Group Classification and Exact Solutions of Nonlinear Wave Equations

Lahno, V.; Zhdanov, Renat; Magda, Olena

doi:10.1007/s10440-006-9039-0

Cited by 46 publications

(60 citation statements)

References 33 publications

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“…In this case, g is given by g = 0 and g = −1/u after by translating u. Substituting g = 0 into the remain equations of system (29) and scaling u, we find that…”

Section: Definition 2 An Evolutionary Vector Fieldmentioning

confidence: 98%

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

Huang

Иванова

2007

Journal of Mathematical Physics

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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.

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“…In this case, g is given by g = 0 and g = −1/u after by translating u. Substituting g = 0 into the remain equations of system (29) and scaling u, we find that…”

Section: Definition 2 An Evolutionary Vector Fieldmentioning

confidence: 98%

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

Huang

Иванова

2007

Journal of Mathematical Physics

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“…As a result, exact solutions differing from (30) were constructed for some of these equations. Below we consider several important equations of form (29) and use their exact solutions for the generation of new exact solutions for equations from class (7) with the coefficients given by (32).…”

Section: Generation Of Exact Solutions By Point Transformationsmentioning

confidence: 99%

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

2008

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A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficientis studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m = 2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m = 2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

Huang

2017

Symmetry

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Abstract:In this paper we make a Lie symmetry analysis of a generalized nonlinear beam equation with both second-order and fourth-order wave terms, which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco. We perform a complete Lie symmetry group classification by using the equivalence transformation group theory for the equation under consideration. Lie symmetry reductions of a nonlinear beam-like equation which are singled out from the classification results are investigated. Some classes of exact solutions, including solitary wave solutions, triangular periodic wave solutions and rational solutions of the nonlinear beam-like equations are constructed by means of the reductions and symbolic computation.

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Group Classification and Exact Solutions of Nonlinear Wave Equations

Cited by 46 publications

References 33 publications

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

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

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

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