Exact Travelling Wave Solutions for Isothermal Magnetostatic Atmospheres by Fan Subequation Method

Abstract

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 𝑢. This equation depends on an arbitrary function of 𝑢 that must be specified. With choices of the different arbitrary functions, we obtain analytical solutions of elliptic equation using the Fan subequation method.

1. Introduction

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 𝐽∧𝐵 force (𝐵 is the magnetic field induction and 𝐽 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 [10] 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 [11].

Step 1. For a given nonlinear partial differential equation 𝑁𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑥,…=0 (2.1) we consider its travelling wave solutions 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡, then (2.1) is reduced to a nonlinear ordinary differential equation 𝑁𝑢(𝜉),−𝑐𝑢^′(𝜉),𝑢^′(𝜉),𝑐²𝑢^′′(𝜉),𝑢^′′(𝜉),…=0, (2.2) where a prime denotes the derivative with respect to the variable 𝜉.

Step 2. Expand the solution of (2.2) in the form 𝑢(𝜉)=^𝑛_𝑖=0𝐴_𝑖𝜙^𝑖,𝐴_𝑛≠0, (2.3) where 𝐴_𝑖(𝑖=0,1,…,𝑛) are constants to be determined later and the new variable 𝜙 satisfies the Fan sub-equation 𝜙^′(𝜉)=𝜖⁴_𝑗=0𝑤_𝑗𝜙^𝑗, (2.4) where 𝜖=±1 and 𝑤_𝑗(𝑗=0,…,4) are constants.

Thus, the derivatives with respect to the variable 𝜉 become the derivatives with respect to the variable 𝜙 as follows: 𝑑𝑢𝑑𝜉=𝜖⁴_𝑗=0𝑤_𝑗𝜙^𝑗𝑑𝑢,𝑑𝑑𝜙²𝑢𝑑𝜉²=12⁴_𝑗=1𝑗𝑤_𝑗𝜙^𝑗−1𝑑𝑢+𝑑𝜙⁴_𝑗=0𝑤_𝑗𝜙^𝑗𝑑²𝑢𝑑𝜙². (2.5)

Step 3. Determine 𝑛 by substituting (2.3) with (2.4) into (2.2) and balancing the linear term of the highest order with the nonlinear term in (2.2).

Step 4. Substituting (2.3) and (2.4) into (2.2) again and collecting all coefficients of 𝜙^𝑖(𝑖=0,1,2,…,𝑛), then setting these coefficients to zero will give a set of algebraic equations with respect to 𝐴_𝑖(𝑖=0,1,…,𝑛).

Step 5. Solve these algebraic equations to obtain 𝐴_𝑖(𝑖=0,1,2,…,𝑛). 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 𝑤₀,𝑤₁,𝑤₂,𝑤₃, and 𝑤₄, 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 𝐽∧𝐵−𝜌∇Φ−∇𝑃=0, (3.1) which is coupled with Maxwells equations 𝐽=∇∧𝐵𝜇,∇⋅𝐵=0, (3.2) 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=𝜌𝑅₀𝑇₀, where 𝑅₀ is the gas constant and 𝑇₀ is the temperature. Then the magnetic field 𝐵 can be written as 𝐵=∇𝑢∧𝑒_𝑥+𝐵_𝑥𝑒_𝑥=𝐵_𝑥,𝜕𝑢,𝜕𝑧−𝜕𝑢.𝜕𝑦 (3.3) The form of (3.3) for 𝐵 ensures that ∇⋅𝐵=0 and there is no mono pole or defect structure.

