Two Modes Bifurcation Solutions of Elastic Beams Equation with Nonlinear Approximation

In this paper we studied two modes bifurcation solutions of elastic beams equation by using Lyapunov-Schmidt method. The bifurcation equation corresponding to the elastic beams equation has been found. Also, we studied two modes of nonlinear approximation of bifurcation solutions of a specified equation and we found the Key function corresponding to the functional related to this equation.


Introduction
It is known that many of the nonlinear problems that appear in Mathematics and Physics can be written in the form of operator equation, where f is a smooth Fredholm map of index zero and X , Y are Banach spaces and O is open subset of X .For these problems, the method of reduction to finite dimensional equation, can be used, where M and N are smooth finite dimensional manifolds.Passage from equation (1.1) into equation (1.2) (variant local scheme of Lyapunov-Schmidt) with the conditions, that equation (1.2) has all the topological and analytical properties of equation (1.1) (multiplicity, bifurcation diagram, etc.) dealing with [3], [8], [11], [12].Definition 1.1.Suppose that E and F are Banach spaces and A : E → F be a linear continuous operator.The operator A is called Fredholm operator, if (i) The kernel of A, Ker (A), is finite dimensional, (ii) The range of A, Im(A), is closed in F , (iii) The Cokernel of A, Coker (A), is finite dimensional.Suppose that f : Ω → F is a nonlinear Fredholm map of index zero.A smooth map f : Ω → F has variational property, if there exist a functional V : where (〈•, •〉 H is the scalar product in Hilbert space H).In this case the solutions of equation f (x, λ) = 0 are the critical points of functional V (x, λ).Suppose that f : E → F is a smooth Fredholm map of index zero, E, F are Banach spaces and where V is a smooth functional on E. Also we assume that E ⊂ F ⊂ H, H is a Hilbert space, then by using method of finite dimensional reduction (Local scheme of Lyapunov-Schmidt) the problem, The function W (ξ, λ) is called Key function.
If N = span{e 1 , . . ., e n } is a subspace of E, where e 1 , . . ., e n are orthonormal basis, then the Key function W (ξ, λ) can be defined in the form, The function W has all the topological and analytical properties of the functional V (multiplicity, bifurcation diagram, etc.) [10].The study of bifurcation solutions of functional V is equivalent to the study of bifurcation solutions of Key function.If f has variational property, then it is easy to check that, The set of all λ for which the function W (ξ, λ) has degenerate critical points, is called Caustic.
The linear Ritz approximation of the functional V is a function W given by the formula, The oscillations and motion of waves of the elastic beams on elastic foundations can be described by means of the following PDE, ∂ 2 y where y is the deflection of beam.It is known that, to study the oscillations of beams, stationary state (u(x) = y(x, t)) should be monitored which is describes by the equation, In this work equation (1.3) has been studied with the following boundary conditions, Equation (1.3) has been studied by Thompson and Stewart [4] they showed numerically the existence of periodic solutions of equation (1.3) for some values of parameters.Bardin and Furta [1] used the local method of Lyapunov-Schmidt and found the sufficient conditions of existence of periodic waves of equation (1.3), also they are introduced the solutions of equation ( 1.3) in the form of power series.Furta and Piccione [9] showed the existence of periodic travelling wave solutions of equation ( 1.3) describing oscillations of an infinite beam, which lies on a nonlinearly elastic support with non-small amplitudes.Sapronov ( [2], [10], [11], [12]) applied the local method of Lyapunov-Schmidt and found the bifurcation solutions of equation (1.3).Abdul Hussain ( [5], [6]) studied equation (1.3) with small perturbation when the nonlinear part has quadratic term and Mohammed [7] studied equation (1.3) in the variational case when the nonlinear part has quadratic term.The goal of this paper to study two modes bifurcation solutions of equation (1.3) with boundary conditions (1.4) by using the procedure of Bardin and Furta [1] and then we used the result of this procedure to study the two modes nonlinear bifurcation solutions of equation (1.3) by using the work of Sapronov.

