REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL

Conservation law plays a very important role in many geometric variational problems and related elliptic systems.In this note,we refine the conservation law obtained by Lamm-Rivière for fourth order systems and de Longueville-Gastel for general even order systems.

Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

In this paper,we study systems of conservation laws in one space dimension.We prove that for classical solutions in Sobolev spaces H^(s),with s>3/2,the data-to-solution map is not uniformly continuous.Our results apply to all nonlinear scalar conservation laws and to nonlinear hyperbolic systems of two equations.

Hierarchy of Lax Integrable Lattice Equations, Darboux Transformation and Conservation Laws

A discrete isospectral problem and the associated hierarchy of Lax integrable lattice equations were investigated. A Darboux transformation for the discrete spectral problem was found. Finally, an infinite number of conservation laws were given for the corresponding hierarchy.

Integrating Factors and Conservation Theorems of Lagrangian Equations for Nonconservative Mechanical System in Generalized Classical Mechanics

The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.

Formal Inferring the Law of Conservation of Energy from Assuming A-Priori-ness of Knowledge in a Formal Axiomatic Epistemology System Sigma

The research purpose is invention (construction) of a formal logical inference of the Law of Conservation of Energy within a logically formalized axiomatic epistemology-and-axiology theory Sigma from a precisely defined assumption of a-priori-ness of knowledge. For realizing this aim, the following work has been done: 1) a two-valued algebraic system of formal axiology has been defined precisely and applied to proper-philosophy of physics, namely, to an almost unknown (not-recognized) formal-axiological aspect of the physical law of conservation of energy;2) the formal axiomatic epistemology-and-axiology theory Sigma has been defined precisely and applied to proper-physics for realizing the above-indicated purpose. Thus, a discrete mathematical model of relationship between philosophy of physics and universal epistemology united with formal axiology has been constructed. Results: 1) By accurate computing relevant compositions of evaluation-functions within the discrete mathematical model, it is demonstrated that a formal-axiological analog of the great conservation law of proper physics is a formal-axiological law of two-valued algebra of metaphysics. (A precise algorithmic definition of the unhabitual (not-well-known) notion “formal-axiological law of algebra of metaphysics” is given.) 2) The hitherto never published significantly new nontrivial scientific result of investigation presented in this article is a formal logical inference of the law of conservation of energy within the formal axiomatic theory Sigma from conjunction of the formal-axiological analog of the law of conservation of energy and the assumption of a-priori-ness of knowledge.

A General Approach to the Construction of Conservation Laws for Birkhoffian Systems in Event Space

For a Birkhoffing system in the event space, a general approach to the construction of conservation laws is presented. The conservation laws are constructed by finding corresponding integrating factors for the parametric equations of the system. First, the parametric equations of the Birkhoffian system in the event space are established, and the definition of integrating factors for the system is given; second the necessary conditions for the existence of conservation laws are studied in detail, and the relation between the conservation laws and the integrating factors of the system is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the results.

HAMILTONIAN STRUCTURE AND INFINITE NUMBER OF CONSERVATION LAWS FOR THE COUPLED DISCRETE KdV EQUATIONS

A new discrete isospectral problem is introduced,from which the coupled discrete KdV hierarchy is deduced and is written in its Hamiltonian form by means of the trace identity.It is shown that each equation in the resulting hierarchy is Liouville integrable.Furthermore,an infinite number of conservation laws are shown explicitly by direct computation.

THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL WEAK SOLUTIONS:SOME ONE-J AND NON-R INITIAL DATA

The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been considered for constant initial data having discontinuities on three rays with vertex at the origin. The solutions are constructed for some one-J and non-R initial data. One kind of new discontinuity, which is labelled as the delta-shock wave, appears in some solutions. The delta-shock wave is a discontinuity plane that is the suport of a generalized function.

An Effective Method for Seeking Conservation Laws of Partial Differential Equations

This paper introduces an effective method for seeking local conservation laws of general partial differential equations （PDEs）. The well-known variational principle does not involve in this method. Alternatively, the conservation laws can be derived from symmetries, which include t/he symmetries of t/he associated linearized equation of t/he PDEs, and the adjoint symmetries of the adjoint eqUation of the PDEs.

