On gauge equivalent integrable systems and<Emphasis Type="Italic">r</Emphasis>-matrices for AKNS hierarchy and a coupled MKdV hierarchy

The new coupled MKdV hierarchy is obtained. By using gauge transformation, the constrained flow, the integrable system and Lax representation for the coupled MKdV hierarchy were first constructed from the AKNS hierarchy and then using the Lax representation, the r-matrix for the constrained flow of the coupled MKdV hierarchy was constructed. The second set of conserved integrals of this constrained flow and their involutivity were also given. Biography: LIU Xiang-lin (1966-), Associate Professor (E-mail: zhangbc@sjzri.edu.cn)     

A New Completely Integrable Liouville's System, its Lax Representation and Bi-Hamiltonian Structure

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓａｎｉｍｐｏｒｔａｎｔｔｏｐｉｃｔｏｓｅａｒｃｈｆｏｒｎｅｗｉｎｔｅｇｒａｂｌｅｓｙｓｔｅｍｓｉｓｓｏｌｉｔｏｎｔｈｅｏｒｙ .Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ｔｗｏｖｅｒｙｅｆｆｅｃｔｉｖｅａｐｐｒｏａｃｈｅｓｔｈａｔｐｒｏｄｕｃｅｉｎｆｉｎｉｔｅ＿ｄｉｍｅｎｓｉｏｎａｌａｎｄｆｉｎｉｔｅ＿ｄｉｍｅｎｓｉｏｎａｌｉｎｔｅｇｒａｂｌｅＨａｍｉｌｔｏｎｉａｎｓｙｓｔｅｍｓｒｅｓｐｅｃｔｉｖｅｌｙｈａｖｅｂｅｅｎｄｅｖｅｌｏｐｅｄ .Ｔｈｅｆｉｒｓｔｏｎｅｉｓｔｈｅｔｒａｃｅ…     

重庆市区域稳定性层次分析模糊综合评价

重庆直辖市建立后 ,其幅员面积由原来的 2 .3万km2 扩大到 8.2万km2 ,原有资源与环境数据已不能满足政府决策的需要 ,因而比较系统地分析评价区域稳定性对重庆直辖市经济发展中长期规划、重大工程选址与布局及移民搬迁 ,实现可持续发展具有十分重要的意义。本文在前人工作的基础上 ,利用层次分析两级模糊综合评判的方法对重庆直辖市全境进行了区域稳定性评价工作 ,编制了区域稳定性评价图。     

The Hamiltonian equations in some mathematics and physics problems

IntroductionHamiltoniansystemsareveryimportantinmathematics,physicsandmechanics.SpeciallyafterFengKang[1]firstproposedthesymplecticnumericalmethodforfinitedimensionalHamiltoniansystemin 1 980s,thestudyofHamiltoniansystemhasbeenfurtherimprovedinboththeoryandapplication .ItisthereforenaturaltoconsidertheproblemaboutwhichkindofpartialdifferentialequationscanberewrittenorrepresentedbyHamiltoniansystem .SuchkindofproblemhasnowbeenwarmlystudiedbymanymathematiciansandphysicianssuchasinRefs.[2 -5] …     

Darboux transformation of generalized coupled KdV soliton equation and its odd-soliton solutions

Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.     

层次分析法在工程岩体分类中的应用

层次分析法是一种考虑定性与定量相结合的多目标决策分析方法。岩体工程分类受多种因素影响,本文选用多种因素进行全面的综合分析,建立了由4个层次组成的岩体工程分类层次分析体系,详细介绍了不同层次的判断矩阵的构造及分类过程。实测数据计算结果表明:岩体工程分类层次分析体系简单、适用、系统性强,评价结果真实、有效。     

Low-rank solution of an optimal control problem constrained by random Navier-Stokes equations

We develop a low-rank tensor decomposition algorithm for the numerical solution of a distributed optimal control problem constrained by two-dimensional time-dependent Navier-Stokes equations with a stochastic inflow. The goal of optimization is to minimize the flow vorticity. The inflow boundary condition is assumed to be an infinite-dimensional random field, which is parametrized using a finite- (but high-) dimensional Fourier expansion and discretized using the stochastic Galerkin finite element method. This leads to a prohibitively large number of degrees of freedom in the discrete solution. Moreover, the optimality conditions in a time-dependent problem require solving a coupled saddle-point system of nonlinear equations on all time steps at once. For the resulting discrete problem, we approximate the solution by the tensor-train (TT) decomposition and propose a numerically efficient algorithm to solve the optimality equations directly in the TT representation. This algorithm is based on the alternating linear scheme (ALS), but in contrast to the basic ALS method, the new algorithm exploits and preserves the block structure of the optimality equations. We prove that this structure preservation renders the proposed block ALS method well posed, in the sense that each step requires the solution of a nonsingular reduced linear system, which might not be the case for the basic ALS. Finally, we present numerical experiments based on two benchmark problems of simulation of a flow around a von Kármán vortex and a backward step, each of which has uncertain inflow. The experiments demonstrate a significant complexity reduction achieved using the TT representation and the block ALS algorithm. Specifically, we observe that the high-dimensional stochastic time-dependent problem can be solved with the asymptotic complexity of the corresponding deterministic problem.     

