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Shijun Zheng

Publications and source records attributed to Shijun Zheng.

At least 19 recordsLinked to original sources
arXiv2025

Blowup rate for rotational NLS with a repulsive potential

In this paper we give an analytical proof of the ``$\log$-$\log$'' blowup rate for mass-critical nonlinear Schrödinger equation (NLS) with a rotation ($Ω\neq 0$) and a repulsive harmonic potential $V_γ(x) = \textrm{sgn}(γ) γ^2 |x|^2$, $γ< 0$ when the initial data has a mass slightly above that of $Q$, the ground state solution to the free NLS. The proof is based on a virial identity and an $\mathcal{R}_γ$-transform, a pseudo-conformal transform in this setting. Further, we obtain a limiting behavior description concerning the mass concentration near blowup time. A remarkable finding is that increasing the value $|γ|$ for the repulsive potential $V_γ$ can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where $γ$ is positive. This kind of phenomenon was earlier observed in the non-rotational case $Ω= 0$ in Carles' work. In addition, we provide numerical simulations to partially illustrate the blowup profile along with the blowup rate using dynamic rescaling and adaptive mesh refinement method.

math.AP↗
arXiv2025

Perturbed Fourier Transform Associated with Schrödinger Operators

We give an exposition on the $L^2$ theory of the perturbed Fourier transform associated with a Schrödinger operator $H=-d^2/dx^2 +V$ on the real line, where $V$ is a real-valued \mbox{finite} measure. In the case $V\in L^1\cap L^2$, we explicitly define the perturbed Fourier transform $\mathcal{F}$ for $H$ and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for $\cF$ that covers the class of short range potentials in $(1+|x|)^{-\frac12-\eps} L^2 $. Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.

math.AP↗
arXiv2024

A New Adversarial Perspective for LiDAR-based 3D Object Detection

Autonomous vehicles (AVs) rely on LiDAR sensors for environmental perception and decision-making in driving scenarios. However, ensuring the safety and reliability of AVs in complex environments remains a pressing challenge. To address this issue, we introduce a real-world dataset (ROLiD) comprising LiDAR-scanned point clouds of two random objects: water mist and smoke. In this paper, we introduce a novel adversarial perspective by proposing an attack framework that utilizes water mist and smoke to simulate environmental interference. Specifically, we propose a point cloud sequence generation method using a motion and content decomposition generative adversarial network named PCS-GAN to simulate the distribution of random objects. Furthermore, leveraging the simulated LiDAR scanning characteristics implemented with Range Image, we examine the effects of introducing random object perturbations at various positions on the target vehicle. Extensive experiments demonstrate that adversarial perturbations based on random objects effectively deceive vehicle detection and reduce the recognition rate of 3D object detection models.

cs.CV↗
arXiv2024

Existence and non-existence of ground state solutions for magnetic NLS

We show the existence and stability of ground state solutions (g.s.s.) for $L^2$-critical magnetic nonlinear Schrödinger equations (mNLS) for a class of unbounded electromagnetic potentials. We then give non-existence result by constructing a sequence of vortex type functions in the setting of RNLS with an anisotropic harmonic potential. These generalize the corresponding results in [3] and [20]. The case of an isotropic harmonic potential for rotational NLS has been recently addressed in [10]. Numerical results on the ground state profile near the threshold are also included.

math.AP↗
arXiv2023

Note on gradient estimate of heat kernel for Schrödinger operators

Let $H=-Δ+V$ be a Schrödinger operator on $\mathbb{R}^n$. We show that gradient estimates for the heat kernel of $H$ with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for $L^p$ and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author's recent proofs. We give a counterexample in one dimension to show that there exists $V$ in the Schwartz class such that the long time gradient heat kernel estimate fails.

math.AP↗
arXiv2023

Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

We consider the focusing NLS with an angular momentum and a harmonic potential, which models Bose-Einstein condensate under a rotating magnetic trap. We give a sharp condition on the global existence and blowup in the mass-critical case. We further consider the stability of such systems via variational method. We determine that at the critical exponent $p=1+4/n$, the mass of $Q$, the ground state for the NLS with zero potential, is the threshold for both finite time blowup and orbital instability. Moreover, we prove a sharp threshold theorem for the rotational NLS with an inhomogeneous nonlinearity. The analysis relies on the existence of ground state as well as a virial identity for the associated kinetic-magnetic operator.

math.AP↗
arXiv2023

Damped nonlinear Schrödinger equation with Stark effect

We study the $L^2$-critical damped NLS with a Stark potential. We prove that the threshold for global existence and finite time blowup of this equation is given by $\|Q\|_2$, where $Q$ is the unique positive radial solution of $ΔQ + |Q|^{4/d} Q = Q$ in $H^1(\mathbb{R}^d)$. Moreover, in any small neighborhood of $Q$, there exists an initial data $u_0$ above the ground state such that the solution flow admits the log-log blowup speed. This verifies the structural stability for the ``$\log$-$\log$ law'' associated to the NLS mechanism under the perturbation by a damping term and a Stark potential. The proof of our main theorem is based on the Avron-Herbst formula and the analogous result for the unperturbed damped NLS.