Equation (3.1) requires the pressure and density to be of the form [4] 𝑃𝑦,𝑧=𝑃(𝑢)𝑒^−𝑧/ℎ=1,𝜌𝑦,𝑧𝑃(𝑢)𝑒𝑔ℎ^−𝑧/ℎ, (3.4) where ℎ=𝑅₀𝑇₀/𝑔 is the scale height. Substituting (3.2)–(3.4) into (3.1), we obtain ∇²𝑢+𝑓(𝑢)𝑒^−𝑧/ℎ=0, (3.5) where 𝑓(𝑢)=𝜇𝑑𝑃.𝑑𝑢 (3.6) Equation (3.6) gives 𝑃(𝑢)=𝑃₀+1𝜇𝑓(𝑢)𝑑𝑢, (3.7) where 𝑃₀ is constant. Substituting (3.7) into (3.4), we obtain 𝑃𝑦,𝑧=𝑃₀+1𝜇𝑓(𝑢)𝑑𝑢𝑒^−𝑧/ℎ,𝜌=1𝑦,𝑧𝑃𝑔ℎ⁰+1𝜇𝑓(𝑢)𝑑𝑢𝑒^−𝑧/ℎ. (3.8) Using transformation 𝑥₁+𝑖𝑥₂=𝑒^−𝑧/𝑙𝑒^{𝑖𝑦/𝑙}, (3.5) reduces to 𝜕²𝑢𝜕𝑥²₁+𝜕²𝑢𝜕𝑥²₂+𝑙²𝑓(𝑢)𝑒^{(2/𝑙−1/ℎ)𝑧}=0. (3.9) These equations have been given in [2].

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 𝑓(𝑢).

4.1. Liouville Equation

We first consider Liouville equation, which is a special case of (3.9), namely, 𝑢_𝑥𝑥+𝑢_𝑡𝑡−𝛼²𝑙²𝑒^−2𝑢=0. (4.1) In order to apply the Fan sub-equation method, we use the wave transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡 and transform (4.1) into the form 1+𝑐²𝑢^′′=𝛼²𝑙²𝑒^−2𝑢. (4.2)

We next use the transformation 𝑣=𝑒^−2𝑢 and obtain the nonlinear ordinary differential equation 1+𝑐²𝑣𝑣^′′−1+𝑐²𝑣^′2+2𝛼²𝑙²𝑣³=0. (4.3) Using Step 3 given above, we get 𝑛=2, therefore the solution of (4.3) can be expressed as 𝑣(𝜉)=𝐴₀+𝐴₁𝜙+𝐴₂𝜙². (4.4) Following Step 4, we obtain a system of nonlinear algebraic equations for 𝐴₀, 𝐴₁, and 𝐴₂: 2𝛼²𝑙²𝐴₀³−𝜖²𝐴₁²𝑤₀−𝑐²𝜖²𝐴₁²𝑤₀+2𝜖²𝐴₀𝑤₀+2𝑐²𝜖²𝐴₀𝐴₂𝑤₀+12𝜖²𝐴₀𝐴₁𝑤₁+12𝑐²𝜖²𝐴₀𝐴₁𝑤₁=0,6𝛼𝑙²𝐴₀²𝐴₁−2𝜖²𝐴₁𝐴₂𝑤₀−2𝑐²𝜖²𝐴₁𝐴₂𝑤₀−12𝜖²𝐴₁²𝑤₁+3𝜖²𝐴₀𝐴₂𝑤₁+3𝑐²𝜖²𝐴₀𝐴₁²+𝜖²𝐴₀𝐴₁𝑤₂+𝑐²𝜖²𝐴₀𝐴₁𝑤₂=0,6𝛼²𝑙²𝐴₀𝐴₁²+6𝛼²𝑙²𝐴₀²𝐴₂−2𝜖²𝐴₂²𝑤₀−12𝜖²𝐴₁𝐴₂𝑤₁−12𝑐²𝜖²𝐴₁𝐴₂𝑤₁+4𝜖²𝐴₀𝐴₂𝑤₂+4𝑐²𝜖²𝐴₀𝐴₂𝑤₂+32𝜖²𝐴₀𝐴₁𝑤₃+32𝑐²𝜖²𝐴₀𝐴₁𝑤₃=0,2𝛼²𝑙²𝐴₁³+12𝛼²𝑙²𝐴₀𝐴₁𝐴₂−𝜖²𝐴₂²𝑤₁−𝑐²𝜖²𝐴₂²𝑤₁+𝜖²𝐴₁𝐴₂𝑤₂+𝑐²𝜖²𝐴₁𝐴₂𝑤₂+12𝜖²𝐴₁²𝑤₃+12𝑐²𝜖²𝐴₁²𝑤₃+5𝜖²𝐴₀𝐴₂𝑤₃+5𝑐²𝜖²𝐴₀𝐴₂𝑤₃+2𝜖²𝐴₀𝐴₁𝑤₄+2𝑐²𝜖²𝐴₀𝐴₁𝑤₄=0,6𝛼²𝑙²𝐴₁²𝐴₂+6𝛼²𝑙²𝐴₀𝐴₂²+52𝜖²𝐴₁𝐴₂𝑤₃+52𝜖²𝑐²𝐴₁𝐴₂𝑤₃+𝜖²𝐴₁²𝑤₄+𝑐²𝜖²𝐴₁²𝑤₄+6𝜖²𝐴₀𝐴₂𝑤₄+6𝑐²𝜖²𝐴₀𝐴₂𝑤₄=0,6𝛼²𝑙²𝐴₁𝐴₂²+𝜖²𝐴₂²𝑤₃+𝑐²𝜖²𝐴₂²𝑤₁+𝜖²𝐴₁𝐴₂𝑤₂+𝑐²𝜖²𝐴₂²𝑤₃+4𝜖²𝐴₁𝐴₂𝑤₄+4𝑐²𝜖²𝐴₁𝐴₂𝑤₄=0,2𝛼²𝑙²𝐴₂³+2𝜖²𝐴₂²𝑤₄+2𝑐²𝜖²𝐴₂²𝑤₄=0. (4.5)

