# New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method

Authors: Jae-Myoung Kim; Changbum Chun
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Based on the characteristics of the truncated Painlevé expansion method and the Exp-function method, new generalized solitary wave solutions are constructed for the KdV-Burgers-Kuramoto equation, which cannot be directly constructed from the Exp-function method. This work highlights the power of the Exp-function method in providing generalized solitary wave solutions of different physical structures.


---

## Body

## 1. Introduction

The investigation of the exact solutions of nonlinear partial differential equations plays an important role in mathematics, physics, and other applied science areas. In recent years, a variety of powerful methods has been proposed and analyzed to construct the explicit exact solutions to nonlinear evolution equations. Among these methods, the Exp-function method proposed by He and Wu [1] is particularly notable in its power and applicability in solving nonlinear problems, and it has been successfully applied to many kinds of nonlinear partial differential equations [2–17]. All of these applications verified that the Exp-function method is a straightforward, efficient, and versatile technique for finding generalized solitary, periodic, and rational solutions for nonlinear evolution equations as well as for revealing intriguing characteristics of various inner-wave interactions.The KdV-Burgers-Kuramoto equation [18, 19] is(1.1)ut+νuux+μuxxx+αuxx+γuxxxx=0,
where ν, μ, α, and γ are constants. This equation is an important mathematical model arising in many different physical contexts to describe many phenomena which are simultaneously involved in nonlinearity, dissipation, dispersion, and instability, especially at the description of turbulence processes. Since the solutions possess their actual physical application [20], various effective methods have been applied to construct the exact solutions of the KdV-Burgers-Kuramoto equation. These methods include the tanh function method [21–23], the homogeneous balance method [24], and the generalized F-expansion method [20]. As a consequence, it is still a significant and interesting task to search for new explicit exact solutions for nonlinear evolution equations. In the present work we extend the Exp-function method in combination with the truncated Painlevé expansion method [25, 26] to obtain new nontrivial exact solitary wave solutions to the KdV-Burgers-Kuramoto equation (1.1). As a result, we have found some new exact solitary wave solutions to (1.1) in the case where the Exp-function method provides trivial solutions only, which will be investigated in more detail in the following section.


## 2. Solitary Wave Solutions by the Exp-Function Method

Using the transformationu=u(η), η=kx+ωt, (1.1) becomes the ordinary differential equation:(2.1)ωu′+kνuu′+k2αu′′+k3μu′′′+k4γu′′′′=0.According to the Exp-function method [1], we assume that the solution of (2.1) can be expressed in the form(2.2)u(η)=∑n=-dcanenη∑m=-qpbmemη=a-de-dη+⋯+acecηb-qe-qη+⋯+bpepη,
where c, d, p, and q are positive integers, an and bm are unknown constants, which are to be determined later.By balancing the linear term of the highest order in (2.1) with the highest order nonlinear term, we get c+3p=2(p+c), and, thus, p=c. Similarly, balancing the linear term of the lowest order in (2.1) with the lowest order nonlinear term yields q=d. Here, the values of c and d can be freely chosen [1]. With the help of Maple, we have obtained only trivial solutions of (1.1) for the cases where (i) p=c=1 and d=q=1, (ii) p=c=1 and d=q=2, and (iii) p=c=2 and d=q=1, but one new solitary wave solution in the case where p=c=2 and d=q=2, which is given by(2.3)u(x,t)=a2e2η+a2b1eη+a2b0+a2b-1e-ηe2η+b1eη+b0+b-1e-η,
where η=kx+ωt, k and ω are arbitrary constants, and the coefficients a2, b-1, b0, b1 are free parameters. We expect that the other cases for the values of c and d will produce nontrivial solitary wave solutions to (1.1). In the next section we will consider finding new exact solitary wave solutions to (1.1) in the case where the Exp-function method provides trivial solutions only, particularly when p=c=1 and q=d=1.