Multidimensional upwind schemes for the shallow water equations

A multidimensional discretisation of the shallow water equations governing unsteady free-surface flow is proposed. The method, based on a residual distribution discretisation, relies on a characteristic eigenvector decomposition of each cell residual, and the use of appropriate distribution schemes. For uncoupled equations, multidimensional convection schemes on compact stencils are used, while for coupled equations, either system distribution schemes such as the Lax–Wendroff scheme or scalar schemes may be used. For steady subcritical flows, the equations can be partially diagonalised into a purely convective equation of hyperbolic nature, and a set of coupled equations of elliptic nature. The multidimensional discretisation, which is second-order-accurate at steady state, is shown to be superior to the standard Lax–Wendroff discretisation. For steady supercritical flows, the equations can be fully diagonalised into a set of convective equations corresponding to the steady state characteristics. Discontinuities such as hydraulic jumps, are captured in a sharp and non-oscillatory way. For unsteady flows, the characteristic equations remain coupled. An appropriate treatment of the coupling terms allows the discretisation of these equations at the scalar level. Although presently only first-order-accurate in space and time, the classical dam-break problem demonstrates the validity of the approach. © 1998 John Wiley & Sons, Ltd.     

与非平衡问题相关的尺度效应:场与微粒

纳米技术的出现,使我们有必要更好地了解,在原子水平上材料微结构的变化是如何影响和控制着材料的宏观性能.这一挑战涉及到许多以前不曾考虑和不曾了解的现象.其中,位错理论的基础现在知道是有问题的.宏观尺度下采用的简化假设,也许不能用于微观和纳米尺度.尺度效应的含义,涉及到物理系统的非均质和非平衡特性.宏观尺度下的均匀与平衡特性,在材料的物理尺度减少到微米量级时就不再保持了.这些基本观点不能够为了方便而随意到处使用,因为这会改变预测的结果.更令人不满的是在建立物理模型时缺乏一致性.由此产生的问题是在确定制造过程中的有关参数时无能为力,导致由于成本过高而不切实际的终端产品.先进的复合材料和陶瓷材料就存在这样的问题.本文将要讨论的是在原子尺度与连续介质尺度下应用理论模型时存在的潜在问题,而不是去揭示自然的真相.主要讨论微粒,均匀连续介质或者两者的结合.尺度效应问题当前的发展趋势,趋向于在有或者没有时间效应的情况下寻找材料微结构的不同特征尺寸.从原子模拟模型中将了解到许多情况,原子模拟计算将揭示计算结果如何随着边界条件和尺度变化而不同.量子力学,连续介质力学和宇宙模型证明,没有普遍适用的方法.当前的主要兴趣也许是针对多尺度物理问题在技术上建立更高的精度,以得到一个更好的表达结果.      

Wave propagation in a fractional viscoelastic Andrade medium: Diffusive approximation and numerical modeling

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.     

Generalized structure of Lax representations for nonlinear evolution equation

I.IntroductionInsolitontheory,itisveryinterestingforustofindtheLaxrepresentationsofnonlinearevolutionequations(NLEEs)andtodiscussthealgebraicstructureofLaxoperatorandotherproperties.MaWenxiustudiedtheLaxrepresentationstructureforthehierarchiesofisospectra…     

An efficient numerical method for reactive flow with general equation of states

This paper presents an efficient method to simulate the reactive flow for general equation of states with the compressible fluid model coupled with reactive rate equation. The important aspect is to deal with mixture of different phases in one cell, which will inevitably happen in the Eulerian method for reactive flow. Physical variables such as the pressure,velocity, and speed of sound in each cell need to be reconstructed for the Harten‐Lax‐Leer‐Contact (HLLC) Riemann solver, which will result in nonlinear algebra equations, and these reconstructed variables are used to obtain the flux. Numerical examples of stable and unstable detonations with different equation of states demonstrate the accuracy and efficiency of this method. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐dimensional shallow water equations by composite schemes

Composite schemes are formed by global composition of several Lax–Wendroff steps followed by a diffusive Lax–Friedrichs or WENO step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. These schemes are applied to the shallow water equations in two dimensions. The Lax–Friedrichs composite is also formulated for a trapezoidal mesh, which is necessary in several example problems. The suitability of the composite schemes for the shallow water equations is demonstrated on several examples, including the circular dam break problem, the shock focusing problem and supercritical channel flow problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Conservation laws and Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients in nonlinear optics

In this paper, by Darboux transformation and symbolic computation we investigate the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients, which come from twin-core nonlinear optical fibers and waveguides, describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the non-Kerr media. Lax pair of the equations is obtained, and the corresponding Darboux transformation is constructed. One-soliton solutions are derived; some physical quantities such as the amplitude, velocity, width, initial phases, and energy are, respectively, analyzed; and finally an infinite number of conservation laws are also derived. These results might be of some value for the ultrashort optical pulse propagation in the non-Kerr media.     