Case 1. When 𝑤₀=𝑤₁=𝑤₃=0, 𝑤₂>0, 𝑤₄<0, (2.4) admits a hyperbolic function solution 𝜙=−𝑤₂𝑤₄sech√𝑤₂𝜉. (4.6) Thus (4.4) yields the following new solitary wave solution of (2.1) of bell-type 𝑣₁(𝜉)=1+𝑐²𝑤₂𝛼²𝑙²sech²√𝑤₂𝜉, (4.7) where 𝑤₂>0, 𝑤₄<0, 𝛼≠0, 𝑙≠0, and 𝑐 are arbitrary constants. Reverting back to the original variables 𝑥 and 𝑡, we obtain the solution of (4.1) in the form 𝑢₁()1𝑥,𝑡=−2ln1+𝑐²𝑤₂𝛼²𝑙²sech²√𝑤₂(𝑥−𝑐𝑡). (4.8)

Case 2. When 𝑤₁=𝑤₃=0, 𝑤₀=𝑤₂²/4𝑤₄, 𝑤₂<0, 𝑤₄>0, (2.4) admits two hyperbolic function solutions 𝜙=±−𝑤₂2𝑤₄tanh−𝑤₂2𝜉, (4.9) and so (4.4) yields one family of solitary travelling wave solutions of (4.1) given by 𝑢₂()1𝑥,𝑡=−2ln−1+𝑐²𝑤₂2𝛼²𝑙²+1+𝑐²𝑤₂𝛼²𝑙²tanh²−𝑤₂2(𝑥−𝑐𝑡), (4.10) where 𝑤₂<0, 𝑤₄>0, 𝛼≠0, 𝑙≠0, and 𝑐 are arbitrary constants.

