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Equation with a lower negative time number in the Davey–Stewartson hierarchy 在Davey-Stewartson层次结构中具有较低负时间数的方程
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S004057792412002X
A. K. Pogrebkov

Earlier, we have presented an integrable system with a negative time variable number for the Davey–Stewartson hierarchy. Here, we develop this approach to construct an integrable equation with a lower time variable number. In addition, we show that the system reduced with respect to this time yields a new integrable equation in (1+1) dimensions.

在此之前,我们提出了一个具有负时变数的可积系统的Davey-Stewartson层次。在此,我们利用此方法来构造具有较低时变数的可积方程。此外,我们证明了系统对这个时间的约简产生了一个新的(1+1)维的可积方程。
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引用次数: 0
A multicomponent generalized nonisospectral super AKNS integrable hierarchy 多分量广义非等谱超AKNS可积层次
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120067
Jinxiu Li, Haifeng Wang

In the nonisospectral case, we introduce the associated spectral problem with a perturbation term. We obtain a generalized nonisospectral super AKNS hierarchy and a coupled generalized nonisospectral super AKNS hierarchy associated with generalized Lie superalgebras (sl(2,1)) and (sl(4,1)). Based on a new type of multicomponent Lie superalgebra (sl(2N,1)), a multicomponent generalized nonisospectral super AKNS hierarchy is obtained. By using the supertrace identity, the super bi-Hamiltonian structures of the resulting superintegrable hierarchies are obtained.

在非等谱情况下,我们引入了带有扰动项的相关谱问题。得到了一个广义非等谱超AKNS结构和一个与广义李超代数(sl(2,1))和(sl(4,1))相关的耦合广义非等谱超AKNS结构。基于一种新的多分量李超代数(sl(2N,1)),得到了一种多分量广义非等谱超AKNS层次。利用超迹恒等式,得到了所得超可积层次的超双哈密顿结构。
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引用次数: 0
On the existence of a nonextendable solution of the Cauchy problem for a ((3+1))-dimensional thermal–electrical model 关于((3+1))维热电模型Cauchy问题不可扩展解的存在性
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120146
M. V. Artemeva, M. O. Korpusov

A thermal–electrical ((3+1))-dimensional model of heating a semiconductor in an electric field is considered. For the corresponding Cauchy problem, the existence of a classical solution nonextendable in time is proved and an a priori estimate global in time is obtained.

考虑了在电场中加热半导体的热电((3+1))维模型。对于相应的柯西问题,证明了经典解在时间上不可扩展的存在性,并得到了一个在时间上全局的先验估计。
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引用次数: 0
On the spectrum of the Landau Hamiltonian perturbed by a periodic smooth electric potential 在朗道哈密顿谱上被周期性平滑电势扰动
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120110
L. I. Danilov

We study the spectrum of the Landau Hamiltonian with a periodic electric potential. In the case of a rational magnetic flux, we present examples of nonconstant zero-mean periodic electric potentials ({Vin C^{infty}(mathbb{R}^2;mathbb{R})}) for which the spectrum has an eigenvalue at the second Landau level.

我们研究了具有周期电位的朗道哈密顿谱。在有理磁通量的情况下,我们提出了非恒定的零平均周期电势({Vin C^{infty}(mathbb{R}^2;mathbb{R})})的例子,其谱在第二朗道水平上具有特征值。
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引用次数: 0
Groups of diagonal gates in the Clifford hierarchy 克利福德层次结构中的对角门组
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120018
Lingxuan Feng, Shunlong Luo

The Clifford hierarchy plays a crucial role in the stabilizer formalism of quantum error correction and quantum computation. Apart form the zeroth level (the discrete Heisenberg–Weyl group) and the first level (the Clifford group), all other levels of the Clifford hierarchy are not groups. However, the diagonal gates at all levels do form groups, and it is desirable to characterize their generators and structures. In this paper, we study the diagonal gates at the second level of the Clifford hierarchy. For this, we introduce the notion of a (T)-gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the (T)-gate, we are able to completely characterize the group structures of the diagonal gates at the second level of the Clifford hierarchy in any (not necessarily prime) dimension. It turns out that the classification depends crucially on the number-theoretic nature of the dimension. The results highlight the special role of the first two primes, (2) and (3), in the prime factorization of the dimension. The (T)-gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.

