The G′/G-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres

H. Jafari; N. Kadkhoda; Anjan Biswas

doi:10.1016/j.jksus.2012.02.002

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Original Article

25 (

); 57-62

doi:

10.1016/j.jksus.2012.02.002

The G′/G-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres

H. Jafari^a, N. Kadkhoda^a, Anjan Biswas^b,*

a Department of Mathematics, University of Mazandaran, Babolsar, Iran

b Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA

*Corresponding author. Tel.: +1 302 857 7913; fax: +1 302 857 7054 biswas.anjan@gmail.com (Anjan Biswas)

Received: 2012-1-23, Accepted: 2012-2-25, Published: 2012-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Available online 5 March 2012

Peer review under responsibility of King Saud University.

Abstract

The equations of magnetohydrostatic equilibria for plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magneto static atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. The distributed current in the model J is directed along the x-axis where x is the horizontal ignorable coordinate. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential u. This equation depends on an arbitrary function of u that must be specified with choices of different arbitrary functions, we obtain analytical nonlinear solutions of the elliptic equation using the $(\frac{G^{'}}{G})$ -expansion method. Finally, the hyperbolic versions of these equations will be solved by the travelling wave hypothesis method.

Keywords

(G′/G)-Expansion method

Magnetostatic equilibria

Nonlinear evolution equations

Travelling waves

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1 Introduction

The equations of magnetostatic equilibria have been used extensively to model the solar magnetic structure (Aslan, 2010; Heyvaerts et al., 1982; Khater et al., 2000, 2008; Kudryashov, 1988, 1990, 1991, 2010a; Low, 1982 ). 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 between J ∧ B force (B, magnetic field induction, J, 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 equilibria have been used to model diverse phenomena, such as the slow evolution stage of solar flares, or the magnetostatic support of prominences (Khater et al., 1997; Zwingmann, 1987). The nonlinear equilibrium problem has been solved in several cases (Khater, 1989; Lerche and Low, 1980; Webb, 1988; Webb and Zank, 1990 ). In this paper, we obtain the exact analytical solutions for the Liouville and sinh-Poisson equations using the $(\frac{G^{'}}{G})$ -expansion method. Because these two models will be special case of magnetostatic atmospheres model. Also here there is force balance between different forces. Recently the $(\frac{G^{'}}{G})$ -expansion method, first introduced by Wang et al. (2008) has become widely used to search for various exact solutions of nonlinear evolution equations (Jafari et al., in press; Kudryashov, 2010b; Li and Wang, 2009; Wang et al., 2008 ). The method is based on the explicit linearization of nonlinear evolution equations for travelling waves with a certain substitution which leads to a second-order differential equation with constant coefficients. Moreover, it transforms a nonlinear equation to a simplest algebraic computation. The outline of this paper is as follows:

First we describe the $(\frac{G^{'}}{G})$ -expansion method and the basic equations. Then we solve Liouville and sinh-Poisson equations with this method.

2 Basic idea of G′/G-expansion method

To illustrate the basic idea of this method, we consider the following nonlinear partial equation with only two independent variables x and t and a dependent variable u

(1)

N (u, u_{t}, u_{x}, u_{tt}, u_{xx}, \dots) = 0

using the travelling wave transformation

(2)

u = u (ξ), ξ = x - ct

Eq. (1) reduces to an ordinary differential equation (ODE) in the form:

(3)

N (u (ξ), - {cu}^{'} (ξ), u^{'} (ξ), c^{2} u^{″} (ξ), u^{″} (ξ), \dots) = 0 .

