Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 597983 | https://doi.org/10.1155/2012/597983

J. F. Alzaidy, "Extended Mapping Method and Its Applications to Nonlinear Evolution Equations", Journal of Applied Mathematics, vol. 2012, Article ID 597983, 14 pages, 2012. https://doi.org/10.1155/2012/597983

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

Academic Editor: Renat Zhdanov
Received02 Apr 2012
Revised15 Jul 2012
Accepted31 Jul 2012
Published05 Sep 2012

Abstract

We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

1. Introduction

The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modeled by the bell-shaped sech solutions and the kink-shaped tanh solutions. Many effective methods have been presented, such as inverse scattering transform method [1], BΓ€cklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], variable separation approach [5], various tanh methods [6–9], homogeneous balance method [10], similarity reductions method [11, 12], (πΊξ…ž/𝐺)-expansion method [13], the reduction mKdV equation method [14], the trifunction method [15, 16], the projective Riccati equation method [17], the Weierstrass elliptic function method [18], the Sine-Cosine method [19, 20], the Jacobi elliptic function expansion [21, 22], the complex hyperbolic function method [23], the truncated PainlevΓ© expansion [24], the F-expansion method [25], the rank analysis method [26], the ansatz method [27, 28], the exp-function expansion method [29], and the sub-ODE method [30].

The main objective of this paper is using the extended mapping method to construct the exact solutions for nonlinear evolution equations in the mathematical physics via the Boussinesq system and the coupled KdV equations.

2. Description of the Extended Mapping Method

Suppose we have the following nonlinear PDE: 𝐹𝑒,𝑒𝑑,𝑒π‘₯,𝑒𝑑𝑑,𝑒π‘₯π‘₯,𝑒π‘₯𝑑,…=0,(2.1) where 𝑒=𝑒(π‘₯,𝑑) is an unknown function, 𝐹 is a polynomial in 𝑒=𝑒(π‘₯,𝑑) and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation method.

Step 1. The traveling wave variable 𝑒(π‘₯,𝑑)=𝑒(πœ‰),πœ‰=π‘˜(π‘₯βˆ’πœ”π‘‘),(2.2) where π‘˜ and πœ” are the wave number and the wave speed, respectively. Under the transformation (2.2), (2.1) becomes an ordinary differential equation (ODE) as 𝑃𝑒,π‘’ξ…ž,π‘’ξ…žξ…ž,π‘’ξ…žξ…žξ…žξ€Έ,…=0.(2.3)

Step 2. If all the terms of (2.3) contain derivatives in 𝜁, then by integrating this equation and taking the constant of integration to be zero, we obtain a simplified ODE.

Step 3. Suppose that the solution (2.3) has the following form: 𝑒(πœ‰)=π‘Ž0+𝑛𝑖=1ξ€·π‘Žπ‘–π‘“π‘–(πœ‰)+π‘π‘–π‘“βˆ’π‘–ξ€Έ+(πœ‰)𝑛𝑖=2π‘π‘–π‘“π‘–βˆ’2(πœ‰)π‘“ξ…ž(πœ‰)+𝑛𝑖=βˆ’1𝑑𝑖𝑓𝑖(πœ‰)π‘“ξ…ž(πœ‰),(2.4) where π‘Ž0, π‘Žπ‘–, 𝑏𝑖, 𝑐𝑖, and 𝑑𝑖 are constants to be determined later, while 𝑓(πœ‰) satisfies the nonlinear ODE: ξ€Ίπ‘“ξ…žξ€»(πœ‰)2=𝑝𝑓4(πœ‰)+π‘žπ‘“2(πœ‰)+π‘Ÿ,(2.5) where 𝑝, π‘ž, and π‘Ÿ are constants.

Step 4. The positive integer β€œπ‘›β€ can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in (2.3). Therefore, we can get the value of 𝑛 in (2.4).