Clifford层次结构在量子纠错和量子计算的稳定器形式中起着至关重要的作用。除了第0层(离散的Heisenberg-Weyl组)和第1层(Clifford组)之外,Clifford层次结构的所有其他层都不是组。然而,所有水平的对角线门确实形成了群,并且希望表征它们的发生器和结构。在本文中,我们研究了Clifford层次的第二层对角门。为此,我们引入了任意维上的(T) -门的概念,推广了素维上的相应概念。通过使用(T) -gate,我们能够在任何(不一定是素数)维度上完全表征Clifford层次的第二级对角线门的群结构。事实证明,分类关键取决于维度的数论性质。结果突出了前两个素数(2)和(3)在量纲质因数分解中的特殊作用。任意维度的(T) -门,除了作为对角门的产生器的关键作用外,在量子理论中可能有独立的兴趣和进一步的应用。
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引用次数: 0
New similarity reductions and exact solutions of the Date–Jimbo–Kashiwara–Miwa equation Date-Jimbo-Kashiwara-Miwa方程的新相似性约简和精确解
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120055
Dongwei Ran, Shaowei Liu

We study the ((2+1))-dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation by the CK direct method. In the literature, no one has used the CK direct method to solve the DJKM equation. However, in the process of solving the DJKM equation by the CK direct method, it is almost impossible to solve for all (beta) and (z), and we therefore use a certain method to find the concrete expressions of (beta) and (z) more easily. Finally, some new one-dimensional similarity reductions and new exact solutions of the DJKM equation are obtained via a large amount of complex and tedious calculations; these solutions can contain some arbitrary functions of (t).

本文用CK直接法研究了((2+1))维非线性Date-Jimbo-Kashiwara-Miwa (DJKM)方程。在文献中,没有人使用CK直接法求解DJKM方程。然而,在用CK直接法求解DJKM方程的过程中,几乎不可能解出所有的(beta)和(z),因此我们使用一定的方法更容易找到(beta)和(z)的具体表达式。最后,通过大量复杂繁琐的计算,得到了DJKM方程新的一维相似约简和精确解;这些解可以包含(t)的一些任意函数。
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引用次数: 0
Generating quantum dynamical mappings 生成量子动力学映射
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-12-25 DOI: 10.1134/S0040577924120122
R. N. Gumerov, R. L. Khazhin

We consider one-parameter families of generating quantum channels. Such families are called the generating quantum dynamical mappings or the generating quantum processes. By the generating channels of composite quantum systems, we understand the channels that allow the channels of constituent subsystems, called the induced channels, to be uniquely defined. Using the criterion for generating and induced linear mappings, we study the properties of bijective quantum channels and the properties of quantum processes consisting of such channels. Using a generating quantum dynamical mapping, we naturally construct the induced dynamical mapping. We show that the properties of continuity and completely positive divisibility of generating quantum dynamical mappings are hereditary for induced dynamical mappings. As an application of the obtained results, we construct continuous completely positive evolutions. For generating quantum dynamical mappings taking values in the set of phase-damping channels, we obtain a criterion for the completely positive divisibility.

我们考虑产生量子信道的单参数族。这些族被称为生成量子动力学映射或生成量子过程。通过复合量子系统的生成通道,我们了解了允许组成子系统的通道(称为诱导通道)被唯一定义的通道。利用线性映射的生成和诱导准则,研究了双射量子通道的性质和由双射量子通道组成的量子过程的性质。利用生成的量子动态映射,我们自然构造了诱导的动态映射。我们证明了生成量子动力映射的连续性和完全正可整除性对于诱导动力映射是遗传的。作为已获得结果的应用,我们构造了连续的完全正演化。对于产生在相位阻尼通道集合中取值的量子动态映射,我们得到了一个完全正可整除的判据。
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引用次数: 0
On the Hirota equation with a self-consistent source 关于具有自洽源的广田方程
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-11-26 DOI: 10.1134/S0040577924110059
A. B. Khasanov, A. A. Reyimberganov

We develop the formalism of the inverse scattering problem method for the Cauchy problem for the defocusing Hirota equation with a self-consistent source. The specific feature of the considered Cauchy problem is that the solution is assumed to approach nonzero limits as the spatial variable approaches the plus and minus infinities. The purpose of the paper is to present two main steps of the formalism: first, the inverse problem for the associated linear Zakharov–Shabat system and, second, the evolution of the associated scattering data. A theorem is proved on the evolution of scattering data of a self-adjoint Zakharov–Shabat system, with the potential given by a solution of the defocusing Hirota equation with a self-consistent source.

