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.
Highlights
In this paper, we study the group properties of higher-order nonlinear wave-type equations.As a basic model, we consider the fourth-order generalized nonlinear beam equation (GNBE) or nonlinear wave equation of the form utt = −[K (u)u xxx ] x + [ D (u)u x ] x + F (u), (1)where K = K (u), D = D (u) and F = F (u) are arbitrary smooth functions
The formal application of this method to equations containing several arbitrary functions (Equation (1) contains three arbitrary functions) usually leads to a large number of equations admitting nontrivial Lie algebras of invariance [44]. This method has been extended by many authors, in which they proposed a numbers of novel techniques, such as algebraic methods based on subgroup analysis of the equivalence group [45,46,47,48] and their generalizations [49,50,51,52,53,54], local transformations and form-preserving transformations [44,55,56,57,58], to solve group classification problem for numerous nonlinear partial differential equations
In this paper we extend the classical Lie-Ovsiannikov method based on equivalence transformations to the generalized nonlinear beam equation
Summary
We study the group properties of higher-order nonlinear wave-type equations. We consider the fourth-order generalized nonlinear beam equation (GNBE) or nonlinear wave equation of the form utt = −[K (u)u xxx ] x + [ D (u)u x ] x + F (u),. Equation (1) generalizes a wide range of the known nonlinear wave equations arising in applications. When K (u) = 1, D (u) = 0 and F (u) = −ku+ + 1, u+ = max{u, 0}, Equation (1) is reduced to the following classical nonlinear beam equation or the fourth-order nonlinear wave equation of the form [5]. Which is a slight generalization of the classical nonlinear beam Equation (3) and is presented by McKenna and Walter [7] in studying travelling wave solutions.
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