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Yang Lan

Publications and source records attributed to Yang Lan.

18 recordsLinked to original sources
arXiv2026

Construction of three-solitons with logarithmic distance for the mass-critical gKdV equations

For the mass-critical generalized Korteweg-de Vries equation, \begin{equation*} \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}, \end{equation*} we prove the existence of three-soliton solutions with logarithmic relative distance and with the choice of signs $(+,-,-)$. The choice of the number three and the signs of solitons are related to the solvability of the ODE system generated by the nonlinear interactions between the three solitons and some non-localized profiles. In particular, these special behaviors are due to strong interactions between the three solitons. That is, the dynamics of each soliton is perturbed at leading order by the presence of other solitons.

math.AP↗
arXiv2026

Construction of two-bubble blow-up solutions for the mass-critical gKdV equations

For the mass-critical generalized Korteweg-de Vries equation, $$ \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}.$$ We prove the existence of a global solution that blows up in infinite time and approaches the sum of two decoupled bubbles with opposite signs. The proof is inspired by the techniques developed for the two-dimensional mass-critical NLS equation in a similar context by Martel-Rapha\"el [37]. The main difficulty originates from the fact that the unstable directions related to scaling are excited by the nonlinear interactions. To overcome this difficulty, a refined approximate solution that involves some non-localized profiles is needed. In particular, a sharp understanding for the interactions between solitons and such profiles is also required.

math.AP↗
arXiv2026

Asymptotic stability of smooth solitons and multi-solitons for the Camassa--Holm equation

We establish the asymptotic stability of smooth solitons and multi-solitons for the Camassa-Holm (CH) equation in the energy space $H^1(\R)$. We show that solutions initially close to a soliton converge, up to translation, weakly in $H^1(\R)$ as time tends to infinity to a (possibly different) soliton. The analysis is based on a Liouville-type rigidity theorem characterizing solutions that remain localized near a soliton trajectory. A central feature of the proof is a complete spectral resolution of the linearized CH operator around a soliton. This linear theory is obtained via the bi-Hamiltonian and integrable structure of the CH equation, through the recursion operator and the completeness of the associated squared eigenfunctions. It provides a substitute for the classical spectral framework used in KdV and gKdV equations, which is unavailable in the nonlocal and variable-coefficient setting of CH. The spectral resolution yields sharp decay estimates for the linearized flow in exponentially weighted spaces, which in turn lead to the nonlinear rigidity result and the asymptotic stability of a single soliton. Combined with known orbital stability results, this approach extends to well-ordered trains of solitons and to the explicit multi-soliton solutions generated by the inverse scattering method. As an additional application, we revisit the linearized problems associated with other integrable dispersive equations, including the KdV and mKdV equations, from the perspective of squared-eigenfunction expansions.

math.AP↗
arXiv2025

Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation

In this article, we construct a minimal mass blow-up solution of the two-dimensional cubic (mass-critical) Zakharov--Kuznetsov equation: \begin{equation*} \partial_t \phi+\partial_{x_1}(\Delta \phi+\phi^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^2. \end{equation*} Let $s>\frac{3}{4}$. Bhattacharya-Farah-Roudenko [2] show that $H^{s}$ solutions with $\|\phi\|_{L^{2}}<\|Q\|_{L^{2}}$ are global in time. For such low regularity solutions, we study the dynamics at the threshold $\|\phi\|_{L^{2}}=\|Q\|_{L^{2}}$ and demonstrate that finite time blow-up singularity formation may occur. This result and its proof are inspired by the recent blow-up result [29] for the mass-critical gKdV equation. This result is also complement of previous result [6] for nonexistence of minimal mass blow-up element in the energy space $H^{1}(\mathbb{R}^{2})$ of the two-dimensional cubic Zakharov--Kuznetsov equation.

math.AP↗
arXiv2024

Nonexistence of minimal mass blow-up solution for the 2D cubic Zakharov-Kuznetsov equation

For the 2D cubic (mass-critical) Zakharov-Kuznetsov equation, \begin{equation*} \partial_t\phi+\partial_{x_1}(\Delta \phi+\phi^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}, \end{equation*} we prove that there exist no finite/infinite time blow-up solution with minimal mass in the energy space. This nonexistence result is in contrast to the one obtained by Martel-Merle-Rapha\"el [17] for the mass-critical generalized Korteweg-de Vries (gKdV) equation. The proof relies on a refined ODE argument related to the modulation theory and a modified energy-virial Lyapunov functional with a monotonicity property.