The $(\frac{G^{'}}{G})$ -expansion method is based on the assumption that the travelling wave solution of Eq. (3) can be expressed by a polynomial in $(\frac{G^{'}}{G})$ as

(4)

u (ξ) = \sum_{i = 0}^{n} A_{i} {(\frac{G^{'}}{G})}^{i}; A_{n} \neq 0;

where G = G(ξ) satisfies the second order linear ODE

(5)

G^{″} + λ G^{'} + μ G = 0

And A_i(i = 0, 1, 2, … , n), λ, μ are constants to be determined later, and G is the solution of (5), the general solutions of (5) are:

(6)

\frac{G^{'}}{G} = \{\begin{matrix} \frac{\sqrt{λ^{2} - 4 μ}}{2} (\frac{c_{1} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{2} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}{c_{2} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{1} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}) - \frac{λ}{2}; λ^{2} - 4 μ > 0 \\ \frac{\sqrt{4 μ - λ^{2}}}{2} (\frac{c_{1} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ - c_{2} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}{c_{2} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + c_{1} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}) - \frac{λ}{2}; λ^{2} - 4 μ < 0 \end{matrix}

u(ξ) can be determined explicitly by using the following steps:

Step (1)

By considering the homogeneous balance between the highest nonlinear terms and the highest order derivatives of u(ξ) in Eq. (3), the positive integer n in (4) is determined.
Step (2)

By substituting (4) with Eq. (5) into (3) and collecting all terms with the same power of $(\frac{G^{'}}{G})$ together, the left hand side of Eq. (3) is converted into a polynomial. After setting each coefficient of this polynomial to zero, we obtain a set of algebraic equations in terms of A_i (i = 0, 1, 2, … , n), c, λ, μ.
Step (3)

Solving the system of algebraic equations and then substituting the results with the general solutions of Eq. (5) into (4) gives travelling wave solutions of (3).

3 Basic equations

The relevance of magnetohydrostatic equations consisting of the equilibrium equation with force balance will be as:

(7)

J \land B - ρ \nabla Φ - \nabla P = 0

which is coupled with Maxwells equations:

(8)

J = \frac{\nabla \land B}{μ}

(9)

\nabla \cdot B = 0

where P, ρ, μ and Φ 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 p = ρR₀ T₀, where R₀ is the gas constant and T₀ is the temperature. Then the magnetic field B can be written by the following:

(10)

B = \nabla u \land e_{x} + B_{x} e_{x} = (B_{x}, \frac{\partial u}{\partial z}, \frac{- \partial u}{\partial y})

The form of (10) for B ensures that ∇ · B = 0, and there is no mono pole or defect structure. Eq. (7) requires the pressure and density be of the form (Low, 1977):

(11)

P (y, z) = P (u) e^{\frac{- z}{h}}, ρ (y, z) = \frac{1}{(gh)} P (u) e^{\frac{- z}{h}}

where

h = \frac{R_{0} T_{0}}{g}

is the scale height and z measures height. Substituting Eqs. (8)–(11) into Eq. (7), we obtain

(12)

\nabla^{2} u + f (u) e^{\frac{- z}{h}} = 0,

where

(13)

f (u) = μ \frac{dP}{du}

Eq. (13) gives

(14)

P (u) = P_{0} + \frac{1}{μ} \int f (u) du

Substituting Eq. (14) into Eq. (11), we obtain

(15)

P (y, z) = (P_{0} + \frac{1}{μ} \int f (u) du) e^{\frac{- z}{h}}

(16)

ρ (y, z) = \frac{1}{gh} (P_{0} + \frac{1}{μ} \int f (u) du) e^{\frac{- z}{h}}

where P₀ is constant. With taking transformation

(17)

x_{1} + {ix}_{2} = e^{\frac{- z}{l}} e^{\frac{iy}{l}}

Eq. (12) reduces to

(18)

\frac{\partial^{2} u}{\partial x_{1}^{2}} + \frac{\partial^{2} u}{\partial x_{2}^{2}} + l^{2} f (u) e^{(\frac{2}{l} - \frac{1}{h}) z} = 0

These equations have been given in Khater et al. (2000).

4 Applications of the G′/G-expansion method

In this section, we will investigate the $(\frac{G'}{G})$ -expansion method for solving specific forms of f(u).