Step 5. Substituting (2.4) into (2.3) with the condition (2.5), we obtain polynomial in 𝑓𝑖(πœ‰)[π‘“ξ…ž(πœ‰)]𝑗, (𝑖=…,βˆ’2,βˆ’1,0,1,2,…;𝑗=0,1). Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for π‘Ž0, π‘Žπ‘–, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, πœ”, and π‘˜.

Step 6. Solving the algebraic equations by use of Maple or Mathematica, we have π‘Ž0, π‘Žπ‘–, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, and π‘˜ expressed by 𝑝, π‘ž, π‘Ÿ.

Step 7. Since the general solutions of (2.5) have been well known for us (see Appendix A), then substituting the obtained coefficients and the general solution of (2.5) into (2.4), we have the travelling wave solutions of the nonlinear PDE (2.1).

3. Applications of the Method

In this section, we apply the extended mapping method to construct the exact solutions for the Boussinesq system and the coupled KdV equations, which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers.

Example 3.1 (the Boussinesq system). We start the Boussinesq system [32] in the following form: 𝑣𝑑=13𝑒π‘₯π‘₯π‘₯+83𝑒𝑒π‘₯,𝑒𝑑=𝑣π‘₯.(3.1) The traveling wave variable (2.2) permits us converting (3.1) into the following ODE: πœ”π‘£ξ…ž+13π‘˜2π‘’ξ…žξ…žξ…ž+83π‘’π‘’ξ…ž=0,πœ”π‘’ξ…ž+π‘£ξ…ž=0.(3.2) Integrating (3.2) with respect to πœ‰ once and taking the constant of integration to be zero, we obtain 1πœ”π‘£+3π‘˜2π‘’ξ…žξ…ž+43𝑒2=0,(3.3)πœ”π‘’+𝑣=0.(3.4) Suppose that the solutions of (3.3) and (3.4) can be expressed by 𝑒(πœ‰)=π‘Ž0+𝑛𝑖=1ξ€·π‘Žπ‘–π‘“π‘–(πœ‰)+π‘π‘–π‘“βˆ’π‘–ξ€Έ+(πœ‰)𝑛𝑖=2π‘π‘–π‘“π‘–βˆ’2(πœ‰)π‘“ξ…ž(πœ‰)+βˆ’π‘›ξ“π‘–=βˆ’1𝑑𝑖𝑓𝑖(πœ‰)π‘“ξ…ž(πœ‰),𝑣(πœ‰)=𝐴0+π‘šξ“π‘–=1𝐴𝑖𝑓𝑖(πœ‰)+π΅π‘–π‘“βˆ’π‘–ξ€Έ+(πœ‰)π‘šξ“π‘–=2πΏπ‘–π‘“π‘–βˆ’2(πœ‰)π‘“ξ…ž(πœ‰)+βˆ’π‘šξ“π‘–=βˆ’1𝐻𝑖𝑓𝑖(πœ‰)π‘“ξ…ž(πœ‰),(3.5) whereπ‘Ž0, π‘Žπ‘–, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝐴𝑖, 𝐡𝑖, 𝐿𝑖, and 𝐻𝑖 are constants to be determined later.

Considering the homogeneous balance between the highest order derivative π‘’ξ…žξ…ž and the nonlinear term 𝑒2 in (3.3), the order of 𝑒 and 𝑣 in (3.4), then we can obtain 𝑛=π‘š=2, hence the exact solutions of (3.5) can be rewritten as, 𝑒(πœ‰)=π‘Ž0+π‘Ž1𝑓(πœ‰)+𝑏11𝑓(πœ‰)+π‘Ž2𝑓2(πœ‰)+𝑏21𝑓2(πœ‰)+𝑐2π‘“ξ…ž(πœ‰)+𝑑1π‘“ξ…ž(πœ‰)𝑓(πœ‰)+𝑑2π‘“ξ…ž(πœ‰)𝑓2,(πœ‰)𝑣(πœ‰)=𝐴0+𝐴1𝑓(πœ‰)+𝐡11𝑓(πœ‰)+𝐴2𝑓2(πœ‰)+𝐡21𝑓2(πœ‰)+𝐿2π‘“ξ…ž(πœ‰)+𝐻1π‘“ξ…ž(πœ‰)𝑓(πœ‰)+𝐻2π‘“ξ…ž(πœ‰)𝑓2,(πœ‰)(3.6) where π‘Ž0, π‘Ž1, π‘Ž2, 𝑏1, 𝑏2, 𝑐2, 𝑑1, 𝑑2, 𝐴0, 𝐴1, 𝐡1, 𝐡2, 𝐿2, 𝐻1, and 𝐻2 are constants to be determined later. Substituting (3.6) with the condition (2.5) into (3.3) and (3.4) and collecting all terms with the same power of 𝑓𝑖(πœ‰)[π‘“ξ…ž(πœ‰)]𝑗, (𝑖=…,βˆ’2,βˆ’1,0,1,2,…;𝑗=0,1). Setting each coefficients of this polynomial to be zero, we get a system of algebraic equations which can be solved by Maple or Mathematica to get the following solutions.