ANALYSIS AND IMPLEMENTATION OF THE GAS-KINETIC BGK SCHEME FOR COMPUTATIONAL GAS DYNAMICS

Gas-kinetic schemes based on the BGK model are proposed as an alternative evolution model which can cure some of the limitations of current Riemann solvers. To analyse the schemes, simple advection equations are reconstructed and solved using the gas-kinetic BGK model. Results for gas-dynamic application are also presented. The final flux function derived in this model is a combination of a gas-kinetic Lax– Wendroff flux of viscous advection equations and kinetic flux vector splitting. These two basic schemes are coupled through a non-linear gas evolution process and it is found that this process always satisfies the entropy condition. Within the framework of the LED (local extremum diminishing) principle that local maxima should not increase and local minima should not decrease in interpolating physical quantities, several standard limiters are adopted to obtain initial interpolations so as to get higher-order BGK schemes. Comparisons for well-known test cases indicate that the gas-kinetic BGK scheme is a promising approach in the design of numerical schemes for hyperbolic conservation laws. © 1997 by John Wiley & Sons, Ltd.     

A family of integrable systems of Liouville and lax representation,Darboux transformations for its constrained flows

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｗｅｌｌ＿ｋｎｏｗｎｂａｓｉｓｏｆｌｏｏｐａｌｇｅｂｒａ Ａ1ｉｓｔｈａｔｈ(ｎ) =λｎ 　 00 -λｎ ,　ｅ(ｎ) =0λｎ0 0 ,ｆ(ｎ) =0 0λｎ 0 ,[ｈ(ｍ) ,ｅ(ｎ) ] =2ｅ(ｍ+ｎ) ,[ｈ(ｍ) ,ｆ(ｎ) ] =　　 -2ｆ(ｍ+ｎ) ,[ｅ(ｍ) ,ｆ(ｎ) ] =ｈ(ｍ+ｎ) .( 1 )Ｍａｎｙｅｑｕａｔｉｏｎｈｉｅｒａｒｃｈｉｅｓｗｅｒｅｏｂｔａｉｎｅｄｂｙｅｓｔａｂｌｉｓｈｉｎｇｌｉｎｅａｒｉｓｏｓｐｅｃｔｒａｌｐｒｏｂｌｅｍｖｉａｔｈｅｕｓｅｏｆ( 1 ) ,ｓｕｃｈａｓＡＫＮＳｈｉｅｒａｒｃｈｙ ,ＫＮｈｉｅｒａ…     

Interactions of the vector breathers for the coupled Hirota system with $$4\times 4$$ 4 × 4 Lax pair

Nonlinear Dynamics - In this paper, we investigate the interactions of the vector breathers for the coupled Hirota system with $$4\times 4$$ Lax pair. Firstly, we give the first-order breather...     

Complete Integrability of Shock Clustering and Burgers Turbulence

We consider scalar conservation laws with convex flux and random initial data. The Hopf–Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial datum is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for N-dimensional approximations of the Lax equations, and yield formulas for the limit N → ∞. The main observation is that the Lax equations and zero-curvature equations are a Markovian analog of known integrable systems (geodesic flow on Lie groups and the N-wave model respectively). This allows us to introduce a variety of methods from the theory of integrable systems.     

Soliton,breather and rogue wave solutions of the coupled Gerdjikov–Ivanov equation via Darboux transformation

Nonlinear Dynamics - The coupled Gerdjikov–Ivanov (cGI) equation, associated with a $$3\times 3$$ matrix Lax pair, is an important integrable system that can be reduced to the third kind of...     

岩石边坡稳定性层次类比评价方法

本文从分析影响边坡稳定性因素的构成关系入手,根据边坡岩体的地质特征,提出了分类层次结构及其权重体系的表征、边坡稳定性排序和根据工点之间的综合相似关系给出边坡稳定性类比评价方法。这种方法能更充分地表达边坡岩体的各种特征及其相互间的关系与联系,统一了可定量的数值数据与不可定量的等级数据,排除了不相关因素的干扰。它以实际典型地质模型为评价基础,能从总体上把握岩体的主要特征,因而得出的评价结果是可靠的。