4.1

4.1 Liouville equation

We first consider Liouville equation and the following equation will be special case of equation (18). Let us assume f(u) has the form (Dungey, 1953; Low, 1975):

(19)

f (u) = - α^{2} A_{0} e^{- \frac{A}{A_{0}}}

where A₀ and α² are constants. Hence

(20)

P (y, z) = (P_{0} + \frac{α^{2} A_{0}^{2}}{2 μ} e^{\frac{- 2 A}{A_{0}}}) e^{\frac{- z}{h}}

Inserting Eq. (19) into Eq. (18) we obtain

(21)

\nabla^{2} A / A_{0} = l^{2} α^{2} e^{- 2 \frac{A}{A_{0}} + (\frac{2}{l} - \frac{1}{h} z)}

where

\nabla^{2} = \frac{\partial^{2}}{\partial x_{1}^{2}} + \frac{\partial^{2}}{\partial x_{2}^{2}}

. Let us set

(22)

\frac{A}{A_{0}} = \frac{z}{L} + w (y, z)

where L is a constant. Then Eq. (21) becomes

(23)

\nabla^{2} w - l^{2} α^{2} e^{- 2 w - (\frac{2}{L} + \frac{1}{h} - \frac{2}{l}) z}

Let us identify l by

(24)

\frac{2}{l} = \frac{2}{L} + \frac{1}{h}

And inserting Eq. (24) into Eq. (23) we obtain a Liouville type equation

(25)

ϕ_{x_{1} x_{1}} + ϕ_{x_{2} x_{2}} - α^{2} l^{2} e^{- 2 ϕ} = 0

In order to apply the

\frac{G^{'}}{G}

, we use the wave transformation ξ = x₁ − cx₂ and change Eq. (25) into the form

(26)

(1 + c^{2}) ϕ^{″} = α^{2} l^{2} e^{- 2 ϕ}

we next use the transformation

(27)

v = e^{- 2 ϕ}

we obtain

(28)

(1 + c^{2}) {vv}^{″} - (1 + c^{2}) {(v^{'})}^{2} + 2 α^{2} l^{2} u^{3} = 0

with balancing according to step (1) we get n = 2, therefore the solution of (28) can be expressed by polynomial in

\frac{G^{'}}{G}

as follows:

(29)

v (ξ) = A_{0} + A_{1} \frac{G^{'}}{G} + A_{2} {(\frac{G^{'}}{G})}^{2}

Substituting Eq. (29) along with (5) into (28) and setting the coefficients of all powers of

\frac{G^{'}}{G}

to zero, we obtain a system of nonlinear algebraic equations for A₀, A₁, A₂. Solving the resulting system with the help of Mathematica, we have the following sets of solutions:

(30)

\{\begin{matrix} A_{0} = \frac{- μ (1 + c^{2})}{l^{2} α^{2}}; \\ A_{1} = \frac{- λ (1 + c^{2})}{l^{2} α^{2}}; \\ A_{2} = \frac{- (1 + c^{2})}{l^{2} α^{2}}; \end{matrix}

where ξ = x₁ − cx₂, λ, α, l are constants. Therefore, substituting (30) into (29), we have

(31)

v (ξ) = - \frac{μ (1 + c^{2})}{l^{2} α^{2}} - \frac{λ (1 + c^{2})}{l^{2} α^{2}} (\frac{G^{'}}{G}) - \frac{(1 + c^{2})}{l^{2} α^{2}} {(\frac{G^{'}}{G})}^{2}

substituting the general solution (6) into (31), according to Eq. (5), we obtain two types of travelling wave solutions of (28) as follows:

where λ² − 4μ > 0, we obtain the general hyperbolic function solutions of (28)

(32)

v_{1} (ξ) = - \frac{(λ^{2} - 4 μ) (1 + c^{2})}{l^{2} α^{2}} [{(\frac{c_{1} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{2} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}{c_{2} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{1} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ})}^{2} - 1]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − c x₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (32) gives the solitary wave solution:

