Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

Abstract

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations f (x)u tt = (H(u)u x ) x + K(u)u x , is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.

Preliminary group classification of quasilinear third-order evolution equations

“…Substituting into the invariance conditions (9) shows that the first realization cannot be admitted by the PDEs (1). This leaves us with the realizations (17) and (18). The most general equations (1) invariant with respect to realizations (17) and (18) are given by the first two members in Table 1.…”

“…There are also many papers on "preliminary group classification" where authors listed some cases with new symmetry but did not claim that the general classification problem was solved completely [6,[16][17] . For this reason, finding an effective approach to simplification is essentially equivalent to showing the feasibility of solving the problem at all [18] .…”

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

“…Background and procedures of the modern Lie group theory are well described in literature [21,22,35,37,50,59]. Without going into the details of the theory, we present only the results below.…”

“…Therefore, the aim of the present work is to find all possible Lie symmetries, which Equation (1) can admit depending on the function triplets (K, D, F), i.e., to solve the so-called group classification problem, which was formulated and solved for a class of nonlinear heat equations in the pioneering work by Ovsiannikov in 1959 [20] and now is the core stone of modern group analysis [21,22]. This problem for the second-order wave equation was probably first solved by Barone et al in [23] and subsequently was extended to other general forms by many authors in the last two decades [2,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], but were all limited to second-order cases.…”

Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid

“…In particular, in the past several years, a numbers of novel techniques, such as algebraic methods based on subgroup analysis of the equivalence group [42][43][44][45], compatibility and direct integration [33,46] (also referred as the LieOvsiannikov method) as well as their generalizations (eg. method of furcate split [47], additional and conditional equivalence transformations [48,49], extended and generalized equivalence transformation group, gauging of arbitrary elements by equivalence transformations [50,51]) have been proposed to solve group classification problem for numerous nonlinear partial differential equations. Although a great deal of classification was solved by these methods, almost all of them are limited to the equations whose order are lower than four (see [51] for details).…”