1D environments. Previous research has shown that this model exhibits a large number of complex spatial and spatiotemporal aggregation patterns, as evidenced by numerical simulations and weakly nonlinear analysis. In this study, we focus on a particular type of localised patterns with odd/even/no symmetries (which are usually part of snaking solution branches with different symmetries that form complex bifurcation structures called snake-and-ladder bifurcations). To numerically investigate the bifurcating solution branches (to eventually construct the full bifurcating structures), we first need to understand the numerical issues that could appear when using different numerical schemes. To this end, in this study, we consider ten different numerical schemes (the upwind scheme, the MacCormack scheme, the Fractional-Step method, and the Quasi-Steady Wave-Propagation algorithm, combining them with high-resolution methods), while paying attention to the preservation of the solution symmetries with all these schemes. We show several numerical issues: first, we observe the presence of two distinct types of numerical solutions (with different symmetries) that exhibit very small errors; second, in some cases, none of the investigated numerical schemes converge, posing a challenge for the development of numerical continuation algorithms for nonlocal hyperbolic systems; lastly, the choice of the numerical schemes, as well as their corresponding parameters such as time-space steps, exert a significant influence on the type and symmetry of bifurcating solutions.

math.NA↗

arXiv2022

Orbital stability of ground states for a Sobolev critical Schrödinger equation

We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrödinger equation with mixed power nonlinearities \begin{equation*} i \partial_t v + Δv + μv |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \end{equation*} where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $μ> 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, we show that the set of ground states is orbitally stable. Our results settle a question raised by N. Soave [35].

math.AP↗

arXiv2021

Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation

We look for solutions to the Schrödinger-Poisson-Slater equation $$- Δu + λu - γ(|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2 \, dx = c \end{equation*} for some prescribed $c>0$. Here $ u \in H^1(\mathbb{R}^3)$, $γ\in \mathbb{R},$ $ a \in \mathbb{R}$ and $p \in (\frac{10}{3}, 6]$. When $γ>0$ and $a > 0$, both in the Sobolev subcritical case $p \in (\frac{10}{3}, 6)$ and in the Sobolev critical case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in (0,c_1)$, the equation admits two solutions $u_c^+$ and $u_c^-$ which can be characterized respectively as a local minima and as a mountain pass critical point of the associated {\it Energy} functional restricted to the norm constraint. In the case $γ>0$ and $a < 0$, we show that, for any $p \in (\frac{10}{3},6]$ and any $c>0$, the equation admits a solution which is a global minimizer. Finally, in the case $γ<0$, $a >0$ and $p=6$ we show that it does not admit positive solutions.

math.AP↗