math.AP↗
arXiv2023

Adaptive Local Adversarial Attacks on 3D Point Clouds for Augmented Reality

As the key technology of augmented reality (AR), 3D recognition and tracking are always vulnerable to adversarial examples, which will cause serious security risks to AR systems. Adversarial examples are beneficial to improve the robustness of the 3D neural network model and enhance the stability of the AR system. At present, most 3D adversarial attack methods perturb the entire point cloud to generate adversarial examples, which results in high perturbation costs and difficulty in reconstructing the corresponding real objects in the physical world. In this paper, we propose an adaptive local adversarial attack method (AL-Adv) on 3D point clouds to generate adversarial point clouds. First, we analyze the vulnerability of the 3D network model and extract the salient regions of the input point cloud, namely the vulnerable regions. Second, we propose an adaptive gradient attack algorithm that targets vulnerable regions. The proposed attack algorithm adaptively assigns different disturbances in different directions of the three-dimensional coordinates of the point cloud. Experimental results show that our proposed method AL-Adv achieves a higher attack success rate than the global attack method. Specifically, the adversarial examples generated by the AL-Adv demonstrate good imperceptibility and small generation costs.

cs.CV↗
arXiv2023

Interpreting Hidden Semantics in the Intermediate Layers of 3D Point Cloud Classification Neural Network

Although 3D point cloud classification neural network models have been widely used, the in-depth interpretation of the activation of the neurons and layers is still a challenge. We propose a novel approach, named Relevance Flow, to interpret the hidden semantics of 3D point cloud classification neural networks. It delivers the class Relevance to the activated neurons in the intermediate layers in a back-propagation manner, and associates the activation of neurons with the input points to visualize the hidden semantics of each layer. Specially, we reveal that the 3D point cloud classification neural network has learned the plane-level and part-level hidden semantics in the intermediate layers, and utilize the normal and IoU to evaluate the consistency of both levels' hidden semantics. Besides, by using the hidden semantics, we generate the adversarial attack samples to attack 3D point cloud classifiers. Experiments show that our proposed method reveals the hidden semantics of the 3D point cloud classification neural network on ModelNet40 and ShapeNet, which can be used for the unsupervised point cloud part segmentation without labels and attacking the 3D point cloud classifiers.

cs.CV↗
arXiv2021

Universal Upper Bound on The Blowup Rate of Nonlinear Schrödinger Equation with Rotation

In this paper, we prove a universal upper bound on the blowup rate of a focusing nonlinear Schrödinger equation with an angular momentum under a trapping harmonic potential, assuming that the initial data is radially symmetric in the weighted Sobolev space. The nonlinearity is in the mass supercritical and energy subcritical regime. Numerical simulations are also presented.

math.AP↗
arXiv2020

Orbital Stability of Standing Waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrödinger equation with a $μ$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denote by $Q_p$ the ground state for the BNLS with $μ=0$. We prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $μ\in ( -λ_0, \iy)$ for some $λ_0=λ_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$\,, we prove the orbital stability on certain mass level below $\|Q^*\|_{L^2}$, provided $μ\in (-\lam_1,0)$, where $\lam_1=\dfrac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows to fill the gap concerning existence of the ground states for the BNLS when $μ$ is negative and $p\in (1,1+\frac8d]$.

math.AP↗
arXiv2020

Spectral multipliers for Schrödinger operators

We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-Δ+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int (1+|x|) |V(x)|dx$ is finite and $H$ has no resonance at zero. In the resonance case we assume $\int (1+|x|^2) |V(x)| dx$ is finite.

math.CA↗
arXiv2019

Nonlinear Schrödinger Equations for Bose-Einstein Condensates

The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with $L^2$ initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.

math.AP↗
arXiv2019

Orbital Stability of Standing Waves for Fractional Hartree Equation with Unbounded Potentials

We prove the existence of the set of ground states in a suitable energy space $Σ^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-Δ+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set of standing waves. The main ingredient is the observation that $Σ^s$ is compactly embedded in $L^2$. This enables us to apply the concentration compactness argument in the works of Cazenave-Lions and Zhang, namely, relative compactness for any minimizing sequence in the energy space.

math.AP↗
arXiv2019

Blowup rate for mass critical rotational nonlinear Schrödinger equations

We consider the blowup rate for blowup solutions to $L^2$-critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "$\log$-$\log$ law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.

math.AP↗
arXiv2008

Interpolation Theorems for Self-adjoint Operators

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schrödinger operator with electro-magnetic potential.

math.AP↗
arXiv2008

Spectral multipliers for Schroedinger operators with Poeschl-Teller potential

We prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials. Our result applies to, in particular, the negative Poeschl-Teller potential $V(x)= -ν(ν+1) \sech^2 x $, $ν\in \N$, for which $H$ has a resonance at zero.

math.AP↗