math.AP↗
arXiv2024

On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations

In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with a suitable space-decay property, we fully describe the asymptotic behavior of the corresponding solution. More precisely, for such initial data, we show that only three possible behaviors can occur: 1) The solution leaves a tube near soliton in finite time; 2) the solution blows up in finite time; 3) the solution is global and locally converges to a soliton. In addition, we show that for initial data near a soliton with non-positive energy and above the threshold mass, the corresponding solution will blow up as described in Case 2. Our proof is inspired by the techniques developed for mass-critical generalized Korteweg-de Vries equation (gKdV) equation in a similar context by Martel-Merle-Rapha\"el. More precisely, our proof relies on refined modulation estimates and a modified energy-virial Lyapunov functional. The primary challenge in our problem is the lack of coercivity of the Schr\"odinger operator which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [13], to perform the virial computations after converting the original problem to the adjoint one. Th coercivity of the Schr\"odinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9].

math.AP↗
arXiv2024

A Large-scale Fine-grained Analysis of Packages in Open-Source Software Ecosystems

Package managers such as NPM, Maven, and PyPI play a pivotal role in open-source software (OSS) ecosystems, streamlining the distribution and management of various freely available packages. The fine-grained details within software packages can unveil potential risks within existing OSS ecosystems, offering valuable insights for detecting malicious packages. In this study, we undertake a large-scale empirical analysis focusing on fine-grained information (FGI): the metadata, static, and dynamic functions. Specifically, we investigate the FGI usage across a diverse set of 50,000+ legitimate and 1,000+ malicious packages. Based on this diverse data collection, we conducted a comparative analysis between legitimate and malicious packages. Our findings reveal that (1) malicious packages have less metadata content and utilize fewer static and dynamic functions than legitimate ones; (2) malicious packages demonstrate a higher tendency to invoke HTTP/URL functions as opposed to other application services, such as FTP or SMTP; (3) FGI serves as a distinguishable indicator between legitimate and malicious packages; and (4) one dimension in FGI has sufficient distinguishable capability to detect malicious packages, and combining all dimensions in FGI cannot significantly improve overall performance.

cs.SE↗
arXiv2023

Stability of multi-solitons for the Benjamin-Ono equation

This paper is concerned with the dynamical stability of the $m$-solitons of the Benjamin-Ono (BO) equation. This extends the work of Neves and Lopes [41], which was restricted to $m=2$ the double solitons case. By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a suitable variational nonlocal elliptic equation. The stability issue is reduced to the spectral analysis of higher order nonlocal operators consist of the Hilbert transform. Such operators are isoinertial and the negative eigenvalues of which are fully classified. Our approach in the spectral analysis consists of an invariance for the multi-solitons and new operator identities motivated by the bi-Hamiltonian structure of the BO equation. Since the BO equation is more likely a two dimensional integrable system, its recursion operator is not explicit which makes our analysis more involved. The key ingredient in the spectral analysis is to employ the completeness in $L^2$ of the squared eigenfunctions of the eigenvalue problem for the BO equation. It is demonstrated here that orbital stability of soliton in $H^{\frac{1}{2}}$ implies that all $m$-solitons are dynamically stable in $H^{\frac{m}{2}}$.

math.AP↗
arXiv2019

Blow-up dynamics for $L^2$-critical fractional Schr\"odinger equations

In this paper, we will consider the $L^2$-critical fractional Schr\"odinger equation $iu_t-|D|^{\beta}u+|u|^{2\beta}u=0$ with initial data $u_0\in H^{\beta/2}(\mathbb{R})$ and $\beta$ close to $2$. We will show that the solution blows up in finite time if the initial data has negative energy and slightly supercritical mass. We will also give a specific description for the blow-up dynamics. This is an extension of the work of F. Merle and P. Rapha\"el for $L^2$-critical Schr\"odinger equations but the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.

math.AP↗
arXiv2018

Ultrafast correlated charge and lattice motion in a hybrid metal halide perovskite

Hybrid organic-inorganic halide perovskites have shown remarkable optoelectronic properties (1-3), believed to originate from correlated motion of charge carriers and the polar lattice forming large polarons (4-7). Few experimental techniques are capable of probing these correlations directly, requiring simultaneous sub-meV energy and femtosecond temporal resolution after absorption of a photon (8). Here we use transient multi-THz spectroscopy, sensitive to the internal motions of charges within the polaron, to temporally and energetically resolve the coherent coupling of charges to longitudinal optical phonons in single crystal CH3NH3PbI3 (MAPI). We observe room temperature quantum beats arising from the coherent displacement of charge from the coupled phonon cloud. Our measurements provide unambiguous evidence of the existence of polarons in MAPI.

