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On the Problem of Energy Concentration 关于能源集中问题
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040077
A. N. Golubyatnikov, D. V. Ukrainskii

Abstract

We discuss the well-known problem of energy concentration, which is the inverse of the strong explosion or the expanding piston problem. Using a number of physical examples, we show that under certain conditions and with certain forces involved in the focusing process, one can achieve the concentration of any amount of energy. First of all, this applies to the gravity force and its manifestation in problems of relativity theory with viscosity and heat conduction taken into account.

摘要 我们讨论了众所周知的能量集中问题,它是强爆炸或膨胀活塞问题的逆问题。通过一些物理实例,我们表明,在一定条件下,如果聚焦过程中涉及一定的力,就可以实现任何数量能量的集中。首先,这适用于万有引力及其在相对论问题中的表现,并考虑了粘度和热传导。
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引用次数: 0
Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points 流量函数有四个拐点情况下广义科特韦格-德弗里斯-伯格斯方程的经典和特殊不连续结构
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040211
V. A. Shargatov, A. P. Chugainova, A. M. Tomasheva

Abstract

We study the structure of the set of traveling wave solutions for the generalized Korteweg–de Vries–Burgers equation with the flux function having four inflection points. In this case there arise two monotone structures of stable special discontinuities propagating at different velocities (such a situation has not been described earlier in the literature). Both structures of special discontinuities are linearly stable. To analyze the linear stability of the structures of classical and special discontinuities, we apply a method based on the use of the Evans function. We also propose a conjecture that establishes the admissibility of classical discontinuities in the case when there are two stable special discontinuities.

摘要 我们研究了通量函数有四个拐点的广义 Korteweg-de Vries-Burgers 方程的行波解集结构。在这种情况下,会出现两个以不同速度传播的稳定特殊不连续的单调结构(这种情况在以前的文献中没有描述过)。这两种特殊不连续结构都具有线性稳定性。为了分析经典不连续面和特殊不连续面结构的线性稳定性,我们采用了一种基于埃文斯函数的方法。我们还提出了一个猜想,即在存在两个稳定的特殊不连续性的情况下,经典不连续性的可接受性。
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引用次数: 0
A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case 在退化情况下证明无限时间间隔上平均定理的拓扑分析方法
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040168
Ivan Yu. Polekhin

Abstract

We present a topological–analytical method for proving some results of the N. N. Bogolyubov averaging method for the case of an infinite time interval. The essence of the method is to combine topological methods of proving the existence of a periodic solution applied to the averaged system with Bogolyubov’s theorem on the averaging on a finite time interval. The proposed approach allows us to dispense with the nondegeneracy condition for the Jacobi matrix from the classical theorems of the averaging method.

摘要 我们提出了一种拓扑分析方法,用于证明 N. N. Bogolyubov 平均法在无限时间间隔情况下的一些结果。该方法的实质是将证明周期解存在的拓扑方法与博格留波夫关于有限时间间隔上的平均定理结合起来。所提出的方法使我们可以省去平均法经典定理中雅各比矩阵的非退化条件。
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引用次数: 0
Separatrix Maps in Slow–Fast Hamiltonian Systems 慢-快哈密顿系统中的分离矩阵图
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040041
Sergey V. Bolotin

Abstract

We obtain explicit formulas for the separatrix map of a multidimensional slow–fast Hamiltonian system. This map is used to partly extend Neishtadt’s results on the jumps of adiabatic invariants at the separatrix to the multidimensional case.

摘要 我们得到了多维慢-快哈密顿系统分离矩阵图的明确公式。利用该映射,我们可以将 Neishtadt 关于分离矩阵处绝热不变式跃迁的结果部分扩展到多维情况。
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引用次数: 0
Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory 浅水理论框架下带有半刚性壁的台球产生的海岸波的非线性效应和上升趋势
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040090
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova

Abstract

By coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth (D(x)), (x=(x_1,x_2)), that are localized in the vicinity of the coastline (Gamma^0={D(x)=0}). In this paper, for the system of nonlinear shallow water equations, we construct asymptotic solutions corresponding to coastal waves in two specific examples. The solutions are presented in the form of parametrically defined functions corresponding to asymptotic solutions of the linearized system, which, in turn, are related to the asymptotic eigenfunctions of the operator (-nablacdot (g D(x)nabla)) that are generated by billiards with semi-rigid walls.