Case 1. Consider π‘Ž0=π‘Ž1=π‘Ž2=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐴2=𝐡1=𝐿1=𝐻1=𝐻2𝐴=0,0=arbitraryconstant,𝑏2=βˆ’9πœ”2π‘Ÿ8π‘ž,𝐡2=9πœ”3π‘Ÿπœ”βˆš8π‘ž,π‘˜=Β±32βˆšπ‘ž.(3.7)

Case 2. Consider π‘Ž0=π‘Ž1=𝑏2=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐡2=𝐡1=𝐿1=𝐻1=𝐻2𝐴=0,0=arbitraryconstant,π‘Ž2=βˆ’9πœ”2𝑝8π‘ž,𝐴2=9πœ”3π‘πœ”βˆš8π‘ž,π‘˜=Β±32βˆšπ‘ž.(3.8)

Case 3. Consider π‘Ž0=π‘Ž1=π‘Ž2=𝑏1=𝑐2=𝑑1=𝐴2=𝐴1=𝐡1=𝐿1=𝐻1𝑏=0,2=βˆ’9πœ”2π‘Ÿ4π‘ž,𝑑2=βˆ“9πœ”2βˆšπ‘Ÿ4π‘ž,𝐡2=9πœ”3π‘Ÿ4π‘ž,𝐻2=Β±9πœ”3βˆšπ‘Ÿ,𝐴4π‘ž0πœ”βˆš=arbitraryconstant,π‘˜=3βˆšπ‘ž.(3.9)

Case 4. Consider π‘Ž0=π‘Ž1=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐡1=𝐿1=𝐻1=𝐻2π‘Ž=0,2=βˆ’9πœ”2𝑝8π‘ž,𝑏2=βˆ’9πœ”2π‘Ÿ8π‘ž,𝐴2=9πœ”3𝑝8π‘ž,𝐡2=9πœ”3π‘Ÿ,𝐴8π‘ž0πœ”βˆš=arbitraryconstant,π‘˜=Β±32βˆšπ‘ž.(3.10)

Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case 4 to illustrate the effectiveness of the extended mapping method.

Substituting (3.10) into (3.6) yields 𝑒(πœ‰)=βˆ’9πœ”2𝑝𝑓8π‘ž2(πœ‰)βˆ’9πœ”2π‘Ÿ18π‘žπ‘“2,(πœ‰)𝑣(πœ‰)=9πœ”3𝑝𝑓8π‘ž2(πœ‰)+9πœ”3π‘Ÿ18π‘žπ‘“2,(πœ‰)(3.11) where πœ”βˆšπœ‰=Β±32βˆšπ‘ž(π‘₯βˆ’πœ”π‘‘).(3.12)

According to Appendix A, we have the following families of exact solutions.