Case 3. When 𝑤₀=𝑤₁=0, 𝑤₃=±2√𝑤₂𝑤₄, 𝑤₂>0, 𝑤₄>0, (2.4) has two kinds of exact solutions: √𝜙=−𝑤₂𝑤₄2𝑤₄sign𝑤₃√1+tanh𝑤₂2𝜉, (4.11) and (4.4) yields one family of solitary travelling wave solutions of (4.1) given by 𝑢₃()1𝑥,𝑡=−2ln±1+𝑐²𝑤₂𝛼²𝑙²sign𝑤₃√1+tanh𝑤₂2()−𝑥−𝑐𝑡1+𝑐²𝑤₂4𝛼²𝑙²√1+tanh𝑤₂2(𝑥−𝑐𝑡)², (4.12) where 𝑤₂>0, 𝑤₄>0, 𝛼≠0, 𝑙≠0, and 𝑐 are arbitrary constants.

Case 4. When 𝑤₁=𝑤₃=0, (2.4) admits three Jacobian elliptic doubly periodic solutions 𝜙=−𝑤₂𝑘²𝑤₄2𝑘²−1cn𝑤₂2𝑘²−1𝜉,𝑘,for𝑤₀=𝑤₂²𝑘²𝑘²−1𝑤₄2𝑘²−1²,𝑤₂>0,𝑤₄<0,𝜙=−𝑤₂𝑤₄2−𝑘²dn𝑤₂2−𝑘²𝜉,𝑘,for𝑤₀=𝑤₂²1−𝑘²𝑤₄𝑘²−2²,𝑤₂>0,𝑤₄<0,𝜙=±−𝑤₂𝑘²𝑤₄𝑘²+1sn−𝑤₂𝑘²+1𝜉,𝑘,for𝑤₀=𝑤₂²𝑘²𝑤₄𝑘²+1²,𝑤₂<0,𝑤₄>0, (4.13) and (4.4), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions 𝑢₄()1𝑥,𝑡=−2ln−1+𝑐²𝑤₂2𝛼²𝑙²+1+𝑐²𝑤₂2𝑘²−14𝛼²𝑙²𝑘²−1cn²𝑤₂2𝑘²()−1𝑥−𝑐𝑡,𝑘, (4.14) with 𝑤₂>0, 𝑤₄<0, 𝛼≠0, 𝑙≠0, 𝑘∈(√2/2,1), and 𝑐 being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions 𝑢₅()1𝑥,𝑡=−2ln−1+𝑐²𝑤₂2𝛼²𝛽²+1+𝑐²𝑤₂𝑘²−24𝛼²𝑙²1−𝑘²dn²𝑤₂2−𝑘²(𝑥−𝑐𝑡),𝑘, (4.15) with 𝑤₂>0, 𝑤₄<0, 𝛼≠0, 𝑙≠0, 𝑘∈(0,1), and 𝑐 being arbitrary constants. Similarly, from (4.4), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions 𝑢₆()1𝑥,𝑡=−2ln−1+𝑐²𝑤₂2𝛼²𝑙²+1+𝑐²𝑤₂𝑘²+14𝛼²𝑙²sn²−𝑤₂𝑘²()+1𝑥−𝑐𝑡,𝑘 (4.16) with 𝑤₂<0, 𝑤₄>0, 𝛼≠0, 𝑙≠0, 𝑘∈(0,1), and 𝑐 being arbitrary constants.

