Linear autonomy conditions for the basic Lie algebra of a system of linear differential equations

Chirkunov, Yu. A.

doi:10.1134/s1064562409030302

Cited by 19 publications

(10 citation statements)

References 1 publication

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“…From the above relations, it follows that the matrices A −1 0 A j (j = 1, 2, 3) satisfy conditions (5). Consequently, for β = ±k and k 2 1 + k 2 2 + k 2 3 = k 2 0 , system (19) multiplied by A −1 0 , is a Friedrichs t-hyperbolic symmetric evolutionary system embedded in the corresponding system of four three-dimensional wave equations and equivalent to system (2) with matrices (15) and (16) or system (2) with matrices (15) and (17).…”

Section: Definition 1 Systemmentioning

confidence: 94%

“…Since in the space C 2 , there is no linear form whose value at each point is a root of the characteristic equation of system (27), its basic group is linearly autonomous [5]. Calculations using this property show that the basic group of system (27) (as for all systems considered in this paper, the question is a factor group with respect to the normal divisor due to the linearity of the system) is generated by ten operators…”

Section: Shear Waves In a Three-dimensional Elastic Mediummentioning

confidence: 96%

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Friedrichs systems for systems of wave equations and shear waves in a three-dimensional elastic medium

Chirkunov

2010

J Appl Mech Tech Phy

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Friedrichs t-hyperbolic symmetric evolutionary systems embedded in systems of wave equations are obtained. The group properties of some of the obtained systems are studied. Shear wave in a threedimensional elastic medium are described using Friedrichs t-hyperbolic symmetric evolutionary systems.Key words: Friedrichs t-hyperbolic symmetric evolutionary systems, equivalence, systems of wave equations, conformal invariance, shear waves in a three-dimensional elastic medium.Introduction. Using quaternions, Gordienko [1, 2] obtained a Friedrichs t-hyperbolic symmetric system which depends on eight arbitrary real constants and is satisfied by first-order derivatives with respect to all independent variables of the solution of the three-dimensional wave equation.In the present work, Friedrichs t-hyperbolic symmetric evolutionary systems for which each of their solution is a solution of wave equations with two and three space variables are found to equivalence transformations. It is shown that such t-hyperbolic Friedrichs symmetric evolutionary systems exist only if the number of equations in the system of wave equations is even. All such Friedrichs systems are listed for systems of any number of wave equations with two space variables and for systems of two, four or six equations wave equations with three space variables. For systems of eight or more equations wave equations with three space variables, the existence of such Friedrichs systems is established and some of them are given. For the system from [1 2], conditions are obtained under which it has the above-mentioned property. The obtained Friedrichs t-hyperbolic symmetric evolutionary systems are used to study shear waves in a three-dimensional elastic medium. In particular, it is shown that a real Friedrichs t-hyperbolic symmetric evolutionary system embedded in a system of four wave equations with three space variables is equivalent to the system of wave equations for scalar potentials describing shear waves. The basic Lie groups of transformations are found which are admitted by the Friedrichs t-hyperbolic symmetric evolutionary system used to describe shear waves. Some partially invariant solutions are obtained in which the presence of functional arbitrariness makes them applicable to the solution of boundary-value problems of propagation of elastic shear waves.

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Section: Definition 1 Systemmentioning

confidence: 94%

Section: Shear Waves In a Three-dimensional Elastic Mediummentioning

confidence: 96%

Friedrichs systems for systems of wave equations and shear waves in a three-dimensional elastic medium

Chirkunov

2010

J Appl Mech Tech Phy

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“…(5) of the higher derivatives at the point of the general situation has a rank 3, then, by the Theorem 2.2 and the Notes 2.3 to this theorem from [9], we have that all operators (6), admitted by the Eq. (5) are linearly autonomous [9][10][11]. It means that ξ ξ η = = =0 0, p pp p 0 .…”

Section: Group Propertymentioning

confidence: 97%

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

Chirkunov

2016

International Journal of Non-Linear Mechanics

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“…In [2], the author proved that if a system of first-order linear differential equations with constant coefficients admits a non-x-autonomous operator then it admits an operator whose coordinates that are responsible for the transformation of functions depend on the functions nonlinearly. We obtain and prove the criteria of the non-x-autonomy of the basic Lie algebra of such a system announced in the author's article [3].…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…It is said in the final part of [1] that "the problem of x-autonomy induces the development of a new fruitful direction of research in the area of group analysis of differential equations." Together with [1][2][3], the present article belongs to this new direction.…”

Section: Introductionmentioning

confidence: 95%

Systems of linear differential equations with non-x-autonomous basic Lie algebra

Chirkunov

2012

J. Appl. Ind. Math.

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Linear autonomy conditions for the basic Lie algebra of a system of linear differential equations

Cited by 19 publications

References 1 publication

Friedrichs systems for systems of wave equations and shear waves in a three-dimensional elastic medium

Friedrichs systems for systems of wave equations and shear waves in a three-dimensional elastic medium

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

Systems of linear differential equations with non-x-autonomous basic Lie algebra

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