(33)

v_{1} (ξ) = \frac{(λ^{2} - 4 μ) (1 + c^{2})}{l^{2} α^{2}} {sech}^{2} (\frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + ξ_{0})

where

\tanh ξ_{0} = \frac{c_{1}}{c_{2}}

, and when λ² − 4μ < 0, the general trigonometric function solutions of (28) will be:

(34)

v_{2} (ξ) = - \frac{(4 μ - λ^{2}) (1 + c^{2})}{l^{2} α^{2}} [{(\frac{c_{1} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ - c_{2} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}{c_{2} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + c_{1} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ})}^{2} + 1]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution Eq. (28) gives the solitary wave solution:

(35)

v_{2} (ξ) = - \frac{(4 μ - λ^{2}) (1 + c^{2})}{l^{2} α^{2}} \sec^{2} (- \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + ξ_{1})

where

\tan ξ_{1} = \frac{c_{1}}{c_{2}}

, when λ² − 4μ > 0 using with transformation v = e^−2ϕ we get:

(36)

ϕ_{1} (ξ) = \frac{- 1}{2} \ln [- \frac{(λ^{2} - 4 μ) (1 + c^{2})}{l^{2} α^{2}} ({(\frac{c_{1} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{2} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}{c_{2} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{1} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ})}^{2} - 1)]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (36) gives the solitary wave solution:

(37)

ϕ_{1} (ξ) = \frac{- 1}{2} \ln [\frac{(λ^{2} - 4 μ) (1 + c^{2})}{l^{2} α^{2}} ({sech}^{2} (\frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + ξ_{0}))]

where

\tanh ξ_{0} = \frac{c_{1}}{c_{2}}

, when λ² − 4μ < 0 using with transformation v = e^−2ϕ we get:

(38)

ϕ_{2} (ξ) = \frac{- 1}{2} \ln [- \frac{(4 μ - λ^{2}) (1 + c^{2})}{l^{2} α^{2}} ({(\frac{c_{1} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ - c_{2} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}{c_{2} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + c_{1} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ})}^{2} + 1)]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (38) give the solitary wave solution:

(39)

ϕ_{2} (ξ) = \frac{- 1}{2} \ln [- \frac{(4 μ - λ^{2}) (1 + c^{2})}{l^{2} α^{2}} \sec^{2} (- \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + ξ_{1})]

where

\tan ξ_{1} = \frac{c_{1}}{c_{2}}

4.2

4.2 Sinh-Poisson equation

Secondly we consider sinh-Poisson equation which plays an important role in the soliton model with BPS bound. Also, this equation will be special case of Eq. (18). If we assume

(40)

f (u) = - \frac{β^{2}}{4} (\frac{A_{0}}{h}) \sinh (ϕ)

As same as above we have

(41)

ϕ_{x_{1} x_{1}} + ϕ_{x_{2} x_{2}} = β^{2} \sinh (ϕ)

where l = 2h. In order to apply the

\frac{G^{'}}{G}

, we use the wave transformation ξ = x₁ − cx₂ and change Eq. (41) into the form

(42)

(1 + c^{2}) ϕ^{″} = β^{2} \sinh (ϕ)

we next use the transformation

(43)

\{\begin{matrix} v = e^{ϕ} \\ \sinh (ϕ) = \frac{e^{ϕ} - e^{- ϕ}}{2} \end{matrix}

we obtain

(44)

2 (1 + c^{2}) {vv}^{″} - 2 (1 + c^{2}) {(v^{'})}^{2} - β^{2} (v^{3} - v) = 0

with balancing according to step (1) we get n = 2, therefore the solution of (44) can be expressed by polynomial in

\frac{G^{'}}{G}

as follow:

(45)

v (ξ) = A_{0} + A_{1} \frac{G^{'}}{G} + A_{2} {(\frac{G^{'}}{G})}^{2}

Where G is the solution of (5) that was displayed in (6), as same as the previous section, we obtain a system of nonlinear algebraic equations for A₀, A₁, A₂, c, λ, μ. By solving the resulting system we obtain the following solution:

(46)

\{\begin{matrix} A_{0} = \frac{λ^{2} (1 + c^{2})}{β^{2}}; \\ A_{1} = \frac{4 λ (1 + c^{2})}{β^{2}}; \\ A_{2} = \frac{4 (1 + c^{2})}{β^{2}}; \end{matrix}

where ξ = x₁ − cx₂, λ, β are constants.