cond-mat.mtrl-sci↗
arXiv2018

On continuation properties after blow-up time for $L^2$-critical gKdV equations

In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_γ(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-γu|u|^{q-1})_x=0$ with the same initial data, where $γ>0$ and $q>5$. A standard argument shows that $u_γ(t)$ is always global in time and for all $t<T$, $u_γ(t)$ converges to $u(t)$ in $H^1$ as $γ\rightarrow0$. We prove in this paper that for all $t\geq T$, $u_γ(t)$ converges to some $v(t)$ as $γ\rightarrow0$, in a certain sense. This limiting function $v(t)$ is a weak solution to the unperturbed $L^2$-critical gKdV, hence can be viewed as a natural extension of $u(t)$ after the blow-up time.

math.AP↗
arXiv2018

Tunable colloidal photonic crystals

Spherical colloids arranged in a crystalline order are known to produce structural colors. The intensity and brilliance of such photonic crystals require high size-monodispersity of the colloids, a low number of lattice defects and disorder, as well as a relatively large refractive index contrast between the scattering colloids and the continuous background. Here we present the unexpected photonic properties of aqueous suspensions of charge-stabilized, 186 nm large, fluorinated colloids with a refractive index of 1.37. Employing reflectivity, optical observation, small angle x-ray scattering measurements and reflectivity modeling, we demonstrate that these suspensions become partially transparent while showing strong, almost angle-independent color in reflection despite the very small refractive index difference. Under certain conditions, additional sharp Bragg reflections are observed. We were able to tune the observed structural colors continuously across the entire visible range by simply changing the volume fraction of these colloidal suspensions, which show a white appearance when dilute, structural color and Bragg peaks when concentrated enough to form Wiegner crystal, and angle-independent color when very concentrated and in a glassy phase.

physics.optics↗
arXiv2017

Liquid crystalline behaviour of self-assembled Laponite PLL-PEG nanocomposites

Synthetic Laponite-clay particles with platelet-like shape display strong aging when dispersed in aqueous solutions, preventing latter from reaching their natural liquid-crystalline equilibrium state. Here we introduce a facile method that successfully prevents this aging behaviour and enables accessing the systems' liquid-crystal and crystalline phases. We graft the comb-like polymer, poly(L-lysine)-g-poly(ethylene glycol) (PLL-PEG), onto the clay surfaces from solution, thereby screening the negative surface charges and thus ensuring steric stabilisation. We show zeta-sizer and rheology measurements, respectively, confirming complete steric coating and that aging of dilute samples is completely suppressed even after a year. Using evaporation as means to concentrate the particles, we observe various liquid crystalline textures under a polarized optical microscope (POM). Upon sequential spreading and drying, we are also able to obtain transparent films with hierarchical architecture.

cond-mat.soft↗
arXiv2017

On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation

In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\partial_t u+(u_{xx}+u^5-γu|u|^{q-1})_x=0$, where $q>5$ and $0<γ\ll1$. For any initial data $u_0\in H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solitons, and our goal is to classify the dynamics near soliton. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $γ^{-2/(q-1)}$, as $γ\rightarrow0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+\infty$; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves. This extends the classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $γ=0$) by Martel, Merle and Raphaël. However, the blow-down behavior (ii) is completely new, and the dynamics of the saturated equation cannot be viewed as a perturbation of the $L^2$ critical dynamics of the unperturbed equation. This is the first example of classification of the dynamics near ground state for a saturated equation in this context. The cases of $L^2$ critical NLS and $L^2$ supercritical gKdV, where similar classification results are expected, are completely open.

math.AP↗
arXiv2017

Blow-up solutions for $L^2$-supercritical gKdV equations with exactly $k$ blow-up points

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there exists an stable self-similar blow-up dynamics for slightly $L^2$-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.

math.AP↗
arXiv2017

Colloidal Motion under the Action of a Thermophoretic Force

We present thermophoretic measurements in aqueous suspensions of three different polystyrene (PS) particles of varying negative charge, size and surface coating. Our measurement technique is based on the observation of the colloidal steady-state distribution using conventional bright-field microscopy, which avoids undesirable effects such as laser-induced convection or local heating. We find that the colloids with the weakest zeta potential exhibit the strongest thermophoretic effect, suggesting that surface functionality leads to a more intricate dependence of the Soret coefficient on hydrodynamic boundary conditions than predicted by existing theoretical approaches. We also study the relaxation of the colloids to steady-state and propose a model to quantify the relaxation speed, based on the time evolution of the colloidal center of mass. Our observations are well described by this model and show that the relaxation speed tends to increase with the magnitude of the thermophoretic force.

cond-mat.soft↗
arXiv2015

Stable self-similar blow-up dynamics for slightly $L^2$-supercritical generalized KdV equations

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up dynamic with self-similar blow-up rate in the energy space $H^1$ and give a specific description of the formation of the singularity near the blow-up time.

math.AP↗