Abstract by coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth (D(x)), (x=(x_1,x_2)), that are localized in the vicinity of the coastline (Gamma^0={D(x)=0})。本文针对非线性浅水方程系统,在两个具体例子中构建了与海岸波相对应的渐近解。这些解以参数定义函数的形式呈现,这些函数与线性化系统的渐近解相对应,而线性化系统的渐近解又与具有半刚性壁的台球产生的算子 (-nablacdot (g D(x)nabla)) 的渐近特征函数相关。
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引用次数: 0
On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface 论粘性流体沿曲面流下的不稳定层表面上的波浪
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040120
A. G. Kulikovskii, J. S. Zayko

Abstract

We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a (delta)-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that (x) is a real variable and to allow time (t) to take complex values.

摘要 我们考虑了曲面上粘性流体层不稳定流动的小扰动线性波的演化。假设扰动源是由定义在一个小域中的初始条件(在极限情况下,以一个 (delta)-function 的形式)或瞬时局部外部冲击给出的。扰动的行为由水层厚度上平均的流体力学方程描述,并考虑了重力和底部摩擦力(圣-维南方程)。我们研究了大时间一维扰动的渐近行为。表面与地平线的倾角由空间变量的缓慢变化函数定义。我们重点研究作为时间和空间变量函数的扰动振幅。为了研究扰动的渐近线,我们使用了著名的基于鞍点技术的方法的简单推广,该方法用于寻找在均匀背景下发展的扰动的渐近线。我们证明,这种方法等同于应用近似 WKB 方法构建微分方程解的方法。在构建渐近线时,可以方便地假设 (x) 是实变量,并允许时间 (t) 取复数值。
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引用次数: 0
Equilibrium Model of Density Flow 密度流平衡模型
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040144
V. Yu. Liapidevskii

Abstract

The flow of a stratified fluid over a slope is considered. In the one-layer shallow water approximation, a mathematical model is constructed for a turbulent flow of a denser fluid over a uniform slope, with the entrainment of the ambient fluid at rest and the sediment entrainment at the wave front taken into account. The main focus is on analyzing the structure of a self-sustaining wave (underwater avalanche) and on estimating its propagation velocity. The mathematical model arises from the equilibrium conditions in a more complete three-parameter model and contains only one numerical parameter that represents a combination of the parameters of the original model characterizing the slope, vortex energy dissipation rate, and entrainment rate. The structure of traveling waves is studied, exact self-similar solutions are constructed, and transition of the flow to a self-similar regime is analyzed numerically. It is shown that depending on the thickness and initial density of the sediment layer, self-similar solutions have different structures and front propagation velocities.

摘要 研究了分层流体在斜坡上的流动。在单层浅水近似条件下,建立了高密度流体在均匀斜坡上湍流的数学模型,并考虑了静止时环境流体的夹带和波浪前沿的沉积物夹带。主要重点是分析自持波(水下雪崩)的结构和估算其传播速度。数学模型源于更完整的三参数模型中的平衡条件,只包含一个数值参数,该参数代表了原始模型中表征坡度、涡流能量耗散率和夹带率的参数组合。研究了行波的结构,构建了精确的自相似解,并对流动向自相似状态的过渡进行了数值分析。结果表明,根据沉积层的厚度和初始密度,自相似解具有不同的结构和前沿传播速度。
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引用次数: 0
Analytic Properties of Solutions to the Equation of Internal Gravity Waves with Flows for Critical Modes of Wave Generation 有流动的内部重力波方程中波产生临界模式的解的解析性质
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040065
V. V. Bulatov

Abstract

Issues related to the statement of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows in critical modes of wave generation are considered. Model physical statements of problems in which critical levels may arise are discussed in the two-dimensional case. Analytic properties of the solutions near critical levels are studied. A system describing a flow of a stratified medium incident on an obstacle behind which outgoing waves may arise is discussed, in which case a singularity at the critical level is formed far away from the obstacle. Asymptotics of the solutions near the critical level are constructed and expressed in terms of the incomplete gamma function.