Therefore, substituting (46) into (45), we have

(47)

v (ξ) = \frac{λ^{2} (1 + c^{2})}{β^{2}} + \frac{4 λ (1 + c^{2})}{β^{2}} (\frac{G^{'}}{G}) + \frac{4 (1 + c^{2})}{β^{2}} {(\frac{G^{'}}{G})}^{2}

substituting the general solution (6) into (47) according to Eq. (5), we obtain two types of travelling wave solutions of (44) as follows:

where λ² − 4μ > 0, we obtain the general hyperbolic function solutions of (44):

(48)

v_{1} (ξ) = \frac{(λ^{2} - 4 μ) (1 + c^{2})}{β^{2}} {(\frac{c_{1} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{2} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}{c_{2} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{1} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ})}^{2}

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (48) gives the solitary wave solution:

(49)

v_{1} (ξ) = \frac{(λ^{2} - 4 μ) (1 + c^{2})}{β^{2}} {[\tanh (\frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + ξ_{0})]}^{2}

where

\tanh ξ_{0} = \frac{c_{1}}{c_{2}}

, and when λ² − 4μ < 0, the general trigonometric function solutions of Eq. (44) will be:

(50)

v_{2} (ξ) = \frac{(4 μ - λ^{2}) (1 + c^{2})}{β^{2}} {(\frac{c_{1} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ - c_{2} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}{c_{2} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + c_{1} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ})}^{2}

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (50) gives the solitary wave solution:

(51)

v_{2} (ξ) = \frac{(4 μ - λ^{2}) (1 + c^{2})}{β^{2}} {[\tan (- \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + ξ_{1})]}^{2}

where

\tan ξ_{1} = \frac{c_{1}}{c_{2}}

, when λ² − 4μ > 0 using with transformation v = e^ϕ we get:

(52)

ϕ_{1} (ξ) = \ln [\frac{(λ^{2} - 4 μ) (1 + c^{2})}{β^{2}} {(\frac{c_{1} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{2} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ}{c_{2} \cosh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + c_{1} \sinh \frac{\sqrt{λ^{2} - 4 μ}}{2} ξ})}^{2}]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (52) gives the solitary wave solution:

(53)

ϕ_{1} (ξ) = \ln [\frac{(λ^{2} - 4 μ) (1 + c^{2})}{β^{2}} {(\tanh (\frac{\sqrt{λ^{2} - 4 μ}}{2} ξ + ξ_{0}))}^{2}]

where

\tanh ξ_{0} = \frac{c_{1}}{c_{2}}

, when λ² − 4μ < 0 using with transformation v = e^ϕ we get:

(54)

ϕ_{2} (ξ) = \ln [\frac{(4 μ - λ^{2}) (1 + c^{2})}{β^{2}} {(\frac{c_{1} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ - c_{2} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ}{c_{2} \cos \frac{\sqrt{4 μ - λ^{2}}}{2} ξ + c_{1} \sin \frac{\sqrt{4 μ - λ^{2}}}{2} ξ})}^{2}]

where c₁ and c₂ are arbitrary constants, and ξ = x₁ − cx₂.

In particular, if we choose $c_{2} \neq 0, c_{1}^{2} < c_{2}^{2}$ , then the solution (54) gives the solitary wave solution:

(55)

ϕ_{2} (ξ) = \ln [\frac{(4 μ - λ^{2}) (1 + c^{2})}{β^{2}} (\tan {(\frac{\sqrt{4 μ - λ^{2}}}{2} ξ + ξ_{1}))}^{2}]

where

\tan ξ_{1} = \frac{c_{1}}{c_{2}}

5 Travelling waves

In this section the travelling wave solution of the regular Liouville equation as well as the sinh-Gordon equation will be obtained. This study is split into the following two subsections.

