Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple Equation Method

Abstract

We construct the traveling wave solutions involving parameters of modified Liouville equation by using a new approach, namely the modified simple equation method. The proposed method is direct, concise, and elementary and can be used for many other nonlinear evolution equations.

1. Introduction

The investigation of the traveling-wave solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. Several direct methods for finding the explicit traveling-wave solutions to nonlinear partial differential equations have been proposed, such as the tanh-function method and its various extension [1], the Jacobi elliptic function expansion method [2], the homogeneous balance method [3, 4], the F-expansion method and its extension [5], the variational iteration method [6], (𝐺^/𝐺)-expansion method [7], and so on. More recently, a new method, named modified simple equation method [8, 9], has been proposed to construct more explicit traveling-wave solutions of modified Liouville equation.

2. Description of the Modified Simple Equation Method

Suppose that a nonlinear equation, say in two independent variables 𝑥 and 𝑡 is given by 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0,(2.1) where 𝑢=𝑢(𝑥,𝑡) is an unknown function, 𝑃 is a polynomial in 𝑢=𝑢(𝑥,𝑡) and its various partial derivatives, in which the highest-order derivatives and nonlinear terms are involved. In the following, the main steps of the modified simple equation method are given.

Step 1. The traveling-wave variable 𝑢(𝑥,𝑡)=𝑢(𝜉),where𝜉=𝐴𝑥+𝐵𝑡(2.2) permits us reducing (2.1) to an ODE for 𝑢=𝑢(𝜉) in the form 𝑃𝑢,−𝑉𝑢^,𝑢^,𝑉²𝑢^,−𝑉𝑢^,𝑢^,…=0.(2.3)

Step 2. Suppose that the solution of ODE (2.1) can be expressed by a polynomial in (𝜓′/𝜓) as follows: 𝑢(𝜉)=^𝑛_𝑖=0𝛼_𝑖𝜓𝜓′^𝑖,(2.4) where 𝛼_𝑖 are arbitrary constants to be determined such that 𝛼_𝑛≠0, while 𝜓(𝜉) is an unknown function to be determined later.

Step 3. We determine the positive integer 𝑛 by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in ODE (2.3).

Step 4. We substitute (2.4) into (2.3), we calculate all the necessary derivatives 𝑢^,𝑢^,…, and then we account the function 𝜓(𝜉). As a result of this substitution, we get a polynomial of 𝜓′/𝜓 and its derivatives. In this polynomial, we equate with zero all the coefficients of it. This operation yields a system of equations which can be solved to find 𝛼_𝑖 and 𝜓(𝜉). Consequently, we can get the exact solution of (2.1).

3. Application of the Method

In this section, we would like to use our method to obtain new and more general exact traveling wave solutions of the modified Liouville equation 𝑤_𝑡𝑡=𝑎²𝑤_𝑥𝑥+𝑏𝑒^𝛽𝑤,(3.1) where 𝑎,𝑏, and 𝛽 are arbitrary constants.

Suppose 𝑒^𝛽𝑤=𝑢(𝑥,𝑡), where the traveling-wave transformation is 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝐴𝑥+𝐵𝑡.(3.2) By using the traveling-wave variable (3.2), (3.1) is converted into an ODE for 𝑢=𝑢(𝜉)𝑢𝑢^−𝑢^2+𝑘𝑢³=0,where𝑘=𝑏𝛽𝑎²𝐴²−𝐵².(3.3) Suppose that the solution of the ODE (3.3) can be expressed by a polynomial in (𝜓′/𝜓) as follows: 𝑢(𝜉)=^𝑛_𝑖=0𝛼_𝑖𝜓𝜓′^𝑖,(3.4) where 𝛼_𝑖 are arbitrary constants provided 𝛼_𝑛≠0.

Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in (3.3), we get 𝑛=2 and hence the solution takes the following form: 𝑢(𝜉)=𝛼₀+𝛼₁𝜓𝜓′+𝛼²𝜓𝜓′²,(3.5) where 𝛼₂≠0. On substituting (3.5) into the ODE (3.3) and equating all the coefficients of 𝜓⁻¹,𝜓⁻²,𝜓⁻³,𝜓⁻⁴,𝜓⁻⁵,𝜓⁻⁶ to zero, we, respectively, obtain 3𝑘𝛼₂₀𝑘𝛼³⁰=0,(3.6)2𝛼₀𝛼₁𝜓′+𝛼₀𝛼₁𝜓^=0,(3.7)𝛼₂𝜓^2−𝛼²¹𝜓^2−3𝛼₀𝛼₁𝜓^𝜓^+3𝑘𝛼²⁰𝛼₂𝜓^2+3𝑘𝛼₀𝛼²¹𝜓^2𝑘𝛼₃₁+2𝛼₀𝛼₂𝜓^𝜓^+𝛼²¹𝜓^𝜓^=0,(3.8)𝜓^3+2𝛼₀𝛼₁𝜓^3−2𝛼₁𝛼₂𝜓^𝜓^2−𝛼²¹𝜓^2𝜓^+3𝛼₁𝛼₂𝜓^2𝜓^−2𝛼₂₂−10𝛼₀𝛼₂𝜓^2𝜓^+6𝑘𝛼₀𝛼₁𝛼₂𝜓^3=0,(3.9)𝜓^2𝜓^2+2𝛼²²𝜓^3𝜓^+𝛼²¹𝜓^4+3𝑘𝛼₀𝛼²²𝜓^4+3𝑘𝛼²¹𝛼₂𝜓^43𝑘𝛼+₁6𝛼₀𝛼₂𝜓^4−5𝛼₁𝛼₂𝜓^3𝜓^=0,(3.10)𝛼²²𝜓𝑘^5𝛼−₃₂2𝛼²²𝜓^4𝜓^+4𝛼₁𝛼₂𝜓^5=0,(3.11)𝜓^6+2𝛼²²𝜓^6=0.(3.12) Equations (3.6), (3.8), and (3.12) give 𝛼₀=0,𝛼₁=0, 𝛼₂=−2/𝑘(𝜓′≠0, otherwise it is the trivial case); those satisfy (3.7) and (3.9), and (3.10), (3.11), respectively, yields 𝜓^2−𝜓𝜓′𝜓^=0,(3.13)_=0.(3.14) Equation (3.14) gives 𝜓^=0. Integrating 𝜓^=0 with respect to 𝜉, we get 𝜓=𝐶₁+𝐶₂𝜉 and the solution of ODE (3.3) takes the following form: 𝑢(𝜉)=−2𝑘𝐶₁+𝐶¹𝐶₂𝜉²=2𝐵²−𝑎²𝐴²𝑏𝛽𝐴𝑥+𝐶𝐵𝑡+𝐶²,where𝐶=𝐶¹𝐶₂.(3.15) And finally the traveling-wave solution of (3.1) is 𝑤=1𝛽ln2𝐵²−𝑎²𝐴²𝑏𝛽𝐴𝑥+𝐶𝐵𝑡+𝐶².(3.16)

4. Conclusion

On comparing this method with the other methods via the tanh-function method, homogeneous balance method, and the (𝐺′/𝐺)-expansion method used in [1, 4, 7], we see that the modified simple equation method is much more simpler than these methods because these methods have used the computer programs, while the modified simple equation method has not used these programs. Also we deduce that the modified simple equation method is effective and standard which allows us to solve complicated nonlinear evolution equations in the mathematical physics.