A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT AXISYMMETRIC VESSELS

We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived are nonconservative and/or nonho- mogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The Riemann solutions may consist of four waves for some cases. The system can also be written as a 3 × 3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.

INTERACTION OF STRONG AND WEAK SINGULARITIES FOR HYPERBOLIC SYSTEM OF CONSERVATION LAWS IN MULTIDIMENSIONAL SPACE

In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.

An analogical study of wave equations,physical quantities,conservation and reciprocity equations between electromagnetic and elastic waves

This paper presents an analogical study between electromagnetic and elastic wave fields,with a one-to-one correspondence principle established regarding the basic wave equations,the physical quantities and the differential operations.Using the electromagnetic-to-elastic substitution,the analogous relations of the conservation laws of energy and momentum are investigated between these two physical fields.Moreover,the energy-based and momentum-based reciprocity theorems for an elastic wave are also derived in the time-harmonic state,which describe the interaction between two elastic wave systems from the perspectives of energy and momentum,respectively.The theoretical results obtained in this analysis can not only improve our understanding of the similarities of these two linear systems,but also find potential applications in relevant fields such as medical imaging,non-destructive evaluation,acoustic microscopy,seismology and exploratory geophysics.

Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.

Computational intelligence interception guidance law using online off-policy integral reinforcement learning

Missile interception problem can be regarded as a two-person zero-sum differential games problem,which depends on the solution of Hamilton-Jacobi-Isaacs(HJI)equa-tion.It has been proved impossible to obtain a closed-form solu-tion due to the nonlinearity of HJI equation,and many iterative algorithms are proposed to solve the HJI equation.Simultane-ous policy updating algorithm(SPUA)is an effective algorithm for solving HJI equation,but it is an on-policy integral reinforce-ment learning(IRL).For online implementation of SPUA,the dis-turbance signals need to be adjustable,which is unrealistic.In this paper,an off-policy IRL algorithm based on SPUA is pro-posed without making use of any knowledge of the systems dynamics.Then,a neural-network based online adaptive critic implementation scheme of the off-policy IRL algorithm is pre-sented.Based on the online off-policy IRL method,a computa-tional intelligence interception guidance(CIIG)law is developed for intercepting high-maneuvering target.As a model-free method,intercepting targets can be achieved through measur-ing system data online.The effectiveness of the CIIG is verified through two missile and target engagement scenarios.

Symmetry and Hojman Conserved Quantity of Tznoff Equations for Unilateral Holonomic System

Symmetry of Tzénoff equations for unilateral holonomic system under the infinitesimal transformationsof groups is investigated.Its definitions and discriminant equations of Mei symmetry and Lie symmetry of Tzénoffequations are given.Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given.Hojman conserved quantity of Tzénoff equations for the system above through special Lie symmetry and Lie symmetryin the condition of special Mei symmetry respectively is obtained.

Residual symmetry, interaction solutions, and conservation laws of the(2+1)-dimensional dispersive long-wave system

We explore the （2＋l）-dimensional dispersive long-wave （DLW） system. From the standard truncated Painleve expansion, the Baicklund transformation （BT） and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the （2＋l）-dimensional DLW system is consistent Riccati expansion （CRE） solvable. If the special form of （CRE）-consistent tanh-function expansion （CTE） is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.

Mei Symmetry and Conserved Ouantity of Tzénoff Equations for Nonholonomic Systems of Non-Chetaev's Type

Mei symmetry of Tzenoff equations for nonholonomic systems of non-Chetaev＇s type under the infinitesimal transformations of groups is studied. Its definitions and discriminant equations of Mei symmetry are given. Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given. Hojman conserved quantity of Tzenoff equations for the systems through Lie symmetry in the condition of special Mei symmetry is obtained.

Integrating Factors and Conservation Laws for Relativistic Mechanical System

In this paper, we present a new method to construct the conservation laws for relativistic mechanical systems by finding corresponding integrating factors. First, the Lagrange equations of relativistic mechanical systems are established, and the definition of integrating factors of the systems is given; second, the necessary conditions for the existence of conserved quantities of the relativistic mechanical systems are studied in detail, and the relation between the conservation laws and the integrating factors of the systems is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the systems are established, and an example is given to illustrate the application of the results.

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of twodimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u＋ 〉 0, while explicit non-selfsirnilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u＋ 〈 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.