## 3. New Solitary Wave Solutions to the KdV-Burgers-Kuramoto Equation

Let us consider the KdV-Burgers-Kuramoto equation defined by (1.1). The truncated Painlevé expansion method will be adopted to find a dependent variable transformation for finding solitary wave solutions to the considered equation. Substituting u(x,t) in the form
(3.1)u(x,t)=u0(x,t)f(x,t)r+u1(x,t)f(x,t)r-1+⋯+ur(x,t)
into (1.1) and finding the order of the pole r=2 for the solution u(x,t) and the functions u0(x,t),u1(x,t),…,ur(x,t), we obtain the dependent variable transformation
(3.2)u(x,t)=60γν∂3ln⁡f∂x3+15αν∂2ln⁡f∂x2+(240μ76ν-15α276νγ)∂ln⁡f∂x,
where γν≠0. Throughout the remainder of this paper we assume that γ and μ satisfy the condition γν≠0.Substitution of this transformation into the KdV-Burgers-Kuramoto equation (1.1) yields a rather complicated system of nonlinear equations in f for which we seek the solution form of the Exp-function method [1]:(3.3)f(x,t)=∑n=-cdanenη∑m=-pqbmemη,
where η=kx+ωt.For the sake of simplicity, we choosep=c=1 and q=d=1. Based on the consideration in (3.2) and (3.3), in this paper we assume that (1.1) admits a generalized solitary wave solution of the form(3.4)u(x,t)=g(x,t)76νγ(b1eη+b0+b-1e-η)3(a1eη+a0+a-1e-η)3,
where
(3.5)g(x,t)=A5e5η+A4e4η+A3e3η+A2e2η+A1eη+A0+B1e-η+B2e-2η+B3e-3η+B4e-4η+B5e-5η,
where
(3.6)A0=-15(-12α2a-12b12b0a0-6α2a-1b12b-1a02+6384γ2k2a12b02a-1b-1-6α2a-1b1b02a02+14592γ2k2a12b-12a-1b1-684γka12b02αb-1a-1+1824γ2k2a12b0b-12a0-6384γ2k2a1b02a-12b1-684γka1b02αa-12b1-2α2a-13b13+6384γ2k2a1b-12a02b1+684γka02b1αb-12a1+684γka02b-1αb12a-1+456γka03b1αb-1b0-1824γ2k2a-12b12b0a0-456γka1b03αa-1a0+192μγa-12b12b0a0-6α2a-12b12b-1a1+12α2a12b0b-12a0+6α2a02b1b-12a1-1824γ2k2a02b-1b02a1+96μγa-1b1b02a02-6384γ2k2a-1b12a02b-1-96μγa-1b02b-1a12-32μγa13b-13-96μγa12b1b-12a-1-192μγa12b0b-12a0+96μγa-12b12b-1a1+96μγa-12b1b02a1+1824γ2k2a-1b1b02a02+32μγa-13b13-96μγa1b1b-12a02-96μγa1b02b-1a02+96μγa-1b12b-1a02-14592γ2k2a1b-1a-12b12+2α2a13b-13+6α2a12b1b-12a-1+6α2a-1b02b-1a12-6α2a-12b1b02a1+6α2a02b-1b02a1)k,A1=-15(9α2a-1b0b-12a12+9α2a02b-12b0a1+96μγa-12b1b0b-1a1+6α2a-1b1b-12a1a0-9α2a-12b1b02a0+α2a03b1b-12-6α2a-12b1b0b-1a1-9α2a-12b12b-1a0+α2a03b-1b02-5α2a-13b12b0+6α2a-1b02b-1a1a0-α2a-12b03a1+5α2a0b-13a12-6α2a-1b1b0b-1a02-α2a-1b03a02+80μγa-13b12b0+304γka03b1αb-12-456γka1b0αa-12b1b-1-1824γ2k2a1b02a-1a0b-1+76αγka03b-1b02+456γka1b-12αa-1b1a0+304γ2k2a-1b03a02-304γ2k2a03b-1b02+16μγa-1b03a02-16μγa03b-1b02-76αγka-1b03a02+96μγa-1b1b0b-1a02-16μγa03b1b-12-912γka1b02αa-1b-1a0-304γ2k2a-13b12b0-14592γ2k2a1b-1a-12b1b0+8208