Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Abstract

Given a class F (θ ) of differential equations with arbitrary element θ , the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member f ∈ F (θ ) the structure of its Lie symmetry group G f , conditional symmetry Q f and conservation law CL f under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1 + 1)-dimensional nonlinear telegraph equations with coefficients depending on the space variableThe usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g = 1 and g = h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g = 1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.

Variational contact symmetries of constrained Lagrangians

“…The well known conservation laws of energy, angular momentum all follow from Noether symmetries. It is well known that conservation laws are important tools which can be used for the determination of integrable manifolds of a dynamical system in mathematical physics, biology, economics and many others [1,2,3,4,5,6,7,8,9].…”

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

“…Since then the nonclassical symmetry method has been applied to various equations and systems in hundreds of published papers, e.g. [27], [37], [16], [28], [17], [54], [13], [11], [12], [15], [51], [4], the latest b being [14], [31], [55], [10].…”

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

“…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.…”