Nonlinear Ritz Approximation for the Camassa-Holm Equation by Using the Modify Lyapunov-Schmidt method

: In this work, the modified Lyapunov-Schmidt reduction is used to find a nonlinear Ritz approximation of Fredholm functional defined by the nonhomogeneous Camassa-Holm equation and Benjamin-Bona-Mahony. We introduced the modified Lyapunov-Schmidt reduction for nonhomogeneous problems when the dimension of the null space is equal to two. The nonlinear Ritz approximation for the nonhomogeneous Camassa-Holm equation has been found as a function of codimension twenty-four.


Introduction:
There are a lot of mathematical, physical, chemical, and engineering phenomena that are shown as nonlinear problems so can be described these problems as a nonlinear Fredholm operator.(, ) = ,  ∈  ⊆ ,  ∈ ,  ∈  𝑛 1 When  is a smooth Fredholm map with zero indexes and S is an open subset of Banach spaces.
One of them is .Write the other one as .To solve these problems may be used the method of reduction to the dimensional equation by solving this equation, (, ) = ,  ∈ ,  ∈ , 2 When  and  are smooth manifolds of finite dimensional and :   →  is a smooth function.The Lyapunov-Schmidt method can reduce Eq. 1 to Eq. 2, in which Eq. 2 has the same properties as Eq. 1, in particular topological properties (multiplicity) and analytical properties (bifurcation diagram), which are found in 1 .So that to study Eq. 1 it is sufficient to study Eq. 2.
Nonlinear problems are one subject of the greatest important subjects of mathematical phenomena possess received a great interest in scientific research in the last decades because of their wide set of geometry and scientific applications.Many of these studies focus on getting the bifurcation solutions of some equations, especially nonlinear partial differential equations (PDEs) that occur in Engineering, Physics, or mathematics.Also, in the Lyapunov-Schmidt method, the solutions in unlimited dimensional spaces coincide with the solutions in limited dimensional spaces.Therefore, the method is an important method in modernistic Mathematics to find analytical solutions.Many researchers have dealt with this method; it was previously called the alternative method by the researcher Krasnoselskii 1956 2 who used it to study Bifurcation for extremely without boundaries while the implicit function theory was unable to be used.Sapronov and his group.For example, in 3 used the homogeneous solution to have the linear Ritz approximation represented by the function (ζ, λ) of the functional in Eq.1.Lyapunov-Schmidt method was also used to study boundary value problems, which can be seen in 4-7 .Abdul Hussain, Mayada 8 and Mizeal 9 , study a bifurcation equation for a nonlinear system given by two algebraic equations.
In the last years, Kadhim 13 studied the bifurcation solution of extremes of the functions of codimensions eight and five at the origin by using Lyapunov-Schmidt reduction (LSR).In previous works, the presence and absence of  shaped solutions were studied using the Lyapunov-Schmidt method and Ritz linear approximation.As for our work, we study the presence and the absence of  +  solutions using the modified Lyapunov-Schmidt method and the nonlinear Ritz approximation.
The goal of this paper is to find the nonlinear Ritz approximation of the functional corresponding to the nonhomogeneous Camassa-Holm equation.

Materials and Methods: Methods:
Proposition 1 4 .Suppose that the triple {, ,  } is an elliptic finite dimensional reduction for the functional  on a set Ω from the smooth Banach manifold M. Then the marginal map  locates a one-to-one corresponding between the critical points for the functional  and the critical points for the key function .

Lyapunov-Schmidt reduction (LSR)
The LSR was first suggested by Schmidt 1908 14 .He discovered this method to get the solutions to operator equations.It is a method employed to solve the problems that possess variational property and the problems that do unpossessed variational property

Modify Lyapunov-Schmidt method for the nonhomogeneous nonlinear differential equations (MLSM)
Modify Lyapunov-Schmidt method is a procedure for obtaining the nonlinear Ritz approximation to a Fredholm functional.MLSM is similar to the Lyapunov-Schmidt reduction but the MLSM is based on finding the particular solution of the operator Eq. 1 in the nonhomogeneous cases as follows: Suppose the nonlinear operator which is Fredholm with zero index :  →  such that  ) =  0 () +  1 (, ) 5 Where  0 () represents a homogenous polynomial with degree  ≥ 3 s.t 0 (0) = 0 &  1 (, ) is a polynomial function of degree < .If  1 ,  2 , …   are the coefficients to the quadratic terms for the function  1 (, ), then can be written the function  1 (, ) in the formula, Where   2 = , 2 <  < .
The functional  has a nonlinear Ritz approximation, it's a function  defined by Taylor's expansion to the functions μ k (ξ) and (x, ξ, β) will be used to determine the nonlinear Ritz approximation for the functional V, by assuming as following: Where   () () and  (𝑗) To calculate the functions (x, ξ, β) & μ k (ξ) equate the coefficients of  = ( 1 ,  2 , … ,   ) in Eq.8 to find the value of   and after some calculation from Eq.9, it is getting a linear ODE in the variable   (, ).Solving the equation which appears one can get the value to   (, ).
In the following section, we give two examples to find a nonlinear Ritz approximation for the functional corresponding to the nonhomogeneous Camassa-Holm Equation and Benjamin-Bona-Mahony equation as an application of the Modify Lyapunov-Schmidt method given above.

Results: Nonlinear Ritz Approximation for the Camassa-Holm Equation (CH)
This section applied MLSM given in the previous section for finding nonlinear Ritz approximation for the functional corresponding to the nonhomogeneous Camassa-Holm equation.
To obtain a nonlinear approximation for the Camassa-Holm equation.Firstly, write Eq. 12 as a nonlinear Fredholm operator as follows: ( ′ ) 2 +  ′′ ) 13 when : E ⟶  is Fredholm operator which is nonlinear of index zero from Banach space  to Banach space , where  =  2 ([0,1], ℝ) is the space of all continuous functions that have derivative of order at most two,  = ([0,1], ℝ) is the space of every continuous function,  = (, ) .The operator  own variational property, so there is a functional  defined by,  (, ) =    (, ) Where V (w, λ, ) = The solution of the linearized Eq.15 verification of the boundary conditions is get by,   =   (),  = 1,2,3, … 16 Substituting Eq. 16 in Eq. 15 has a characteristic equation identical to the above solution in the form,  −  2  2 = 0 The equation above gives in the characteristic lines (  −  ), wherefore, a point of characteristic lines it's the points of (, ) such that Eq.10 own nontrivial solutions.Can be found at the bifurcation point 18 in the space of parameters (, ) from the point of intersection of the  − .As a result, (0,0) is a bifurcation point for Eq.10.And localized parameters for ,  gives by,  ̂= 0 + Γ 1 ,  ̂= 0 + Γ 2 .where Γ 1 , Γ 2 are parameters which small lead to the below modes over the bifurcation.

Proof:
To determine the key function of V (w, γ, ) wall substituting Eq.27 in the functional The geometry of the bifurcation of critical points and the principal asymptotic of the branches of bifurcating points for the function (, ) are entirely determined by its principal part (, ).The function (, ) has all the topological and analytical properties of functional V (w, γ, ).The spreading of the critical points of the function (, ) depends on the change of parameter  and will be discussed in this paper as follows: The study of the discriminant set of function (, ) it not easy to find so, we will use maple 16 to find the discriminant set of the above function (, ), in particular, we will fix the values of  1 , γ i ,  = 1,2, . .,11. and then to find all sections of discriminant set in the  2  1  2 − surfaces, so we have three cases.