γ2k2a12b-12a-1b0+14592γ2k2a1b-12a-1b1a0-96μγa-1b1b-12a1a0+1824γ2k2a-1b0a02b1b-1+2432γ2k2a03b1b-12+144μγa-12b12b-1a0+684γka-12b12αb-1a0-144μγa1b0b-12a02+144μγa-12b1b02a0-80μγa12b-13a0-96μγa-1b02b-1a1a0+16μγa-12b03a1-144μγa-1b0b-12a12+912γka-1b1αa02b-1b0+76γka-13b12αb0-8208γ2k2a-12b12a0b-1-304γka1b03αa-12-76γka12b-13αa0+304γ2k2a12b-13a0-684γka12b0αb-12a-1-2432γ2k2a1b03a-12)k,A2=-15(-12α2a-12b1b0b-1a0-4α2a-13b1b02-4α2a-13b12b-1+12α2a-1b0b-12a1a0-2α2a-12b03a0+2α2a03b-12b0+4α2a12b-13a-1+912αγka-12b1b0b-1a0+4α2a02b-13a1-304γ2k2a-12b03a0+304γ2k2a-13b1b02+2432γ2k2a12b-13a-1-76γka02b-13αa1+76γka-13b1αb02-304γka12b-13αa-1-684γka1b02αa-12b-1+8208γ2k2a-1b1a02b-12-32μγa03b-12b0-304γ2k2a1b-13a02+64μγa-13b1b02-1824γ2k2a-12b1b0a0b-1-76γka-12b03αa0-912αγka-1b0b-12a1a0-64μγa1b-13a02+32μγa-12b03a0+304γka-13b12αb-1+304γ2k2a03b-12b0+76γka03b-12αb0+64μγa-13b12b-1-64μγa12b-13a-1-192μγa-1b0b-12a1a0+192μγa-12b1b0b-1a0-2432γ2k2a-13b12b-1+1824γ2k2a1b-12a-1b0a0+684γka-1b1αa02b-12-8208γ2k2a1b-1a-12b02)k,A3=-15(α2a03b-13-α2a-13b03-456γka1b-13αa0a-1+3α2a-12b0b-12a1+48μγa-12b1b-12a0-96μγa1b-13a0a-1+16μγa-13b03-48μγa-1b0b-12a02+96μγa-13b1b0b-1+6α2a0b-13a1a-1+1824γ2k2a-1b0a02b-12-1824γ2k2a-12b02a0b-1-48μγa-12b0b-12a1-3α2a-12b02b-1a0-16μγa03b-13-684γka1b-12αa-12b0+684γka-12b1αa0b-12+48μγa-12b02b-1a0+6384γ2k2a-12b1a0b-12-6384γ2k2a1b-12a-12b0+456γka-13b1αb0b-1+3α2a-1b0b-12a02-6α2a-13b1b0b-1-3α2a-12b1b-12a0)k,A4=-15(-2432γ2k2a1b-13a-12-32μγa1b-13a-12+2432γ2k2a-13b1b-12+32μγa-13b1b-12+304αγka-13b1b-12-2α2a-13b02b-1+2α2a02b-13a-1-304γ2k2a-13b02b-1+76γka-13b02αb-1+304γ2k2a02b-13a-1-76γka02b-13αa-1+2α2a1b-13a-12-32μγa02b-13a-1-2α2a-13b1b-12+32μγa-13b02b-1-304αγka1b-13a-12)k,A5=-15(76αγka-13b0b-12-76αγka0b-13a-12+α2a0b-13a-12+16μγa-13b0b-12+304γ2k2a-13b0b-12-304γ2k2a0b-13a-12-16μγa0b-13a-12-α2a-13b0b-12)k,B1=-15(-96μγa12b1b0b-1a-1-16μγa12b03a-1+304γ2k2a03b1b02+9α2a12b1b-12a0-9α2a-1b12b0a02+684γka12b-12αb1a0+6α2a12b1b0b-1a-1-9α2a-12b12b0a1-α2a03b1b02+8208γ2k2a12b-12a0b1-304γ2k2a1b03a02-80μγa13b0b-12+α2a12b03a-1+α2a1b03a02+6α2a1b1b0b-1a02+9α2a12b02b-1a0-6α2a-1b1b02a1a0-5α2a0b13a-12-α2a03b12b-1-6α2a-1b12b-1a1a0+5α2a13b0b-12-912γka1b02αa-1b1a0+144μγa-12b12b0a1+304γka03b-1αb12-144μγa12b1b-12a0-76αγka1b03a02-16μγa1b03a02+16μγa03b1b02+2432γ2k2a12b03a-1+76αγka03b1b02-304γ2k2a-12b13a0+96μγa-1b12b-1a1a0-96μγa1b1b0b-1a02+80μγa-12b13a0+144μγa-1b12b0a02+912γka1b-1αa02b1b0-76γka-12b13αa0+456γka1b-1αa0b12a-1+1824γ2k2a1b02a-1a0b1-1824γ2k2a1b0a02b1b-1+76γka13b0αb-12-304γka12b03αa-1-456γka12b0αb1b-1a-1-684γka1b0αa-12b12+14592γ2k2a12b0b-1a-1b1