From the Solution of the Tsarev System to the Solution of the Whitham Equations

We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2, ... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so-called hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g>0, we construct the unique solution of the Tsarev system, which matches the genus g+1 and g–1 solutions on the transition boundaries.     

Schlesinger System,Einstein Equations and Hyperelliptic Curves

We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of the Schlesinger system, Ernst equation, and self-dual SU(2)-invariant Einstein equations.     

Backlund Transformations for the Initial Problems of Nizhnich and Nizhnich-Novikov-Veselov Equations

The homogeneous balance method is a method for solving general partial differential equations (PDEs). Inthis paper we solve a kind of initial problems of the PDEs by using the special Backlund transformations of the initialproblem. The basic Fourier transformation method and some variable-separation skill are used as auxiliaries. Two initialproblems of Nizhnich and the Nizhnich-Novikov-Veselov equations are solved by using this approach.     

Symmetry Reduction and Cauchy Problems for a Class of Fourth-Order Evolution Equations

We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations to Cauchy problems for systems of ordinary differential equations （ODEs）. We classify a class of fourth-order evolution equations which admit certain higher-order generalized conditional symmetries （GCSs） and give some examples to show the main reduction procedure. These reductions cannot be derived within the framework of the standard Lie approach, which hints that the technique presented here is something essential for the dimensional reduction of evolution equations.     

Symmetry Reduction of Initial-Value Problems for a Class of Third-order Evolution Equations

Symmetry reduction of a class of third-order evolution equations that admit certain generalized conditional symmetries （GCSs） is implemented. The reducibility of the initial-value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations （ODEs） is characterized via the GCS and its Lie symmetry. Complete classification theorems are obtained and some examples are taken to show the main reduction procedure.     

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.     

The Self-Dual Einstein–Weyl Metric and Classical Solution of Painlevé VI

In the natural correspondence between the self-dual Bianchi type IX metrics and solutions of Painlevé VI, the self-dual Ricci-flat metrics or the nontrivial self-dual Einstein–Weyl metrics correspond to the classical solutions of Painlevé VI that determine isomonodromic deformations with reducible monodromy.     

稳态问题混合边界积分方程的高精度求积法与分裂外推

提出了求积法解稳态问题的混合边界积分方程,它拥有高精度,低复杂度.通过并行地解粗网格上的离散方程,根据误差的多参数渐近展开,应用分裂外推算法得到高精度的近似解,同时获得后验误差估计.     

Gaudin Model,KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.     

Jacobi Elliptic Function Solutions of Some Nonlinear Evolution Equations

In this letter, the modified Jacobi elliptic function expansion method is extended to solve M-coupled KdV equation, M-coupled Ito equation, vKdV equation, and AKNS equation. Some new Jacobi elliptic function solutions are obtained by using Mathematica. When the modulus m → 1, those periodic solutions degenerate as the corresponding soliton solutions.     

Conformally Invariant Generalization of Einstein Equations and the Causality Principle

The paper presents an improved derivation of the dynamic equations for the conformally invariant generalization of Einstein's equations. The consistency of the variational procedure with the causality principle is studied. The well-posedness of the Cauchy problem in the synchronous coordinate system is proved as applied to the generalized equations. The possibility of generalized equations at finding quantitative relations between observed values is noted.     

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad {with}\quad x(t_0)=x_0,$$ can be presented by the matrix exponential as $\exp({\int_{t_0}^tA(s)\,ds})x_0,$ if the commutativity condition for the coefficient matrix $A(t)$ holds: $$\Big[\int_{t_0}^tA(s)\,ds,A(t)\Big]=0.$$ A natural question is whether this is true without the commutativity condition. To give a definite answer to this question, we present two classes of illustrative examples of coefficient matrices, which satisfy the chain rule $$ \frac d {dt}\, \exp({\int_{t_0}^t A(s)\, ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.     

基于受迫振动实验中几个问题的理论推导

本文论述了受迫振动方程建立与求解,相关物理量的理论推导,并附以测量实例进行分析,解决了同学们该实验疑难之处,补充了教材之空缺。     

Supports of Measure Solutions for Spatially Homogeneous Boltzmann Equations

We prove that the support of the unique measure solution for the spatially homoge-neous Boltzmann equation in R3 is the whole space, if the initial distribution is not a Dirac measure and has 4-order moment. More precisely, we obtain the lower bound of exponential type for the probability of any small ball in ℝ³ relative to the measure solution.     

Solutions and Conditional Lie-Backlund Symmetries of Quasi-linear Diffusion-Reaction Equations

New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.     

Elliptic Equation and New Solutions to Nonlinear Wave Equations

The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.     

On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation

In this paper we find a class of solutions of the sixth Painlevé equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We describe the critical behavior close to the critical points in terms of two parameters and we find the relation among the parameters at the different critical points (connection problem). We also study the critical behavior of Painlevé transcendents in the elliptic representation.     

Approximate Solutions of Perturbed Nonlinear Schroedinger Equations

By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.