Preliminary group classification of quasilinear third-order evolution equations

Abstract

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six nonequivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

“…From equation (38) we can get ϕ ∈ {e qx , |x| r (r = 0), 1} modĜ ∼ h . For ϕ = e qx it follows from the determining equations (38)- (40)…”

“…Its main idea rely on the description of inequivalent realizations of Lie algebras in certain set of vector fields of the equation under consideration [9,93], which was original from S. Lie [44,62] and recently rediscovered by Winternitz and Zhdanov et al [34,93]. The method has been applied to classifying a number of nonlinear differential equations [2,9,33,34,[40][41][42]59,60,[93][94][95], including the class is normalized (see [81] for rigorous definitions of normalized classes and related notions). The second approach is based on the investigation of compatibility and the direct integration, up to the equivalence relation generated by the corresponding equivalence group, of determining equations implied by the infinitesimal invariance criterion [73].…”

Preliminary group classification of $(2+1)$-dimensional linear ultraparabolic Kolmogorov - Fokker - Planck equations

“…Among operators (21), only the operators 𝜕 𝑡 , 𝜕 𝑥 and 𝑢𝜕 𝑢 satisfy SDE ( 14)- (17). However, operator 𝜕 𝑥 leads us to the condition 𝐶 𝑥 = 0 and is excluded from the consideration.…”

“…In constructing three-dimensional solvable invariance algebras for class of equations ( 1), it is sufficient to add one operator (19) to the operators in algebra 2𝑔 1 1 and to find all possible nonequivalent realizations satisfying the corresponding commutation conditions and SDE ( 14)- (17). At that, we shall make use the transformation t = 𝑡 + 𝑇 (𝑦), x = 𝑋(𝑥, 𝑦), ȳ = 𝑌 (𝑦), 𝑣 = 𝜙(𝑥, 𝑦)𝑢…”

“…This is why for the preliminary group classification we shall make use of the method by Zhdanov-Lahno proposed in work [11] (for more details see [12]). As of today, the mentioned method is employed in considering wide classes of differential equations [13]- [17].…”