-14592γ2k2a1b-1a-1b12a0-144μγa12b02b-1a0+96μγa-1b1b02a1a0+16μγa03b12b-1-2432γ2k2a03b12b-1+304γ2k2a13b0b-12-8208γ2k2a1b0a-12b12)k,B2=-15(-4α2a02b13a-1-2α2a03b12b0-4α2a-12b13a1+4α2a13b1b-12+4α2a13b02b-1+1824γ2k2a12b0b-1a0b1+64μγa-1b13a02-8208γ2k2a1b-1a02b12+12α2a12b1b0b-1a0+2α2a12b03a0-12α2a-1b12b0a1a0-912αγka-1b12b0a1a0-304γ2k2a13b02b-1-76γka02b13αa-1-1824γ2k2a1b0a-1b12a0+304γ2k2a12b03a0+8208γ2k2a12b02a-1b1+304γ2k2a-1b13a02+304γka13b-12αb1+76γka13b02αb-1-192μγa12b1b0b-1a0+912αγka12b1b0b-1a0-76γka12b03αa0+192μγa-1b12b0a1a0-304γka-12b13αa1+64μγa-12b13a1-32μγa12b03a0-64μγa13b02b-1-64μγa13b1b-12-304γ2k2a03b12b0+76γka03b12αb0-2432γ2k2a-12b13a1+684γka1b-1αa02b12+2432γ2k2a13b-12b1-684γka12b02αb1a-1+32μγa03b12b0)k,B3=-15(α2a13b03-α2a03b13+3α2a12b1b02a0-3α2a02b12b0a1+3α2a12b12b-1a0-48μγa12b1b02a0+96μγa-1b13a1a0-16μγa13b03-48μγa12b12b-1a0-3α2a12b12b0a-1+6α2a13b1b0b-1-96μγa13b1b0b-1-6384γ2k2a12b-1a0b12-456γka-1b13αa0a1-6α2a0b13a1a-1-1824γ2k2a1b0a02b12+456γka13b0αb1b-1+1824γ2k2a12b02a0b1-684γka12b0αa-1b12+684γka12b-1αa0b12+48μγa12b12b0a-1+6384γ2k2a12b0a-1b12+48μγa1b12b0a02+16μγa03b13)k,B4=-15(-304αγka-1b13a12+304αγka13b12b-1-32μγa13b12b-1+32μγa02b13a1-304γ2k2a02b13a1+2432γ2k2a-1b13a12-2432γ2k2a13b12b-1-32μγa13b1b02+32μγa-1b13a12+76γka13b02αb1-76γka02b13αa1-2α2a02b13a1-2α2a-1b13a12+304γ2k2a13b02b1+2α2a13b12b-1+2α2a13b1b02)k,B5=-15(-16μγa13b12b0-304γ2k2a13b12b0+304γ2k2a0b13a12-α2a0b13a12+76αγka13b12b0-76αγka0b13a12+α2a13b12b0+16μγa0b13a12)k.Substituting (3.4) into (1.1) and equating the coefficients of all powers of enη to zero yields a system of algebraic equations for a1, a0, a-1, b1, b0, b-1, k and ω. Solving the system with the help of Maple, we obtain the following cases.Case 1.
Consider(3.7){a1=0,a0=a-1b1b0,a-1=a-1,b1=b1,b0=b0,b-1=0,k=k,ω=ω}.Case 2.
Consider(3.8){k2(15α2-76αkγ-76γ2k2-316μγ)76γa1=a0b0b-1,a0=a0,a-1=0,b1=0,b0=b0,b-1=b-1,k=k,ω=k2(15α2-76αkγ-76γ2k2-316μγ)76γ}.Case 3.
Consider(3.9){μ(240μγ±76α-μγ-15α2)76γ2a1=a0b0b-1,a0=a0,a-1=0,b1=0,b0=b0,b-1=b-1,k=±-μγγ,ω=μ(240μγ±76α-μγ-15α2)76γ2}.Case 4.
Consider(3.10){20848μα2γ+39936μ2γ2+707α4-1168γαμ1923α2-64μγ-41α31923α2-64μγ184832γ3a1=a0b0b-1,a0=a0,a-1=0,b1=0,b0=b0,b-1=b-1,k=14-19αγ±437α2γ2-1216γ3μ38γ2,ω=-20848μα2γ+39936μ2γ2+707α4184832γ3-1168γαμ1923α2-64μγ-41α31923α2-64μγ184832γ3}.Substituting these cases into (3.4) we obtain the following new solitary wave solutions to (1.1):
(3.11)u(x,t)=15k76γνc3e6η+c2e5η+c1e4η+c0e3η+c-1e2η+c-2eη+c-3(b0+b1eη)6,c-3=(α2-16μγ)b16,c-2=(6α2-96μγ)b15b0,c-1=(15α2-240μγ)b02b14,c0=(20α2-320μγ)b03b13,c1=(15α2-240μγ)b12b04,c2=(6α2-96μγ)b05b1,c3=(α2-16μγ)b06,
