A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrödinger equations with nonlocal smooth nonlinearity are also given in this paper.     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

Consistent Riccati Expansion Method and Its Applications to Nonlinear Fractional Partial Differential Equations

In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.     

A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions

Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S ² × S ¹. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).     

Equivalence Theorem for Higher Order Equations

We show that the theory of an nth-order fieldequation, minimally coupled to electromagnetism, iscompletely equivalent to the theory of n independentsecond-order equations, also minimally coupled to the electromagnetic field. The equivalence is shownto hold as an algebraic identity between the respectivematrix elements for a given order of the perturbativesolution. A general functional proof is also given. The equivalence shows that the higherorder theory is both renormalizable andunitary.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the ‘fluctuation-dissipation theorem’ is satisfied, and this verifies that satisfying ‘fluctuation-dissipation theorem’ indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

Block Spins for Partial Differential Equations

We investigate the use of renormalization group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse grained or “perfect” Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1 + 1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg–Landau equation, the Swift–Hohenberg equation, and the damped Kuramoto–Sivashinsky equation. We find that the renormalization group is superior to conventional sampling-based discretizations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultraviolet cutoff in the problem. We discuss limitations and open problems of this approach.     

Generalized Partial Differential Equation and Fermat's Last Theorem

The equivalence of Fermat's Last Theorem and the non-existence of solutions of a generalized n th order homogeneous hyperbolic partial differential equation in three dimensions and periodic boundary conditions defined in a cubic lattice is demonstrated for all positive integer, n > 2. For the case n = 2, choosing one variable as time, solutions are identified as either propagating or standing waves. Solutions are found to exist in the corresponding problem in two dimensions.     

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.     

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.     

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.     

时变偏微分方程的贝叶斯稀疏识别方法

在数据驱动的建模中,通过测量或模拟得到时空数据,我们发现基于拉普拉斯先验的贝叶斯稀疏识别方法能有效地恢复时变偏微分方程的稀疏系数.本文将贝叶斯稀疏识别方法运用于各种时变偏微分方程模型(KdV方程、Burgers方程、Kuramoto-Sivashinsky方程、反应-扩散方程、非线性薛定谔方程和纳维-斯托克斯方程)的方...     

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday.     

A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form ut+α(t)(p1(u))x+β(t)(p2(u))xx+γ(t)(p3(u))xxx+η(t)(p4(u))xxxx=0. A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.     

A Laplace Decomposition Method for Nonlinear Partial Differential Equations with Nonlinear Term of Any Order

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial differential equations with nonlinear term of any order, utt ＋ auxx ＋ bu ＋ cup ＋ du^2p-1 = 0, which contains some important equations of mathematical physics. Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained, including numerical hyperbolic function solutions and doubly periodic ones. Illustrative figures and comparisons between the numerical and exact solutions with different values of p are used to test the efficiency of the proposed method, which shows good results are azhieved.     

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.     

Numerical Linear Algebra and Solvability of Partial Differential Equations

It was observed long ago that the obstruction to the accurate computation of eigenvalues of large non-self-adjoint matrices is inherent in the problem. The basic idea is that the resolvent of a highly non-normal operator can be very large far away from the spectrum. This leads to an easily observable fact that algorithms for locating eigenvalues will typically find some ``false eigenvalues''. These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE P u = f. Almost immediately after that discovery, Hörmander provided an explanation of Lewy's example showing that almost all operators with non-constanct complex valued coefficients are not locally solvable. In modern language, that was done by considering the essentially dual problem of existence of non-propagating singularities. The purpose of this article is to review this work in the context of ``almost eigenvalues'' and from the point of view of semi-classical analysis.