On the characteristic Lie algebras for equations u xy = f(u, u x )

Method for searching higher symmetries for quad-graph equations

“…Thus the condition rank(A) = rank(B) is necessary for existence of a symmetry (13). Let us deduce some differential consequences of equation (19). Just applying the reasonings above to this equation we obtain…”

“…Substitute it to the difference equation (19) and obtain the relation log −C 3 (m, n + 1) C 3 (m, n) = 0, solution to which is C 3 (m, n) = (−1) n C 4 (m), where C 4 (m) is an arbitrary function of m. As a result function g takes the form;…”

Lie Algebras of Slow Growth and Klein-Gordon PDE

“…In spite of the rather large number of papers where this algebraic object is studied [25,18,28,27,22], it can not be said that there exists any completely unambiguous definition of characteristic Lie algebra χ of a hyperbolic non-linear PDE. We use the definition of characteristic Lie algebra proposed in the initial papers [25,18].…”

“…The question of describing such algebras is very important, because, the characteristic Lie algebras of the one-dimensional and twodimensional systems (2) are different by the definition. This circumstance, as well as some gaps in proofs of [18,17] led to the appearance of [27,22], where the problem of an explicit description of characteristic Lie algebras χ(sinh u) and χ(e u +e −2u ) was posed and solved. It was solved from the point of view of constructing infinite bases and structure relations (different from the bases and relations proposed in this article).…”

“…It was solved from the point of view of constructing infinite bases and structure relations (different from the bases and relations proposed in this article). However the extremely important relationship between the characteristic Lie algebras of χ(sinh u) and χ(e u +e −2u ) of the sinh-Gordon and Tzitzeica equations and affine Kac-Moody algebras A escaped the attention of the authors in [27,22]. In addition, we wrote the generators of these algebras in terms of Bell polynomials, which helped us to determine and relate various gradings of χ(sinh u) and χ(e u +e −2u ).…”