Group-Invariant Solutions and Conservation Laws of One-Dimensional Nonlinear Wave Equation

Based on classical Lie symmetry method, the one-dimensional nonlinear wave equation is investigated. By using four-dimensional subalgebras of the equation, the invariant groups and commutator table are constructed. Furthermore, optimal system of the equation is obtained, and the exact solutions can be gained by solving reduced equations. Finally, a complete derivation of the conservation law is given by using conservation multipliers.     

Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether’s Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka–Volterra equation.     

Essential Conservation Laws for Equations with Infinite Symmetries

We consider partial differential equations of variational problems with infinite symmetry groups. We study local conservation laws associated with arbitrary functions of one variable in the group generators. We show that only symmetries with arbitrary functions of dependent variables lead to an infinite number of conservation laws. We also calculate local conservation laws for the potential Zabolotskaya-Khokhlov equation for one of its infinite subgroups.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 190–198, July, 2005.     

非线性退化波方程的Riemann问题

研究了弹性力学中一退化波方程的Riemann问题．其应力函数非凸非凹,从而使得激波条件退化．通过引入广义激波条件下的退化激波,构造性地得到了各种情形下Riemann问题的整体解．     

Min-Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

In this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n -sphere under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with. This proves rigidity for compact, connected, locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r) , where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball . As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three-space ℍ³ . In fact, we extend it to obtain rigidity for supersolutions to certain Monge-Ampère equations. © 2019 Wiley Periodicals, Inc.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

Asymptotics from Scaling for Nonlinear Wave Equations

We present a scaling technique which transforms the evolution problem for a nonlinear wave equation with small initial data to a linear wave equation with a distributional source. The exact solution of the latter uniformly approximates the late-time behavior of solutions of the nonlinear problem in timelike and null directions.     

Fully Nonlinear Parabolic Equations and the Dini Condition

Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolic equations under the Dini condition, which improve and generalize a result due to Kovats, are obtained by the use of the approximation lemma. Received November 21, 2000, Accepted April 27, 2001     

非线性守恒方程的两种稳定化谱元方法

考虑非线性守恒方程的高阶数值解法,介绍了基于谱元法的两种稳定性方法,一种是谱粘性消去法(SVV),另一种是过滤法.在SVV方法中,我们推广并分析了传统的基于单区域的SVV算子的定义.在过滤法中,我们分析了SVV-H elm holtz过滤算子的性质.文中从分析和计算两方面对两种方法进行了比较,建立了两者之间的关系.最后通过一系列数值试验说明方法的有效性.     

Nonlinear Wave Equations on the Two-Dimensional Sphere

We study a nonlinear wave equation on the two-dimensional sphere with a blowing-up nonlinearity. The existence and uniqueness of a local regular solution are established. Also, the behavior of the solutions is examined. We show that a large class of solutions to the initial value problem quench in finite time.     

Initial and Nonlinear Oblique Boundary Value Problems for Fully Nonlinear Parabolic Equations

We consider the initial and nonlinear oblique derivative bouodary value problem for fully nonlinear uniformly parabolic partial differential equations of second order. The parabolic operators satisfy natural structure conditions which have been introduced by Krylov. The nonlinear boundary operalors satisfy certain natural structure conditions also. The existence and uniqueness of classical solution are proved when the initial boundary values and the coefficients of the equation are suitable smooth.     

Regularized Conservation Laws and Nonlinear Geometric Optics

 This paper is devoted to the study of Cauchy problems for regularized conservation laws in Colombeau algebras of generalized functions. The existence and uniqueness of generalized solutions to these Cauchy problems are obtained. Further, we develop a generalized variant of nonlinear geometric optics for the regularized problems. Consistency with the classical results is shown to hold for scalar conservation laws with bounded variation initial data in one space variable. Received 6 November 1996; in revised form 5 August 1997     

广义拟线性双曲型方程的对称势及其守恒定律

基于Lie群方法,研究广义拟线性双曲型方程的对称势和不变解.为了得到显式的不变解,关注物理上有趣的有对称势的情况.然后,利用局部的Lagrange函数逼近,在3种物理上引起注意的情况下,得到该方程的守恒定律.     

完全非线性椭圆方程的古典解

当完全非线性一致椭圆方程中的非线性算子具有凸(凹)性时,其解有C2,α正则性.该条件可以减弱,只假定非线性算子几乎处处是局部C1,β的,其中0＜β＜1是固定的常数,仍可证明该方程的解具有C2,α正则性.     

Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully non-linear elliptic equations. Received: June 2006, Accepted: January 2007     

非线性波方程的初边值问题

本文利用Leray-Schauder不动点定理证明了非线性波方程utt-[a0 a2(ux)^β]uxx-a1uxxtt=f(x,t,ux,ut,uxt)的初边值问题广义解的存在唯一性。     

Blow-up Surfaces for Nonlinear Wave Equations,I

We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example □u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/?²) + v, where {? = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.