Family 1. Ifπ‘Ÿ=1,π‘ž=βˆ’(1+π‘š2),𝑝=π‘š2,𝑓(πœ‰)=𝑠𝑛(πœ‰), then we get 𝑒(πœ‰)=9πœ”28ξ€·1+π‘š2ξ€Έξ€Ίπ‘š2sn2(πœ‰)+ns2ξ€»,(πœ‰)𝑣(πœ‰)=βˆ’9πœ”38ξ€·1+π‘š2ξ€Έξ€Ίπ‘š2sn2(πœ‰)+ns2ξ€»,(πœ‰)(3.13) where πœ”βˆšπœ‰=Β±3√2𝑖1+π‘š2(π‘₯βˆ’πœ”π‘‘).(3.14)

Family 2. If π‘Ÿ=1βˆ’π‘š2, π‘ž=2π‘š2βˆ’1, 𝑝=βˆ’π‘š2, 𝑓(πœ‰)=cn(πœ‰), then we get 𝑒(πœ‰)=9πœ”28ξ€·2π‘š2ξ€Έξ‚ƒπ‘šβˆ’12cn2ξ€·(πœ‰)βˆ’1βˆ’π‘š2ξ€Έnc2ξ‚„,(πœ‰)𝑣(πœ‰)=βˆ’9πœ”3π‘š28ξ€·2π‘š2ξ€Έξ€Ίβˆ’1cn2ξ€·(πœ‰)βˆ’1βˆ’π‘š2ξ€Έnc2ξ€»,(πœ‰)(3.15) where πœ”βˆšπœ‰=Β±32√2π‘š2βˆ’1(π‘₯βˆ’πœ”π‘‘).(3.16)

Family 3. If π‘Ÿ=π‘š2βˆ’1, π‘ž=2βˆ’π‘š2, 𝑝=βˆ’1, 𝑓(πœ‰)=dn(πœ‰), then we get 𝑒(πœ‰)=9πœ”28ξ€·2βˆ’π‘š2ξ€Έξ€Ίdn2ξ€·π‘š(πœ‰)βˆ’2ξ€Έβˆ’1nd2ξ€»,(πœ‰)𝑣(πœ‰)=βˆ’9πœ”38ξ€·2βˆ’π‘š2ξ€Έξ€Ίdn2ξ€·π‘š(πœ‰)βˆ’2ξ€Έβˆ’1nd2ξ€»,(πœ‰)(3.17) where πœ”βˆšπœ‰=Β±32βˆšπ‘š2βˆ’1(π‘₯βˆ’πœ”π‘‘).(3.18)

Family 4. If π‘Ÿ=π‘š2, π‘ž=βˆ’(1+π‘š2), 𝑝=1, 𝑓(πœ‰)=dc(πœ‰), then we get 𝑒(πœ‰)=9πœ”28ξ€·1+π‘š2ξ€Έξ€Ίdc2(πœ‰)+π‘š2cd2ξ€»,(πœ‰)𝑣(πœ‰)=βˆ’9πœ”38ξ€·1+π‘š2ξ€Έξ€Ίdc2(πœ‰)+π‘š2cd2ξ€»,(πœ‰)(3.19) where πœ”βˆšπœ‰=Β±3√2𝑖1+π‘š2(π‘₯βˆ’πœ”π‘‘).(3.20)

Family 5. If π‘Ÿ=1, π‘ž=2βˆ’π‘š2, 𝑝=1βˆ’π‘š2, 𝑓(πœ‰)=sc(πœ‰), then we get 𝑒(πœ‰)=βˆ’9πœ”28ξ€·2βˆ’π‘š2ξ€Έξ€Ίξ€·1βˆ’π‘š2ξ€Έsc2(πœ‰)+cs2ξ€»,(πœ‰)𝑣(πœ‰)=9πœ”38ξ€·2βˆ’π‘š2ξ€Έξ€Ίξ€·1βˆ’π‘š2ξ€Έsc2(πœ‰)+cs2ξ€»,(πœ‰)(3.21) where πœ”βˆšπœ‰=Β±32√2βˆ’π‘š2(π‘₯βˆ’πœ”π‘‘).(3.22)