摘要 研究了在波产生的临界模式下,在具有水平剪切流的分层介质中描述线性内重力波的动力学问题。在二维情况下,讨论了可能出现临界水平问题的物理模型陈述。研究了临界水平附近解的分析特性。讨论了一个描述入射到障碍物上的分层介质流的系统,在该障碍物后面可能会产生出射波,在这种情况下,临界水平的奇点会在远离障碍物的地方形成。构建了临界水平附近解的渐近线,并用不完全伽马函数表示。
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引用次数: 0
Exact Solutions of Second-Grade Fluid Equations 二级流体方程的精确解
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040156
A. G. Petrova, V. V. Pukhnachev, O. A. Frolovskaya

Abstract

The second-grade fluid equations describe the motion of relaxing fluids such as aqueous solutions of polymers. The existence and uniqueness of solutions to the initial–boundary value problems for these equations were studied by D. Cioranescu, V. Girault, C. Le Roux, A. Tani, G. P. Galdi, and others. However, their studies do not contain information about the qualitative properties of solutions of these equations. Such information can be obtained by analyzing their exact solutions, which is the main goal of this work. We study layered flows and a model problem with a free boundary, construct an analog of T. Kármán’s solution, which describes the stationary motion of a second-grade fluid in a half-space induced by the rotation of the plane bounding it, and propose a generalization of V. A. Steklov’s solution of the problem on unsteady helical flows of a Newtonian fluid to the case of a second-grade fluid.

摘要 二级流体方程描述了聚合物水溶液等松弛流体的运动。D. Cioranescu、V. Girault、C. Le Roux、A. Tani、G. P. Galdi 等人研究了这些方程初界值问题解的存在性和唯一性。然而,他们的研究并不包含有关这些方程的解的定性属性的信息。这些信息可以通过分析其精确解来获得,而这正是本研究的主要目标。我们研究了层流和自由边界的模型问题,构建了 T. Kármán 的类似解,该解描述了第二级流体在其边界平面旋转引起的半空间中的静止运动,并提出了将 V. A. Steklov 的牛顿流体非稳态螺旋流动问题解推广到第二级流体的情况。
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引用次数: 0
On Linear Equations of Dynamics 论动力学线性方程
IF 0.5 4区 数学 Q3 Mathematics Pub Date : 2023-12-20 DOI: 10.1134/s0081543823040119
V. V. Kozlov

Abstract

We consider linear autonomous systems of second-order differential equations that do not contain first derivatives of independent variables. Such systems are often encountered in classical mechanics. Of particular interest are cases where external forces are not potential. An important special case is given by the equations of nonholonomic mechanics linearized in the vicinity of equilibria of the second kind. We show that linear systems of this type can always be represented as Lagrange and Hamilton equations, and these equations are completely integrable: they admit complete sets of independent involutive integrals that are quadratic or linear in velocity. The linear integrals are Noetherian: they appear due to nontrivial symmetry groups.

摘要 我们考虑的是不包含自变量一阶导数的二阶微分方程线性自治系统。这种系统在经典力学中经常遇到。尤其令人感兴趣的是外力不是潜在的情况。一个重要的特例是在第二类平衡点附近线性化的非荷尔蒙力学方程。我们证明,这种类型的线性方程组总是可以表示为拉格朗日方程和汉密尔顿方程,而且这些方程是完全可积分的:它们允许独立的渐开线积分的完整集合,这些独立的渐开线积分是速度的二次积分或线性积分。这些线性积分是诺特积分:它们因非对偶对称群而出现。
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引用次数: 0
期刊
Proceedings of the Steklov Institute of Mathematics
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