Nonlinear Approximation Solutions
Suppose that f : E → F is a nonlinear Fredholm operator of index zero from Banach space E to Banach space F defined by, where is the space of all continuous functions which have derivative of order at most four, In this case the solutions of equation (1.3) is equivalent to the solutions of the operator equation, We note that the operator f has variational property that is; there exist a functional where (〈•, •〉 H is the scalar product in Hilbert space H) and In this case the solutions of equation (2.2) are the critical points of the functional V (u, λ), where the critical points of the functional V (u, λ) are the solutions of Euler-Lagrange equation, Thus, the study of equation (1.3) with the conditions (1.4) is equivalent to the study extremely problem, Analysis of bifurcation can be finding by using method of Lyapunov-Schmidt to reduce into finite dimensional space.By localized parameters, δ 1 , δ 2 are small parameters.
The reduction lead to the function in two variables, It is well known that in the reduction of Lyapunov-Schmidt the function W (ξ, δ) is smooth.This function has all the topological and analytical properties of functional V [10].In particular, for small δ there is one-to-one corresponding between the critical points of functional V and smooth function W , preserving the type of critical points (multiplicity, index Morse, etc.) [10].By using the scheme of Lyapunov-Schmidt, the linearized equation corresponding to the equation (2.2) has the form: This equation give in the αβ-plane characteristic lines.The point of characteristic lines are the points of (α, β) in which equation (2.2) has nonzero solutions.The point of intersection of characteristic lines in the αβ-plane is a bifurcation point [10].The result of this intersection lead to bifurcation along the modes e 1 = c 1 sin(x), e 2 = c 2 sin(2x).For the equation (2.2) the point (α, β) = (5, 4) is a bifurcation point [10].Localized parameters, Lead to the bifurcation along the modes e 1 , e 2 , where e 1 = e 2 = 1 and dx 2 +β, then the space E can be decomposed in direct sum of two subspaces, N and the orthogonal complement to N , Similarly, the space F decomposed in direct sum of two subspaces, N and orthogonal complement to N , There exists projections p : E → N and I − p : E → E such that pu = w and (I − p)u = v, (I is the identity operator).Hence every vector u ∈ E can be written in the form, Similarly, there exists projections (2.3) Accordingly, equation (2.2) can be written in the form, By the implicit function theorem, there exist a smooth map Φ : N → E, such that and then the linear Ritz approximation of the functional V is a function W given by, The nonlinear Ritz approximation of the functional V is a function W given by, W (ξ, δ) = V (ξ 1 e 1 + ξ 2 e 2 + Φ(ξ 1 e 1 + ξ 2 e 2 , δ), δ), v(x, ξ, λ) = Φ(w, δ).
Function (2.17) has all the topological and analytical properties of functional V .Also, the function is symmetric in the variables ξ 1 and ξ 2 ( W (ξ 1 , ξ 2 ) = W (−ξ 1 , −ξ 2 )) it have 121 critical points.So it is not easy to determine the Caustic of function (2.17) and study the bifurcation solutions of this function.The point u(x) = ξ 1 e 1 + ξ 2 e 2 + v(x, ξ, λ) is a critical point of the functional V (u, λ) iff the point ξ is a critical point of the function W (ξ, δ) [10].This mean that the existence of the solutions of equation (2.2) depend on the existence of the critical points of the functional V (u, λ) and then on the existence of the critical points of the function W (ξ, δ).From this notation, we can find a nonlinear approximation of the solutions of equation (2.2) corresponding to each critical point of the function W (ξ, δ).Caustic of the function W (ξ, δ) and the distribution of the critical points in the plane of parameters (Bifurcation diagram) depending on the corner singularities of smooth maps will be discuses in other paper.
Key words and phrases.Bifurcation solutions; Local scheme of Lyapunov-Schmidt; Singularities of smooth maps.The number dim(Ker A) − dim(Coker A) is called Fredholm index of the operator A.