共查询到20条相似文献,搜索用时 78 毫秒
1.
A. S. Lewis P. A. Parrilo M. V. Ramana 《Proceedings of the American Mathematical Society》2005,133(9):2495-2499
In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov.
2.
Jinbing Chen 《Studies in Applied Mathematics》2011,127(2):153-190
An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω2) . Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω2) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem. 相似文献
3.
J. Golenia O. Ye. Hentosh A. K. Prykarpatsky 《Central European Journal of Mathematics》2007,5(1):84-104
The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied. 相似文献
4.
V.V. Palin 《Applicable analysis》2013,92(8):1233-1264
We study the large-time behaviour of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consolidated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. 相似文献
5.
In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces. 相似文献
6.
Valery G. Yakhno Handan Çerdik Yaslan Tatiana M. Yakhno 《Central European Journal of Mathematics》2012,10(1):188-203
A new method for computation of the fundamental solution of electrodynamics for general anisotropic nondispersive materials
is suggested. It consists of several steps: equations for each column of the fundamental matrix are reduced to a symmetric
hyperbolic system; using the Fourier transform with respect to space variables and matrix transformations, formulae for Fourier
images of the fundamental matrix columns are obtained; finally, the fundamental solution is computed by the inverse Fourier
transform. Applying the suggested approach, the fundamental solution components are computed in general anisotropic media.
Computational examples confirm robustness of the suggested method. 相似文献
7.
This article concerns the question, Which subsets of ?m can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also, before having much hope of representing engineering problems as LMIs by automatic methods, one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition that we call “rigid convexity,” which must hold for a set ?? ? ?m in order for ?? to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m = 2. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [15]. As shown by Lewis, Parillo, and Ramana [11], our main result also establishes (in the case of three variables) a 1958 conjecture by Peter Lax on hyperbolic polynomials. © 2006 Wiley Periodicals, Inc. 相似文献
8.
H. Çerdik Yaslan 《Applied Mathematical Modelling》2013,37(18-19):8409-8418
The fundamental solution (FS) of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are studied in the paper. Equations of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are written in the form of a symmetric hyperbolic system of the first order. Using the Fourier transform with respect to the space variables and matrix transformations we obtain explicit formulae for Fourier images of the FS columns; finally, the FS is computed by the inverse Fourier transform. As a computational example applying the suggested approach FS components are computed for icosahedral QCs. 相似文献
9.
Integrability and equivalence relationships of six integrable coupled Korteweg–de Vries equations 下载免费PDF全文
Deng‐Shan Wang Jiang Liu Zhifei Zhang 《Mathematical Methods in the Applied Sciences》2016,39(12):3516-3530
In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665‐669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth‐order Lax pairs instead of the fifth‐order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto‐Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
10.
In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with “small” initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with “small” initial data. 相似文献
11.
We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere. 相似文献
12.
A system of conservation laws admitting an additional convex conservation law can be written as a symmetric t‐hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non‐conservative hyperbolic models. We generalize the Godunov–Friedrichs–Lax approach to this new class of models. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
13.
We discuss the linearization of a nonautonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries, which turn out to be nonautonomous and dependent on an arbitrary function of the dependent variables defined at two lattice points. These generalized symmetries are differential–difference equations, which admit peculiar Bäcklund transformations in some cases. 相似文献
14.
Xiaohan Long Chuanjun Chen 《Numerical Methods for Partial Differential Equations》2007,23(6):1321-1342
Implicit–explicit multistep characteristic methods are given for convection‐dominated diffusion equations. Multistep difference along characteristics of the one‐order hyperbolic part of the equation is used for discretization in time, and finite element method is used to discrete the space variables. The resulting schemes are consistent, stable and very efficient. Optimal‐rate of convergence is proved. Also, a note is given for a paper published earlier© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 相似文献
15.
The hyperbolic complex (HC) space is congruent with Minkowski space time.HC is a special kind of non-Euclidean space with continuous odd-points. The Clifford algebraic spinor and the Dirac wave equation
can be introduced in the hyperbolic complex space. The Clifford algebraic spinor contains eight independent elements and the
Dirac wave equations 64 coefficients. For Dirac particles 4×8 and for antiparticles 4×8 variables which are Hermitian conjugate
to each other (on four dimensional space-time). 相似文献
16.
By transforming the usual Lax pairs of isospectral and non-isospectral matrix Kdv hierarchies into Lax pairs Riccati form, a unified explicit from of Backlund transformations and superposition formulas for these two kinds of hierarchies of equations can be ohtaincd. 相似文献
17.
孙同军 《高等学校计算数学学报(英文版)》2001,10(1)
1 IntroductionADI Galerkin methods were first formulated for the solution of nonlinear parabolic andlinear second-order hyperbolic problems on rectangular regions by Douglas and Dupont[1 ] .These methods combine alternating-direction method and Galerkin finite element methodtogether.So,they have the advantage of reducing the solution of a multidimensional problemto the solution of sets of independent one-dimensional problems,decreasing the amount ofcalculation,natural parallelism and highe… 相似文献
18.
其中u=u(x,t),f=f(u),φ=φ(x)是M维向量函数。拟线性双曲方程存在击波现象,既使初始值无限光滑,也会出现间断解。(1)与(2)的弱解一般而言是不唯一的,满足熵条件的弱解是有物理意义的广义解。 双曲型方程求数值解时,需要考察所得数值解满足熵条件的问题。Lax和Wendroff曾证明,当网格步长△_t,△_x趋于零时,若守恒型差分格式的解几乎处处有界收敛到函数 相似文献
19.
The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group. 相似文献
20.
Cheikh Ould Mohamed El-Hafedh Elbar Ould Ely Telmoudy Mohamed Vall Ould Moustapha 《Comptes Rendus Mathematique》2013,351(19-20):747-752
In this note, we give the solutions of the Cauchy problems for the Euler–Poisson–Darboux equations (EPD) with modified conditions in the hyperbolic space with application to the wave equation. 相似文献