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Exact Travelling Wave Solutions for Isothermal Magnetostatic Atmospheres by Fan Subequation Method
The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential . This equation depends on an arbitrary function of that must be specified. With choices of the different arbitrary functions, we obtain analytical solutions of elliptic equation using the Fan subequation method.
The equations of magnetostatic equilibria have been used extensively to model the solar magnetic structure [1–4]. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. The force balance consists of the force ( is the magnetic field induction and is the electric current density), the gravitational force, and gas pressure gradient force. However, in many models, the temperature distribution is specified a priori and direct reference to the energy equations is eliminated. In solar physics, the equations of magnetostatic have been used to model diverse phenomena, such as the slow evolution stage of solar flares, or the magnetostatic support of prominences [5, 6]. The nonlinear equilibrium problem has been solved in several cases [7–9].
Recently, Fan and Hon  developed an algebraic method, belonging to the sub-equation method to seek more new solutions of nonlinear partial differential equations (NLPDEs) that can be expressed as polynomial in an elementary function which satisfies a more general sub-equation, called Fan sub-equation, than other sub-equations like Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. As we know, the more general analytical exact solutions of the sub-equation are proposed, the more general corresponding exact solutions of NLPDEs will be obtained. Thus, it is very important how to obtain more new solutions to the sub-equation. Fortunately, the Fan sub-equation method can construct more general exact solutions to the sub-equation that can capture all the solutions of the Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. Some works using the Fan's technique are presented in [1, 11–16].
In this paper, we obtain the exact travelling wave solutions for the Liouville and sinh-Poisson equations using the Fan sub-equation method. These two models are special cases of magnetostatic atmospheres model. Also in these cases there is force balance between differents forces.
2. The Basic Idea of Fan Subequation Method
In this section, we outline the main steps of Fan sub-equation method .
Step 1. For a given nonlinear partial differential equation we consider its travelling wave solutions , , then (2.1) is reduced to a nonlinear ordinary differential equation where a prime denotes the derivative with respect to the variable .
Step 2. Expand the solution of (2.2) in the form where are constants to be determined later and the new variable satisfies the Fan sub-equation where and are constants.
Thus, the derivatives with respect to the variable become the derivatives with respect to the variable as follows:
Step 5. Solve these algebraic equations to obtain . Substituting these results into (2.3) yields the general form of travelling wave solutions.
Step 6. For each solution to (2.4) which depends on the special conditions chosen for the , and , it follows from (2.3) obtained from the above steps that the corresponding exact solution of (2.2) can be constructed.
3. Basic Equations
The relevant magnetohydrostatic equations consist of the equilibrium equation which is coupled with Maxwells equations where , , , and are the gas pressure, the mass density, the magnetic permeability, and the gravitational potential, respectively. It is assumed that the temperature is uniform in space and that the plasma is an ideal gas with equation of state , where is the gas constant and is the temperature. Then the magnetic field can be written as The form of (3.3) for ensures that and there is no mono pole or defect structure.
Equation (3.1) requires the pressure and density to be of the form  where is the scale height. Substituting (3.2)–(3.4) into (3.1), we obtain where Equation (3.6) gives where is constant. Substituting (3.7) into (3.4), we obtain Using transformation , (3.5) reduces to These equations have been given in .
4. Applications of the Fan Subequation Method
In this section, we will employ the Fan sub-equation method for solving (3.9) for specific forms of the function .
4.1. Liouville Equation
We next use the transformation and obtain the nonlinear ordinary differential equation Using Step 3 given above, we get , therefore the solution of (4.3) can be expressed as Following Step 4, we obtain a system of nonlinear algebraic equations for , , and :
Case 1. When , , , (2.4) admits a hyperbolic function solution Thus (4.4) yields the following new solitary wave solution of (2.1) of bell-type where , , , , and are arbitrary constants. Reverting back to the original variables and , we obtain the solution of (4.1) in the form
Case 4. When , (2.4) admits three Jacobian elliptic doubly periodic solutions and (4.4), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants.
4.2. The -Poisson Equation
Secondly, we consider sinh-Poisson equation which plays an important role in soliton model with BPS Bound. Also, this equation is a special case of (3.9) and is given by In order to apply the Fan sub-equation method, we use the wave transformation and convert (4.17) into the form We next use the transformation and obtain the equation Applying Step 3, we get , therefore the solution of (4.19) can be expressed as Then using Step 4, we obtain a system of nonlinear algebraic equations for , , and :
Case 3. When , , , , (2.4) has two kinds of exact solutions and (4.20) yields one family of solitary travelling wave solutions solitary travelling wave solutions of (4.17) given by where ,, and are arbitrary constants.
Case 4. When , (2.4) admits three Jacobian elliptic doubly periodic solutions and (4.20), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions with , , , , and being arbitrary constants. Similarly, from (4.20), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants. Likewise, from (4.20), respectively, we can get two families of Jacobian elliptic doubly periodic wave solutions with , , , , , and being arbitrary constants.
5. Concluding Remarks
In this paper, the Fan sub-equation method has been successfully used to obtain some exact travelling wave solutions for the Liouville and sinh-Poisson equations. These exact solutions include the hyperbolic function solutions, trigonometric function solutions. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. Thus, this study shows that the Fan sub-equation method is quite efficient and practically well suited for use in finding exact solutions for nonlinear partial differential equations. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.
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Copyright © 2012 Hossein Jafari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.