Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically.     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically. Received: February 3, 2004; revised: June 2, 2004     

Symmetry Classification Using Noncommutative Invariant Differential Operators

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G_f or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations v_x = u, v_t = B(u) u_x - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

Lie group analysis of the axisymmetric flow

The non-linear equations of motion describing the incompressible axisymmetric flow in a flexible and extensible circular cylindrical tube is considered. By employing Lie theory, the full one-parameter infinitesimal transformation group leaving the equations of motion invariant is derived along with its associated Lie algebra. Subgroups of the full group are then used to obtain a reduction by one in the number of independent variable in the system. These reductions are continued until a system of ordinary differential equations is reached. A series type approximate solution of these ordinary differential equations is obtained which leads to a series type approximate solutions in R²{0} to momentum equations.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

Autonomy,Controllability, and Extension Modules

The so‐called behavioral approach to systems theory, developed by Willems, provides a unified framework for the mathematical treatment of linear systems. In the behavioral context, a linear system is nothing but the solution space of a linear system of (partial) difference or differential equations. For simplicity, the coefficients are supposed to be constant. Oberst proved a duality theorem that builds upon an earlier result of Palamodov. It says that for certain signal spaces of interest, e.g., the smooth functions or the distributions, the solutions spaces of linear systems of partial differential equations, are dual to certain polynomial modules associated to them. Then the solution space and the module contain the same information, and algebraic properties of the module translate to analytic properties of the solution space. Powerful tools from commutative algebra may be used to derive them. As a prominent example, we study two properties that lie at the very heart of systems and control theory: autonomy and controllability. We summarize the characterizations given by several authors, and unify them in the language of extension modules, an algebraic concept which yields a full classification of these systems theoretic notions.     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

Exceptional symmetry reductions of Burgers' equation in two and three spatial dimensions

Lie point symmetry analysis of the general class of nonlinear diffusion-convection equations in two and three dimensions has shown that only for Burgers' equation (that isD(u)=const,K(u)=quadratic) is a full symmetry reduction to an ordinary differential equation possible. The optimal system of symmetry operators is determined to ensure that a minimal complete set of reductions is obtained. For each reduced partial differential equation, classical Lie group analysis has been performed and further reductions obtained. In this manner, all possible reductions to an ordinary differential equation are found, leading to exact solutions to both the two and three dimensional Burgers' equation.     

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier–Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (\(1+1\))-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

     

Group Foliation Approach to the Complex Monge–Ampère Equation

We apply the group foliation method to find noninvariant solutions of the complex Monge–Ampère equation (CMA₂). We use the infinite symmetry subgroup of the CMA₂ to foliate the solution space into orbits of solutions with respect to this group and correspondingly split the CMA₂ into an automorphic system and a resolvent system. We propose a new approach to group foliation based on the commutator algebra of operators of invariant differentiation. This algebra together with Jacobi identities provides the commutator representation of the resolvent system. For solving the resolvent system, we propose symmetry reduction, which allows deriving reduced resolving equations.     

Differential-operator representations of Weyl group and singular vectors in Verma modules

Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.