Family 6. If π‘Ÿ=1/4, π‘ž=(1/2)(1βˆ’2π‘š2), 𝑝=1/4, 𝑓(πœ‰)=ns(πœ‰)Β±cs(πœ‰), then we get 𝑒(πœ‰)=9πœ”2ξ€·1βˆ’2ns2ξ€Έ(πœ‰)8ξ€·1βˆ’2π‘š2ξ€Έ,𝑣(πœ‰)=βˆ’9πœ”3ξ€·1βˆ’2ns2ξ€Έ(πœ‰)8ξ€·1βˆ’2π‘š2ξ€Έ,(3.23) where πœ”βˆšπœ‰=Β±32(1/2)1βˆ’2π‘š2ξ€Έ(π‘₯βˆ’πœ”π‘‘).(3.24)

Family 7. If π‘Ÿ=(1/4)(1βˆ’π‘š2), π‘ž=(1/4)(1+π‘š2), 𝑝=(1/4)(1βˆ’π‘š2), 𝑓(πœ‰)=nc(πœ‰)Β±sc(πœ‰), then we get 𝑒(πœ‰)=βˆ’9πœ”2ξ€·1βˆ’π‘š2ξ€Έξ€·sc2(πœ‰)+nc2ξ€Έ(πœ‰)4ξ€·1+π‘š2ξ€Έ,𝑣(πœ‰)=9πœ”3ξ€·1βˆ’π‘š2ξ€Έξ€·sc2(πœ‰)+nc2(ξ€Έπœ‰)4ξ€·1+π‘š2ξ€Έ,(3.25) where πœ”βˆšπœ‰=Β±3√1+π‘š2(π‘₯βˆ’πœ”π‘‘).(3.26) Similarly, we can write down the other families of exact solutions of (3.1) which are omitted for convenience.

Example 3.2 (the coupled KdV equations). In this subsection, consider the coupled KdV equations [32]: 𝑒𝑑=𝑒π‘₯π‘₯π‘₯+6𝑒𝑒π‘₯+6𝑣𝑣π‘₯,𝑣𝑑=𝑣π‘₯π‘₯π‘₯+6𝑒𝑣π‘₯+6𝑣𝑒π‘₯.(3.27) Substituting (2.2) into (3.27) yields πœ”π‘’ξ…ž+π‘˜2π‘’ξ…žξ…žξ…ž+3(𝑒2+𝑣2)ξ…ž=0,πœ”π‘£ξ…ž+π‘˜2π‘£ξ…žξ…žξ…ž+6(𝑒𝑣)ξ…ž=0.(3.28) Integrating (3.2) with respect to πœ‰ once and taking the constant of integration to be zero, we obtain πœ”π‘’+π‘˜2π‘’ξ…žξ…žξ€·π‘’+32+𝑣2ξ€Έ=0,(3.29)πœ”π‘£+π‘˜2π‘£ξ…žξ…ž+6(𝑒𝑣)=0.(3.30) Suppose that the solutions of (3.27) can be expressed by 𝑒(πœ‰)=π‘Ž0+𝑛𝑖=1ξ€·π‘Žπ‘–π‘“π‘–(πœ‰)+π‘π‘–π‘“βˆ’π‘–ξ€Έ+(πœ‰)𝑛𝑖=2π‘π‘–π‘“π‘–βˆ’2(πœ‰)π‘“ξ…ž(πœ‰)+βˆ’π‘›ξ“π‘–=βˆ’1𝑑𝑖𝑓𝑖(πœ‰)π‘“ξ…ž(πœ‰),𝑣(πœ‰)=𝛼0+π‘šξ“π‘–=1𝛼𝑖𝑓𝑖(πœ‰)+π›½π‘–π‘“βˆ’π‘–ξ€Έ+(πœ‰)π‘šξ“π‘–=2π›Ύπ‘–π‘“π‘–βˆ’2(πœ‰)π‘“ξ…ž(πœ‰)+βˆ’π‘šξ“π‘–=βˆ’1𝑒𝑖𝑓𝑖(πœ‰)π‘“ξ…ž(πœ‰),(3.31) where π‘Ž0, π‘Žπ‘–, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝛼𝑖, 𝛽𝑖, 𝛾𝑖, and 𝑒𝑖 are constants to be determined later.
Balancing the order of π‘’ξ…žξ…ž and 𝑣2 in (3.29), the order of π‘£ξ…žξ…ž and 𝑒𝑣 in (3.30), then we can obtain 𝑛=π‘š=2, so (3.31) can be rewritten as 𝑒(πœ‰)=π‘Ž0+π‘Ž1𝑓(πœ‰)+𝑏11𝑓(πœ‰)+π‘Ž2𝑓2(πœ‰)+𝑏21𝑓2(πœ‰)+𝑐2π‘“ξ…ž(πœ‰)+𝑑1π‘“ξ…ž(πœ‰)𝑓(πœ‰)+𝑑2π‘“ξ…ž(πœ‰)𝑓2,(πœ‰)𝑣(πœ‰)=𝛼0+𝛼1𝑓(πœ‰)+𝛽11𝑓(πœ‰)+𝛼2𝑓2(πœ‰)+𝛽21𝑓2(πœ‰)+𝛾2π‘“ξ…ž(πœ‰)+𝑒1π‘“ξ…ž(πœ‰)𝑓(πœ‰)+𝑒2π‘“ξ…ž(πœ‰)𝑓2,(πœ‰)(3.32) where π‘Ž0, π‘Ž1, π‘Ž2, 𝑏1, 𝑏2, 𝑐2, 𝑑1, 𝑑2, 𝛼0, 𝛼1, 𝛽1, 𝛽2, 𝛾2, 𝑒1, and 𝑒2 are constants to be determined later. Substituting (3.31) with the condition (2.5) into (3.29) and (3.30) and collecting all terms with the same power of 𝑓𝑖(πœ‰)[π‘“ξ…ž(πœ‰)]𝑗, (𝑖=…,βˆ’2,βˆ’1,0,1,2,…;𝑗=0,1). Setting each coefficient of this polynomial to be zero, we get a system of algebraic equations which can be solved by Maple or Mathematica to get the following solutions.

