Symmetries and Infinite Dimensional Lie Algebra of the Variable Coefficient KDV Equation

With the help of the conditional strong symmetry, we obtain the K symmetries, r symmetries, infinite dimensional Lie algebra and the corresponding infinite conservation quantities of the variable coefficient KdV equation u_t = h₁(t)(u_xxx +6uu_x) +4h₂(t)u_x - h₀(t)(2u+ xu_x) with h₀, h₁ and h₂ being three arbitrary functions of t.     

Lie Symmetries of Ishimori Equation

The Ishimori equation is one of the most important(2+1)-dimensional integrable models,which is an integrable generalization of(1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations.Based on importance of Lie symmetries in analysis of differential equations,in this paper,we derive Lie symmetries for the Ishimori equation by Hirota's direct method.     

Lie Algebras of Approximate Symmetries

Abstract

Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.     

Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein–Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove local in time existence, uniqueness and well-posedness of classical solutions. The zero order term of our system contains an expression which might not be a C ^∞ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. In Brauer and Karp (J Diff Eqs 251:1428–1446, 2011) we constructed an initial data set for these of systems in the same type of weighted Sobolev spaces. We obtain the same lower bound for the regularity as Hughes et al. (Arch Ratl Mech Anal 63(3):273–294, 1977) got for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Lie Point Symmetries of Differential-Difference Equations

In this paper, the classical Lie group approach is extended to find some Lie point symmetries of differentialdifference equations. It reveals that the obtained Lie point symmetries can constitute a Kac-Moody-Virasoro algebra.     

Lie Symmetries of Magnetoacoustic Gravitational Wave Equations

A model of magnetoacoustic gravitational waves is proposed. The Lie method is used to study the group properties of model equations, the algebra of invariance, and the problem of group classification.     

Lie Symmetries for Hamiltonian Systems Methodological Approach

This paper proposes an algorithm for the Lie symmetries investigation in the case of a 2D Hamiltonian system. General Lie operators are deduced firstly and, in the the next step, the associated Lie invariants are derived. The 2D Yang-Mills mechanical model is chosen as a test model for this method. PACS: 05.45.-a; 02.30.Ik     

The Symmetries of Fermion Fluids at Low Dimensions

We point out that the quasiparticle spectrum of the Landau Fermi liquid theory has an extra Z₂ symmetry, local in momentum space, which is not generic to the Hamiltonian with interactions. Thus the Fermi liquid is in this sense a (quantum) zero-temperature critical point.     

Lie Symmetries for a Conformally Flat Radiating Star

We consider a relativistic radiating spherical star in conformally flat spacetimes. In particular we study the junction condition relating the radial pressure to the heat flux at the boundary of the star which is a nonlinear partial differential equation. The Lie symmetry generators that leave the equation invariant are identified and we generate an optimal system. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. We identify new categories of exact solutions to the boundary conditions. Two classes of solutions are of interest. The first class depends on a self similar variable. The second class is separable in the spacetime variables.     

Lie Symmetries and Non-Noether Conserved Quantities of Nonholonomic Systems

A new conservation theorem of the nonholonomic systems is studied. The conserved quantity is onlyconstructed in terms of a general Lie group of transformation vector of the dynamical equations. Firstly, we establish thedynamical equations of the nonholonomic systems and the determining equations of Lie symmetry. Next, the theore mof non-Noether conserved quantity is deduced. Finally, we give an example to illustrate the application of the result.     

Mei Symmetries and Lie Symmetries for Nonholonomic Controllable Mechanical Systems with Relativistic Rotational Variable Mass

The Mei symmetries and the Lie symmetries for nonholonomic controllable mechanical systems with relativistic rotational variable mass are studied. The differential equations of motion of the systems are established. The definition and criterion of the Mei symmetries and the Lie symmetries of the system are studied respectively. The necessary and sufficient condition under which the Mei symmetry is Lie symmetry is given. The condition under which the Mei symmetries can be led to a new kind of conserved quantity and the form of the conserved quantity are obtained. An example is given to illustrate the application of the results.     

Generalized Lie Symmetries and the Integrability of Generalized Emden-Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden-Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden-Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.     

Lie Point Symmetries and Exact Solutions of Couple KdV Equations

The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.     

Lie Symmetries and Conserved Quantities for Super-Long Elastic Slender Rod

DNA is a nucleic acid molecule with double-helical structures that are special symmetrical structures attracting great attention of numerous researchers. The super-long elastic slender rod, an important structural model of DNA and other long-train molecules, is a useful tool in analysing the symmetrical properties and the stabilities of DNA. We study the Lie symmetries of a super-long elastic slender rod by using the methods of infinitesimal transformation. Based on Kirchhoff＇s analogue, generalized Hamilton canonical equations are analysed. The infinitesimal transformations with respect to the radian coordinate, the generalized coordinate, and the quasimomentum of the model are introduced. The Lie symmetries and conserved quantities of the model are presented.     

Approximate Symmetries and Conservation Laws with Applications

The relationship between the approximateLie-Backlund symmetries and the approximate conservedforms of a perturbed equation is studied. It is shownthat a hierarchy of identities exists by which thecomponents of the approximate conserved vector or theassociated approximate Lie-Backlund symmetries aredetermined by recursive formulas. The results areapplied to certain classes of linear and nonlinear waveequations as well as a perturbed Korteweg-de Vriesequation. We construct approximate conservation laws forthese equations without regard to aLagrangian.     

Point Symmetries of Controlled Systems and Their Applications

Abstract

A problem of finding point symmetries of controlled systems is discussed, basic theorems and algorithms are formulated. The application to some problems of flight dynamics is suggested.     

First Integrals and Parametric Solutions for Equations Integrable Through Lie Symmetries

Abstract

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computation of first integrals gives in the most general case, the parametric form of the general solution. For some polynomial functions we obtain a time parametrisation quadrature which can be solved in terms of “known” functions.