4.2. The sinh-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 𝑢_𝑥𝑥+𝑢_𝑡𝑡=𝛽²sinh(𝑢). (4.17) In order to apply the Fan sub-equation method, we use the wave transformation 𝜉=𝑥−𝑐𝑡 and convert (4.17) into the form 1+𝑐²𝑢^′′=𝛽²sinh(𝑢). (4.18) We next use the transformation 𝑣=𝑒^𝑢 and obtain the equation 21+𝑐²𝑣𝑣^′′−21+𝑐²𝑣^′2−𝛽²𝑣³−𝑣=0. (4.19) Applying Step 3, we get 𝑛=2, therefore the solution of (4.19) can be expressed as 𝑣(𝜉)=𝐴₀+𝐴₁𝜙+𝐴₂𝜙². (4.20) Then using Step 4, we obtain a system of nonlinear algebraic equations for 𝐴₀, 𝐴₁, and 𝐴₂: −𝑙²𝐴₀³−2𝜖²𝐴₁²𝑤₀−2𝑐²𝜖²𝐴₁²𝑤₀+4𝜖²𝐴₀𝐴₂𝑤₀+4𝑐²𝜖²𝐴₀𝐴₂𝑤₀+𝜖²𝐴₀𝐴₁𝑤₁+𝑐²𝜖²𝐴₀𝐴₁𝑤₁=0,−3𝑙²𝐴₀²𝐴₁−4𝜖²𝐴₁𝐴₂𝑤₀−4𝑐²𝜖²𝐴₁𝐴₂𝑤₀−𝜖²𝐴₁²𝑤₁−𝑐²𝜖²𝐴₁²𝑤₁+6𝜖²𝐴₀𝐴₂𝑤₁+6𝑐²𝜖²𝐴₀𝐴₂𝑤₁+2𝜖²𝐴₀𝐴₁𝑤₂+2𝑐²𝜖²𝐴₀𝐴₁𝑤₂=0,−3𝑙²𝐴₀𝐴₁²−3𝑙²𝐴₀²𝐴₂−4𝜖²𝐴₂²𝑤₀−4𝑐²𝜖²𝐴₂²𝑤₀−𝜖²𝐴₁𝐴₂𝑤₁−𝑐²𝜖²𝐴₁𝐴₂𝑤₁+8𝜖²𝐴₀𝐴₂𝑤₂+8𝑐²𝜖²𝐴₀𝐴₂𝑤₂+3𝜖²𝐴₀𝐴₁𝑤₃+3𝑐²𝜖²𝐴₀𝐴₁𝑤₃=0,−𝑙²𝐴₁³−6𝑙²𝐴₀𝐴₁𝐴₂−2𝜖²𝐴₂²𝑤₁−2𝑐²𝜖²𝐴₂²𝑤₁+2𝜖²𝐴₁𝐴₂𝑤₂+2𝑐²𝜖²𝐴₁𝐴₂𝑤₂+𝜖²𝐴₁²𝑤₃+𝑐²𝜖²𝐴₁²𝑤₃+10𝜖²𝐴₀𝐴₂𝑤₃+10𝑐²𝜖²𝐴₀𝐴₂𝑤₃+4𝜖²𝐴₀𝐴₁𝑤₄+4𝑐²𝜖²𝐴₀𝐴₁𝑤₄=0,−3𝑙²𝐴₁²𝐴₂−3𝑙²𝐴₀𝐴₂²+5𝜖²𝐴₁𝐴₂𝑤₃+5𝑐²𝜖²𝐴₁𝐴₂𝑤₃+2𝜖²𝐴₁²𝑤₄+2𝑐²𝜖²𝐴₁²𝑤₄+12𝜖²𝐴₀𝐴₂𝑤₄+12𝑐²𝜖²𝐴₀𝐴₂𝑤₄=0,−3𝑙²𝐴₁𝐴₂²+2𝜖²𝐴₂²𝑤₃+2𝑐²𝜖²𝐴₂²𝑤₃+8𝜖²𝐴₁𝐴₂𝑤₄+8𝑐²𝜖²𝐴₁𝐴₂𝑤₄=0,−𝑙²𝐴₂³+4𝜖²𝐴₂²𝑤₄+4𝑐²𝜖²𝐴₂²𝑤₄=0. (4.21)

Case 1. When 𝑤₀=𝑤₁=𝑤₃=0, 𝑤₂>0, 𝑤₄<0, (2.4) admits a hyperbolic function solution 𝜙=−𝑤₂𝑤₄sech√𝑤₂𝜉 (4.22) and (4.20) yields the following new solitary wave solution of (4.17) of bell-type 𝑢₁()−4𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²sech²√𝑤₂(𝑥−𝑐𝑡), (4.23) where 𝑤₂>0, 𝑤₄<0, 𝑙≠0, and 𝑐 are arbitrary constants.