These travelling wave solutions are going to be very useful in various situations and circumstances. For example, in the context of plasma Physics, these travelling waves very easily study the behaviour of the weakly nonlinear ion acoustic waves in the presence of an uniform magnetic field. Thus, these solutions will be extremely useful for problems that are related to nonlinear quantum ion-acoustic waves in magnetized plasma containing cold ions and hot isothermal electrons.

5.1

5.1 Liouville equation

The form of the Liouville equation that will be studied in this section is given by Wazwaz (2009)

(56)

q_{tt} - k^{2} q_{xx} + {ae}^{2 q} = 0

Occasionally, this is referred to as the hyperbolic Liouville equation. The travelling wave assumption that is going to be made is given by

(57)

q (x, t) = g (x - vt)

where g(s) is the wave profile and v is the velocity of the wave and

(58)

s = x - vt

Substituting the hypothesis given by Eq. (50) into Eq. (49) yields

(59)

g^{″} + \frac{a}{v^{2} - k^{2}} e^{2 g} = 0

Now, multiplying both sides of (52) by g′ and integrating yields

(60)

{(g^{'})}^{2} = \frac{a}{v^{2} - k^{2}} (1 - e^{2 g})

and on separating variables this leads to

(61)

\int \frac{dg}{\sqrt{1 - e^{2 g}}} = \sqrt{\frac{a}{v^{2} - k^{2}}} \int ds

Eq. (54) integrates to

(62)

\tanh^{- 1} (\sqrt{1 - e^{2 g}}) = \sqrt{\frac{a}{v^{2} - k^{2}}} (x - vt)

which yields, after simplification,

(63)

q (x, t) = \frac{1}{2} \ln (sech [B (x - vt)])

where

(64)

B = \sqrt{\frac{a}{v^{2} - k^{2}}}

In order for the travelling wave to exist, it is necessary that the constraint condition given by

(65)

a (v^{2} - k^{2}) > 0

must hold, that follows from Eq. (57).

5.2

5.2 Sinh-Gordon equation

In this subsection, the travelling wave hypothesis will be applied to solve the hyperbolic version of the sinh-Poisson equation that is also known as the sinh-Gordon equation. The equation of study is therefore going to be (Wazwaz, 2009)

(66)

q_{tt} - k^{2} q_{xx} - b \sinh q = 0

The starting hypothesis stays the same as in Eq. (50), which when substituted into Eq. (59) gives after simplification

(67)

{(g^{'})}^{2} = \frac{2 b}{v^{2} - k^{2}} (1 + \cosh g)

Now, separation of variables imply

(68)

\int \frac{dg}{\sqrt{1 + \cosh g}} = \sqrt{\frac{2 b}{v^{2} - k^{2}}} \int ds

which integrates to

(69)

4 \tan^{- 1} (e^{\frac{g}{2}}) = \sqrt{\frac{2 b}{v^{2} - k^{2}}} (x - vt)

Simplification leads to the travelling wave solution as

(70)

q (x, t) = 2 \ln (sech [B (x - vt)])

where

(71)

B = \frac{1}{2} \sqrt{\frac{b}{v^{2} - k^{2}}}

Similarly, the constraint condition is given by

(72)

b (v^{2} - k^{2}) > 0

that follows from Eq. (64).

6 Concluding remarks

This study shows that the $(\frac{G^{'}}{G})$ -expansion method is quite efficient and practically well suited for use in finding exact solutions for the Liouville and sinh-Poisson equations. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. In this paper, the $(\frac{G^{'}}{G})$ -expansion 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 and trigonometric function solutions. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. Finally, the travelling wave solutions of the hyperbolic Liouville equation, or the regular Liouville equation as well as the sinh-Gordon equation are also obtained.

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