where η=kx+ωt, k and ω are determined in Case 1, b0 and b1 are arbitrary constants. Consider
(3.12)u(x,t)=-15k76γνc3e6η+c2e5η+c1e4η+c0e3η+c-1e2η+c-2eη+c-3(b0eη+b-1)6,c-3=(α2-16μγ)b06,c-2=(6α2-96μγ)b05b-1,c-1=(15α2-240μγ)b-12b04,c0=(20α2-320μγ)b03b-13,c1=(15α2-240μγ)b-14b02,c2=(6α2-96μγ)b0b-15,c3=(α2-16μγ)b-16,
where η=kx+ωt, k and ω are determined in Case 2, b0 and b-1 are arbitrary constants. Consider
(3.13)u(x,t)=-15k76γνc3e6η+c2e5η+c1e4η+c0e3η+c-1e2η+c-2eη+c-3(b0eη+b-1)6,c-3=(α2-16γμ)b06,c-2=(6α2-96γμ)b05b-1,c-1=(15α2-240μγ)b04b-12,c0=(20α2-320μγ)b03b-13,c1=(15α2-240μγ)b-14b02,c2=(6α2-96γμ)b-15b0,c3=(α2-16γμ)b-16,
where η=kx+ωt, k and ω are determined in Case 3, b0 and b-1 are arbitrary constants. Consider
(3.14)u(x,t)=-15k2888γνc3e6η+c2e5η+c1e4η+c0e3η+c-1e2η+c-2eη+c-3(b0eη+b-1)6,c-3=(α2-16μγ)b06,c-2=(6α2-96μγ)b05b-1,c-1=(15α2-240μγ)b04b-12,c0=(20α2-320μγ)b03b-13,c1=(15α2-240μγ)b02b-14,c2=(6α2-96μγ)b0b-15,c3=(α2-16μγ)b-16,
where η=kx+ωt, k and ω are determined in Case 4, b0 and b-1 are arbitrary constants.The correctness of solutions (3.11)–(3.14) is ensured by testing them on computer with the aid of symbolic computation software Maple. It should be noted that these solutions have not been found in the literature. We also applied the Exp-function method to the KdV-Burgers-Kuramoto equation (1.1) to obtain its solitary wave solutions. As a result, we found only trivial solutions for the cases (i)p=c=q=d=1, (ii)p=c=2,q=d=1, (iii)p=c=1, q=d=2, and one nontrivial solution for the case p=c=2, q=d=2, and we expect that more nontrivial solutions will be found for other cases. This indicates that the solutions obtained in this work cannot be directly constructed by the Exp-function method particularly in the case where p=c=q=d=1.


## 4. Conclusion

In this paper, we have successfully implemented the Exp-function method based on the truncated Painlevé expansion method and obtained new generalized solitary wave solutions of the KdV-Burgers-Kuramoto equation. Our approach yields new travelling wave solutions with some free parameters. The approach considered in this paper and the Exp-function method complement each other in that our approach may provide nontrivial solutions in the cases that the Exp-function method may provide only trivial solutions. The result reveals that the Exp-function method is a promising tool because it can provide a variety of new soliton solutions with distinct physical structures.



---

*Source: 892420-2012-05-20.xml*