Lie symmetries for systems of evolution equations

Abstract

The Lie symmetries for a class of systems of evolution equations are studied. The evolution equations are defined in a bimetric space with two Riemannian metrics corresponding to the space of the independent and dependent variables of the differential equations. The exact relation of the Lie symmetries with the collineations of the bimetric space is determined.Lie symmetries is a powerful method for the determination of solutions in the theory of differential equations. A Lie symmetry is important as it provides invariants which can be used to write a new differential equation with less degree of freedom. Furthermore, a solution of such an equation is also a solution of the original differential equation [1,2]. The reduction process is the main application of Lie symmetries, however, it is not a univocal approach. Symmetries can be used for the determination of conservation currents [3], for the classification of differential equations [4][5][6][7][8][9][10] and for the reconnaissance of some well-known systems [11][12][13][14][15].In the recent literature, it has been shown that there is a close relation between the Lie symmetries of a second order differential equation and the geometry of the space where motion occurs. For example, the conservation of energy and angular momentum in Newtonian Physics is a result of the Lie point symmetries, generated by the Killing vectors of translations and rotations respectively. The general result for a holonomic autonomous dynamical system moving in a Riemannian space is that the Lie point symmetries of the equations of motion *

Constructing the CKVs of Bianchi III and V spacetimes

“…In [19], [20] it has been shown that in these case the Lie point symmetries are generated by the special projective algebra and the Noether point symmetries by the Homothetic algebra of the kinetic metric. Similar results have been found for some partial differential equations of special interest in curved spacetimes, as the wave and the heat equation (see [21], [22], [23] and references therein) where it has been shown that in these cases the Lie point symmetries involve the CKVs.…”

“…We conclude that the Bianchi III spacetime A + λ which reduces to a HV when A (τ ) is an exponential in which case the line element is ds 2 (III) = −e mκτ dτ 2 + e mκτ dx 2 + e m(κ−1)τ dy 2 + e m(κ−λ)τ e −2x dz 2 (23) or in equivalent form…”

“…We conclude that the Bianchi III spacetime ds 2 (III) = e mλτ A 2 (τ ) −dτ 2 + dx 2 + e −mτ dy 2 + e −mλτ e −2x dz 2 (22) admits the proper CKV L 1 with conformal factor ψ (III) (L 1 ) = 2 m A,τ A + λ which reduces to a HV when A (τ ) is an exponential in which case the line element is ds 2 (III) = −e mκτ dτ 2 + e mκτ dx 2 + e m(κ−1)τ dy 2 + e m(κ−λ)τ e −2x dz 2 (23) or in equivalent form…”

A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

“…Lie symmetry method provides an effective tool for deriving the analytic solutions of the nonlinear partial differential equations (NLPDEs) [1][2][3][4]. In recent years, many authors have studied the nonlinear fractional differential equations (NLFDEs) because these equations express many nonlinear physical phenomena and dynamic forms in physics, electrochemistry, and viscoelasticity [5][6][7][8][9].…”

Θεωρία Νηματικών Δεσμών Και Η Εφαρμογή Τους Στις Συμμετρίες

“…Ωστόσο, δεν υπάρχει ένα τέτοιο ισχυρό αποτέλεσμα για την περίπτωση των δυναμικών συμμετριών (Lie και Noether). ΄Εχουν εξαχθεί, παρόμοια αποτελέσματα, για την περίπτωση των μερικών διαφορικών εξισώσεων δεύτερης τάξης, [185][186][187][188] καθώς και για την περίπτωση διαταραγμένων συναρτήσεων Lagrange, βλέπε [189]. Ακόμα, αποτελέσματα που να αφορούν συναρτήσεις Lagrange που παρουσιάζουν ανωμαλία (singular Lagrangians), οι οποίες αφορούν συστήματα με δεσμούς μπορούν να βρεθούν στα [190,191].…”