Global well-posedness for Dirac equation with concentrated nonlinearity

Scattering of cubic Dirac equations with a general class of Hartree-type nonlinearity for the critical Sobolev data

Recently low-regularity behaviour of solutions to cubic Dirac equations with the Hartree-type nonlinearity has been extensively studied in somewhat a specific assumption on the structure of the nonlinearity. The key approach of previous results was to exploit the null structure in the nonlinearity and the decay of the Yukawa potential. In this paper, we aim to go beyond; we investigate the strong scattering property of cubic Dirac equations with quite a general class of the Hartree-type nonlinearity, which covers the Coulomb potential as well as the Yukawa potential, and the bilinear form, in …

具有集中非线性的周期非线性薛定谔方程的适定性

Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity

研究环面上具有集中非线性的非线性薛定谔方程的解理论，通过基于紧致性结果和利用沃尔泰拉积分方程结构的方法，建立全局能量守恒解存在唯一性及局部适定性，是该方程首个严格解理论。

The modified scattering for Dirac equations of scattering-critical nonlinearity

In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Ma…

Well-posedness of the Two-dimensional Nonlinear Schrödinger Equation with Concentrated Nonlinearity

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.

Asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three. II

We investigate the asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $δ$) interaction with strength $α$, and letting the strength $α$ depend on the wavefunction in a prescribed way. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $Φ_ω$, and the linearization around them admits two imaginary eigenvalues which in principle could corre…

Conditional large-data global well-posedness of Dirac equation with Hartree-type nonlinearity

We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers. Secondly, we obtain large data scatte…

具有球面集中非线性的三维 NLS 方程的适定性

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

本文研究了非线性集中于 $\mathbb{S}^2$ 的三维 NLS 方程，证明了任意 $C^2$ 幂非线性的局部适定性，并在小数据或散焦条件下获得整体适定性，指出该支撑维度需引入新方法。

具有非强制非线性项的非线性狄拉克方程的定态周期解 II：分裂

Stationary periodic solutions for nonlinear Dirac equations with non-coercive nonlinearity II: splitting

该研究针对具有Soler型非线性项的非线性狄拉克方程，通过分离圆因子降维，利用强制扰动与定量特征空间分离等方法，得到对应正谱阈值附近的非平凡周期解。

关于具有点集中非线性的NLS方程的综述

A general review on the NLS equation with point-concentrated nonlinearity

该论文全面综述了具有点集中非线性的NLS方程的研究，涵盖模型定义、适定性、驻波稳定性、爆破分析及从标准NLS方程的推导。

On the standing waves of the Schroedinger equation with concentrated nonlinearity

We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobo…

具有Soler型非线性项的非线性Dirac方程的周期解

Periodic Solutions for a Nonlinear Dirac Equation with Soler-Type Nonlinearity

针对三维环面上带Soler型非线性的非线性Dirac方程，通过引入强制扰动恢复Palais–Smale条件，结合谱估计与联结论证，证明了质量以下区间内任意谱参数对应的非平凡连续可微周期解存在性。

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation \begin{equation*} i\frac{\partial }{\partial t} ψ^\varepsilon(t)= -Δψ^\varepsilon(t) + g(\varepsilon,μ,|(ρ^\varepsilon,ψ^\varepsilon(t))|^{2μ}) (ρ^\varepsilon,ψ^\varepsilon(t)) ρ^\varepsilon \end{equation*} where $ρ^{\varepsilon} \to δ_0$ weakly and the function $g$ embodies the nonlinearity and the scaling and has to be fine tun…

具有集中非线性的Dirac方程孤立波的光谱稳定性与不稳定性

Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity

本文研究具有集中Soler型非线性的Dirac方程孤立波的光谱稳定性，证明仅打破SU(1,1)对称性而不破坏宇称对称性的扰动仍保持稳定性，而同时打破两种对称性的扰动会通过嵌入特征值分岔导致弱相对论孤立波失稳。

Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in $ \R^{1+3}$

We prove that the initial value problem for the Dirac equation $ \left ( -iγ^μ\partial_μ+ m \right) ψ = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline ψψ)\right) ψ\quad \text{in } \ \R^{1+3} $ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$…

Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrödinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength $α$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_α$, where the strength $α$ depends on the wavefunction: $i\dot u= H_αu$, $α=α(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $α$ depend on $u$ according to th…

Improved multilinear estimates and global regularity for general nonlinear wave equations in $(1+3)$ dimensions

This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity so that we can treat it as a toy model and simultaneously with enough generality so that we can apply our result to wave and Dirac equations with various nonlinearities. The challenging point is that nonlinearity possesses singularity near the origin. Our strategy is to relax such a singularity by exploiting fully an angular momentum operator. In this manner…

The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

In the present paper we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} ψ^{\varepsilon}(t) =-Δψ^{\varepsilon}(t) + \frac{1}εV\left(\frac{x}ε\right)|ψ^{\varepsilon}(t)|^{2μ}ψ^{\varepsilon}(t) \quad \quad ε>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ . \] This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit $ε\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlineari…

具有非局部非线性项的共形不变狄拉克方程

Conformally Invariant Dirac Equation with Non-Local Nonlinearity

本文研究受4维共形爱因斯坦-狄拉克问题启发的共形不变狄拉克方程，通过分析能量泛函的性质、刻画基态解、证明Aubin型不等式及研究线性扰动，确保了问题解的存在性并得到相关结果。

周期量子图上狄拉克方程的谱性质与束缚态

Spectral properties and bound states of the Dirac equation on periodic quantum graphs

本文研究了周期量子图上狄拉克方程的束缚态存在性，通过变分方法证明了至少一个束缚态的存在，并在非线性偶函数条件下证明了无限多个几何不同的束缚态。

Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

We study the point spectrum of the linearization at a solitary wave solution $ϕ_ω(x)e^{-\mathrm{i}ωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(ψ^*βψ)βψ$ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $ω\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(τ)=|τ|^k+O(|τ|^K)$ for $τ\to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($ω\lesssim m$) for the charge-subcr…

Global $H^s, s>0$ Large Data Solutions of 2D Dirac Equation with Hartree Type Interaction

Local and global well - posedness of the solution to the two space dimensional Dirac equation with Hartree type nonlinearity is established with the initial datum in the space $H^s(\mathbb{R}^2, \mathbb{C}^2)$ with $s >0.$.

Stability of the standing waves of the concentrated NLSE in dimension two

In this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the st…

Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity

We use the perturbation theory to build solitary wave solutions $ϕ_ω(x)e^{-iωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\barψψ)βψ$, with $f(τ)=|τ|^k+o(|τ|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $ω\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrödinger charge-critical, one has $Q'(ω)<0$ for $ω\lesssim m$, i…

An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.

具有非齐次非线性的非线性Dirac方程的归一化解

Normalized solutions for a nonlinear Dirac equation with an inhomogeneous nonlinearity

研究具有奇异权重的非线性Dirac方程归一化解的存在性，通过约束极小极大方法和加权紧嵌入，在小质量范围内统一建立了次临界、临界和超临界非线性情形的存在性。

Local well-posedness for the nonlinear Dirac equation in two space dimensions

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s > 1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.

Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based …

半线性Dirac方程的有界时间逆散射

Bounded Time Inverse Scattering for Semilinear Dirac Equation

该研究针对带光滑非线性项的半线性Dirac方程，首次证明有界时间逆散射问题的唯一性，通过构造趋近初始时间线的碰撞序列模拟边界碰撞，无需假设非线性结构即可确定相关区域内的非线性项。

Semilinear fractional elliptic equations with gradient nonlinearity involving measures

We study the existence of solutions to the fractional elliptic equation (E1) $(-Δ)^αu+εg(|\nabla u|)=ν$ in a bounded regular domain $Ω$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $Ω^c$, where $ε=1$ or $-1$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(1/2,1)$, $ν$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $ν$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $ε=1$.