Case 2. When 𝑤₁=𝑤₃=0, 𝑤₀=𝑤₂²/4𝑤₄, 𝑤₂<0, 𝑤₄>0, (2.4) admits two hyperbolic function solutions 𝜙=±−𝑤₂2𝑤₄tanh−𝑤₂2𝜉, (4.24) and (4.20) yields one family of solitary travelling wave solutions of (4.17) given by 𝑢₂()2𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²−21+𝑐²𝑤₂𝑙²tanh²−𝑤₂2(𝑥−𝑐𝑡), (4.25) where 𝑤₂<0, 𝑤₄>0, 𝑙≠0, and 𝑐 are arbitrary constants.

Case 3. When 𝑤₀=𝑤₁=0, 𝑤₃=±2√𝑤₂𝑤₄, 𝑤₂>0, 𝑤₄>0, (2.4) has two kinds of exact solutions √𝜙=−𝑤₂𝑤₄2𝑤₄sign𝑤₃√1+tanh𝑤₂2𝜉, (4.26) and (4.20) yields one family of solitary travelling wave solutions solitary travelling wave solutions of (4.17) given by 𝑢₃()±2𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²sign𝑤₃√1+tanh𝑤₂2()−𝑥−𝑐𝑡1+𝑐²𝑤₂𝑙²√1+tanh𝑤₂2(𝑥−𝑐𝑡)², (4.27) where 𝑤₂>0,𝑤₄>0, 𝑙≠0 and 𝑐 are arbitrary constants.

Case 4. When 𝑤₁=𝑤₃=0, (2.4) admits three Jacobian elliptic doubly periodic solutions 𝜙=−𝑤₂𝑘²𝑤₄2𝑘²−1cn𝑤₂2𝑘²−1𝜉,𝑘,for𝑤₀=𝑤₂²𝑘²𝑘²−1𝑤₄2𝑘²−1²,𝑤₂>0,𝑤₄<0,𝜙=−𝑤₂𝑤₄2−𝑘²dn𝑤₂2−𝑘²𝜉,𝑘,for𝑤₀=𝑤₂²1−𝑘²𝑤₄𝑘²−2²,𝑤₂>0,𝑤₄<0,𝜙=±−𝑤₂𝑘²𝑤₄𝑘²+1sn−𝑤₂𝑘²+1𝜉,𝑘,for𝑤₀=𝑤₂²𝑘²𝑤₄𝑘²+1²,𝑤₂<0,𝑤₄>0, (4.28) and (4.20), respectively, yields two families of Jacobian elliptic doubly periodic wave solutions 𝑢₄()2𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²+21+𝑐²2𝑘²−1𝑤₂𝑙²𝑘²−1cn²𝑤₂2𝑘²()−1𝑥−𝑐𝑡,𝑘, (4.29) with 𝑤₂>0, 𝑤₄<0, 𝑙≠0, 𝑘∈(√2/2,1), and 𝑐 being arbitrary constants. Similarly, from (4.20), respectively, we can obtain two families of Jacobian elliptic doubly periodic wave solutions 𝑢₅()2𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²−21+𝑐²𝑤₂2−𝑘²𝑙²1−𝑘²dn²𝑤₂2−𝑘²(𝑥−𝑐𝑡),𝑘, (4.30) with 𝑤₂>0, 𝑤₄<0, 𝛼≠0, 𝑙≠0, 𝑘∈(0,1), and 𝑐 being arbitrary constants. Likewise, from (4.20), respectively, we can get two families of Jacobian elliptic doubly periodic wave solutions 𝑢₆()2𝑥,𝑡=ln1+𝑐²𝑤₂𝑙²−21+𝑐²𝑤₂𝑘²+1𝑙²sn²−𝑤₂𝑘²(),+1𝑥−𝑐𝑡,𝑘 (4.31) with 𝑤₂<0, 𝑤₄>0, 𝛼≠0, 𝑙≠0, 𝑘∈(0,1), and 𝑐 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.