Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras

We study the problem of group classification of quasilinear elliptic equations in a two-dimensional space. The list of all equations of this type admitting solvable Lie algebras of symmetry operators is obtained. Together with the results obtained earlier by the authors, these results give a complete solution of the problem of group classification of quasilinear elliptic equations.     

Abelian subalgebras in some particular types of Lie algebras

It is well-known that there exists a close link between Lie Theory and Relativity Theory. Indeed, the set of all symmetries of the metric in our four-dimensional spacetime is a Lie group. In this paper we try to study this link in depth, by dealing with three particular types of Lie algebras: h_n algebras, g_n algebras and Heisenberg algebras. Our main goal is to compute the maximal abelian dimensions of each of them, which will allow us to move a step forward in the advancement of this subject.     

Noetherianness of wreath products of Abelian Lie algebras with respect to equations of universal enveloping algebra

It is proved that a wreath product of two Abelian finite-dimensional Lie algebras over a field of characteristic zero is Noetherian w.r.t. equations of a universal enveloping algebra. This implies that an index 2 soluble free Lie algebra of finite rank, too, has this property.     

Examples of two-dimensional pro-p-groups and Lie algebras

Several examples of pro-p-groups and profinite-nilpotent Lie algebras of dimension two are given. In particular, a negative answer is given to one of the questions raised by L. V. Kuz'min. An essential role in the construction of these examples is played by the author's criterion regarding dimension two and by H. Koch's results. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 263–271, 1987.     

Nonlinear equations and graded Lie algebras

The survey is devoted to a new method of constructing a broad class of exactly and completely integrable one- and two-dimensional nonlinear dynamical systems and finding explicit solutions of them.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 101–136, 1984.     

Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1+1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2,R)s_⊕W, where W is the three-dimensional Heisenberg-Weyl algebra. We construct a complete symmetry group of a 1+2 evolution equation ut=(Fy(u)ux) for some functions F using the point symmetries admitted by the equation. The 1+2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377-392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1+2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.     

Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cocalibrated G₂-structures and cocalibrated ${{\rm G}_2^*}$ -structures are the natural initial values for Hitchin’s evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin₀(3, 4), respectively. In this article, we classify 7-D real Lie algebras with a codimension one Abelian ideal which admit such structures. Moreover, we classify the 7-D complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated ${({\rm G}_2)_{\mathbb{C}}}$ -structures.     

Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition

The problem of group classification of quasilinear elliptic-type equations in the two-dimensional space is considered. We obtain the lists of all equations of this class that admit the semisimple Lie algebras of symmetry operators and the Lie algebras of symmetry operators with nontrivial Levi decomposition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1532–1545, November, 2007.     

Integrable equations on ℤ-graded Lie algebrasLie algebras

We consider evolution systems admitting L-A-pairs in ℤ-graded Lie algebras. We relate several hierarchies of integrable systems to a single L operator. The different hierarchies corresponds to different decompositions of the zero component of a ℤ-graded algebra into the sum of two subalgebras. This allows us to construct new examples of multi-component integrable system following the Burgers, mKdV, NLS and Boussinesq equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 3, pp. 375–383 September, 1997.