我们针对具有自洽源的广达(Hirota)离焦方程的考奇问题,提出了反散射问题法的形式主义。所考虑的 Cauchy 问题的具体特征是,当空间变量接近正无穷大和负无穷大时,假设解接近非零极限。本文旨在介绍形式主义的两个主要步骤:第一,相关线性 Zakharov-Shabat 系统的逆问题;第二,相关散射数据的演化。本文证明了一个关于自共轭 Zakharov-Shabat 系统散射数据演化的定理,该系统的势由具有自洽源的离焦 Hirota 方程的解给出。
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引用次数: 0
Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models 点物质黑洞和非交换黑洞模型中的哈密顿映射和量子扰动方程
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-11-26 DOI: 10.1134/S0040577924110138
Jun Yan

The Hamiltonian mapping of the Dirac equation with a quantum perturbation in (2)D black hole models are investigated. We derive the Hamiltonians of the modified Rabi model for the point matter black hole and noncommutative black hole, and obtain perturbed expressions for the velocity and acceleration of the simulating Dirac particles. The fluctuation equations for the charge–current density in two types of black holes are derived based on the solutions of the Dirac equation. The results indicate that the Hamiltonians contain the correction terms with different powers of the creation and annihilation operators, and the accelerations have some additional Dirac Zitterbewegung terms. In addition, the coordinate operators in the fluctuation equations of the charge–current density have different powers. These characteristics can be used to distinguish different types of black holes in the analogical gravity models.

研究了具有量子扰动的狄拉克方程在(2)D黑洞模型中的哈密顿映射。我们推导了点物质黑洞和非交换黑洞的修正拉比模型的哈密顿,并得到了模拟狄拉克粒子的速度和加速度的扰动表达式。根据狄拉克方程的解,推导出了两种黑洞中电荷电流密度的波动方程。结果表明,汉密尔顿方程包含不同幂次的创造和湮灭算子修正项,加速度包含一些附加的狄拉克齐特贝克项。此外,电荷电流密度波动方程中的坐标算子也具有不同的幂。这些特征可用来区分类比引力模型中的不同类型黑洞。
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引用次数: 0
Lie group geometry: Riemann and Ricci tensors and normal forms of Lie algebras 李群几何:黎曼和利玛窦张量及列代数的正常形式
IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Pub Date : 2024-11-26 DOI: 10.1134/S0040577924110011
A. V. Borovskikh

In the context of the connection discovered in a preceding paper between left-invariant objects (both geometric and dynamical) defined on a Lie group and the algebra of right automorphisms (the dual algebra), we consider the representation of the main geometric characteristics via this algebra and the corresponding metric form. These characteristics are shown to be constant (independent of a point) and defined only by the structure constants of the dual algebra and the coefficients of the metric form. Due to this connection, it is possible to introduce the concept of normal forms of a Lie algebra. Reducing any algebra and any metric to normal form in fact consists in reducing two quadratic forms to canonical form: first, the metric is reduced to the sum of squares of linear differential forms, and then the constant matrix characterizing the Ricci tensor is reduced to diagonal form (with the principal curvatures appearing on the diagonal). It turns out that there are only two different normal forms for three-dimensional Lie algebras, each depending on three parameters associated with three principal curvatures in the general case.

根据前一篇论文中发现的定义在李群上的左不变对象(包括几何和动力学)与右自变量代数(对偶代数)之间的联系,我们考虑通过该代数和相应的度量形式来表示主要几何特征。这些特征被证明是恒定的(与点无关),并且仅由对偶代数的结构常数和度量形式的系数定义。由于这种联系,我们有可能引入李代数正常形式的概念。将任何代数和任何度量形式还原为正则形式,实际上就是将两个二次方形式还原为规范形式:首先,度量形式被还原为线性微分形式的平方和,然后,表征利玛窦张量的常数矩阵被还原为对角线形式(主曲率出现在对角线上)。事实证明,三维李代数只有两种不同的法线形式,在一般情况下,每种形式都取决于与三个主曲率相关的三个参数。
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Theoretical and Mathematical Physics
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