Group classification of nonlinear wave equations

Lahno, V.; Zhdanov, Renat

doi:10.1063/1.1884886

Cited by 27 publications

(36 citation statements)

References 30 publications

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“…Let Q 3 = ∂ t . Imposing the second commutation relation from (20) gives (up to equivalence under ε) the operator Q 1 either equal to 2t∂ t or 2t∂ t + x∂ x .…”

Section: Theoremmentioning

confidence: 99%

“…Using the commutation relations (20), we find that the non-equivalent realizations of sl(2, R) within the class of operators (8) …”

Section: Theoremmentioning

confidence: 99%

“…They have utilized the method to derive the complete group classification of the general quasilinear heat conductivity equation and nonlinear wave equation in two independent variables [15,[20][21] . A review of the method has also been given by them to clarify the group classification of the general nonlinear evolution equation of order 2 [22] .…”

Section: Introductionmentioning

confidence: 99%

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Preliminary group classification of quasilinear third-order evolution equations

Huang

Zhang

2009

Appl. Math. Mech.-Engl. Ed.

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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.

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“…Let Q 3 = ∂ t . Imposing the second commutation relation from (20) gives (up to equivalence under ε) the operator Q 1 either equal to 2t∂ t or 2t∂ t + x∂ x .…”

Section: Theoremmentioning

confidence: 99%

“…Using the commutation relations (20), we find that the non-equivalent realizations of sl(2, R) within the class of operators (8) …”

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Preliminary group classification of quasilinear third-order evolution equations

Huang

Zhang

2009

Appl. Math. Mech.-Engl. Ed.

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“…To overcome this difficulty, Zhdanov and Lahno [25] developed a different purely algebraic approach enabling to classify NLEEs when the arbitrary functions in the equations depend on two or more arguments or the equation admits infinite-dimensional equivalence groups. The method suggested in [25] has been used to classify the nonlinear heat conductivity equations [3], the general second-order NLEEs [26], the nonlinear Schrödinger equations [27], the KdV-type equations [11], the nonlinear wave equations [15,16] and the fourth-order quasi-linear evolution equations [12,13].…”

Section: Introductionmentioning

confidence: 99%

Lie group classification of the N-th-order nonlinear evolution equations

Shen

Huang³

et al. 2011

Sci. China Math.

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In this paper, Lie group classification to the N -th-order nonlinear evolution equation t, u, ux, . . . , u is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three-and four-dimensional solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group so(3) as the symmetry group of the equation, and only two realizations of sl(2, R) are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Huang

Zhou

2011

Acta Appl Math

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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.

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Group classification of nonlinear wave equations

Cited by 27 publications

References 30 publications

Preliminary group classification of quasilinear third-order evolution equations

Preliminary group classification of quasilinear third-order evolution equations

Lie group classification of the N-th-order nonlinear evolution equations

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

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