A Remark on Nonlinear Dirac Equations

For a $n$-dimensional spin manifold $M$ with a fixed spin structure and a spinor bundle $ΣM$, we prove an $ε$-regularity theorem for weak solutions to the nonlinear Dirac equation of cubic nonlinearity. This, in particular, answers a regularity question raised by Chen-Jost-Wang when $n=2$.

Global well-posedness for the Schroedinger equation coupled to a nonlinear oscillator

The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$.

Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity

We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: $i\partial_tψ=-iα\partial_xψ+mβψ-f(ψ^\astβψ)βψ$, $ψ(x,t)\in\C^2$, $x\in\R$, $f\in C^\infty(\R)$, $m>0$, where $α$, $β$ are $2\times 2$ hermitian matrices which satisfy $α^2=β^2=1$, $αβ+βα=0$. We study the spectral stability of solitary wave solutions $ϕ_ω(x)e^{-iωt}$. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at solitary waves of arbitrarily small amplitude, in the limit $ω\to m$. We prove that if $f(s)…

Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-γ} * \langle ψ, βψ\rangle)βψ$ with $c\in \mathbb R\setminus\{0\} $, $0 < γ< 2$. Our aim is to show the small data global well-posedness and scattering in $H^s$ for $s > γ-1$ and $1 < γ< 2$. The difficulty stems from the singularity of the low-frequency part $|ξ|^{-(2-γ)}χ_{\{|ξ|\le 1\}}$ of potential. To overcome it we adapt $U^p-V^p$ space argument and bilinear estimates of \cite{yang, tes2d} arising from the null structure. We also provide nonexistence result for scattering in the long…

Strongly localized semiclassical states for nonlinear Dirac equations

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_tψ= ic\hbar\sum_{k=1}^3α_k\partial_kψ- mc^2βψ- M(x)ψ+ f(|ψ|)ψ,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|ψ|)ψ$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=…

Dirac concentrations in Lotka-Volterra parabolic PDEs

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends s…

Time-frequency analysis of the Dirac equation

The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for…

Boltzmann-Fermi-Dirac 方程的流体动力学极限：声学与 Stokes-Fourier

Hydrodynamic limits for the Boltzmann-Fermi-Dirac equation: Acoustic and Stokes-Fourier

本文在二维及以上环面上证明了 BFD 方程的声学和 Stokes-Fourier 流体动力学极限，通过改进弱解框架处理三次非线性，得到涨落的唯一极限。

一维双重非线性Schrödinger方程的指定质量基态

Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one

研究直线上两族聚焦非线性Schrödinger方程在固定质量下基态的存在性与唯一性，针对幂次为次临界、临界及双重临界等情形给出了相应条件与结论。

非线性狄拉克方程非协化势的稳态周期解

Stationary periodic solutions to Nonlinear Dirac equations with non-coercive potentials

本文研究了非线性狄拉克方程在非协化势下的稳态周期解问题，通过引入协化扰动和Sobolev范数估计，得到了非平凡解的存在性。

Scattering and non-scattering of the Hartree-type nonlinear Dirac system at critical regularity

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4π}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-d…

Solutions with peaks for a coagulation-fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In a companion paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of …

Harnack inequalities and quantization properties for the n-Liouville equation

We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtain…

Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet

In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} Δu =\frac{4e^u}{1+e^u} -4π\sum^{N}_{i=1}δ_{p_i}+4π\sum^{M}_{j=1}δ_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where $\{δ_{p_i}\}_{i=1}^N$ (resp. $\{δ_{q_j}\}_{j=1}^M$ ) are Dirac masses concentrated at the points $p_i, i=1,\cdots, N$, (resp. $q_j, i=1,\cdots, M$) %$δ_{p_j}$ is Dirac mass concentrated at the point $p_j$ and $N-M>1$. This equation presents a governing equation of Gauged Sigma model for Heisenberg ferromagnet and we prove that it has a sequence of solutions $u_β$ having behaviors as $-2πβ\l…

Solutions with peaks for a coagulation-fragmentation equation. Part II: aggregation in peaks

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solu…

构造狄拉克方程的吉布斯测度

Construction of a Gibbs measure for the zonal Dirac equation

本文提出了一种构造球面上狄拉克方程吉布斯测度的框架，并通过紧性论证证明了该测度的存在性。

一维Vlasov-Dirac-Benney型方程的局部解析适定性

Local analytic well-posedness for one-dimensional Vlasov$\unicode{x2013}$Dirac$\unicode{x2013}$Benney-type equations

本文研究了一维非线性Vlasov方程，其中自洽力场由密度生成，力由实分析非线性函数的空间导数给出。对于小的解析初始数据，证明了局部时间内的解析解的存在性和唯一性。同时给出了弱稳态解的能量表示，并讨论了围绕空间均匀稳态剖面的扰动。证明依赖于在解析函数完备度量空间中的压缩映射论证。

紧流形上能量超临界薛定谔方程的概率全局适定性

Probabilistic global-wellposedness for the energy-supercritical Schrödinger equations on compact manifolds

针对所有维度下带一般非线性幂次的能量超临界薛定谔方程，在一般紧黎曼流形（含环面、佐尔流形）上构造奇异阶Sobolev空间不变测度，证明解的几乎必然全局适定性及时间增长界。

Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions

We consider the Cauchy problem for coupled systems of wave and Klein-Gordon equations with quadratic nonlinearity in three space dimensions. We show global existence of small amplitude solutions under certain condition including the null condition on self-interactions between wave equations. Our condition is much weaker than the strong null condition introduced by Georgiev for this kind of coupled system. Consequently our result is applicable to certain physical systems, such as the Dirac-Klein-Gordon equations, the Dirac-Proca equations, and the Klein-Gordon-Zakharov equations.

非线性Dirac方程的归一化解

Normalized solutions for a nonlinear Dirac equation

该研究证明了非线性Dirac方程存在满足特定形式非线性项条件的归一化定态解，通过将解作为限制在L²单位球面的泛函临界点得到，频率w为对应拉格朗日乘子。

Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations

In a recent article [10], the authors proved that the non-relativistic Schrödinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.

On a class of semilinear fractional elliptic equations involving outside Dirac data

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-Δ)^α u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-Δ)^α +u^p} u=δ_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (-Δ)^α$ with $α\in(0,1)$ denotes the fractional Laplacian operator in the principle value sense, $B_1(e_N)$ is the unit ball centered at $e_N=(0,\cdots,0,1)$ in $\mathbb{R}^N$ with $N\ge 2$ and $δ_0$ is the Dirac mass concentrated at the origin. We pr…

非线性 Dirac 方程在 Bopp-Podolsky 电动力学中解的存在性

Existence of solutions for nonlinear Dirac equations in the Bopp-Podolsky electrodynamics

本文研究非线性 Dirac-Bopp-Podolsky 系统，给出物理动机，并在适当假设下利用极小极大技巧证明至少存在一对解。

On Global Attraction to Solitary Waves for the Klein-Gordon Equation Coupled to Nonlinear Oscillator

The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to\pm\infty$ to the set of ``nonlinear eigenfunctions'' $ψ(x)e\sp{-iωt}$.

关于半凸低阶部分波非线性狄拉克方程的对称性扰动

Perturbation from symmetry for semiconvex lower-order partial-wave nonlinear Dirac equations

研究\(\mathbb{R}^3\)中最低部分波通道的强不定非线性狄拉克泛函，通过施加半凸性条件等方法，利用多项式变形定理得出受限部分波泛函的多个高能临界点，这些点在通道不变性下是完整狄拉克方程的弱解，且凸扰动无幅度小性限制。