Case 1. Consider π‘Ž1=π‘Ž2=𝑏1=𝑐2=𝑑1=𝑑2=𝛼1=𝛼2=𝛽1=𝑒1=𝑒2=𝛾2π‘Ž=0,0πœ”=βˆ’ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺβˆ’3π‘π‘Ÿ,𝑏2=βˆ’πœ”π‘Ÿ4βˆšπ‘ž2,π›Όβˆ’3π‘π‘Ÿ0πœ”=βˆ’ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺβˆ’3π‘π‘Ÿ,𝛽2=βˆ’πœ”π‘Ÿ4βˆšπ‘ž2βˆšβˆ’3π‘π‘Ÿ,π‘˜=Β±πœ”24βˆšπ‘ž2.βˆ’3π‘π‘Ÿ(3.33)

Case 2. Consider π‘Ž1=𝑏1=𝑏2=𝑐2=𝑑1=𝑑2=𝛼1=𝛽1=𝛽2=𝑒1=𝑒2=𝛾2π‘Ž=0,0πœ”=βˆ’ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺβˆ’3π‘π‘Ÿ,π‘Ž2=βˆ’πœ”π‘4βˆšπ‘ž2,π›Όβˆ’3π‘π‘Ÿ0=πœ”ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺβˆ’3π‘π‘Ÿ,𝛼2=πœ”π‘4βˆšπ‘ž2βˆšβˆ’3π‘π‘Ÿ,π‘˜=Β±πœ”24βˆšπ‘ž2.βˆ’3π‘π‘Ÿ(3.34)

Case 3. Consider π‘Ž1=𝑏1=𝑐2=𝑑1=𝑑2=𝛼1=𝛽2=𝑒1=𝑒2=𝛾2π‘Ž=0,0πœ”=βˆ’ξƒ©π‘ž123+βˆšπ‘ž2ξƒͺ+12π‘π‘Ÿ,π‘Ž2=βˆ’πœ”π‘4βˆšπ‘ž2+12π‘π‘Ÿ,𝑏2=βˆ’πœ”π‘Ÿ4βˆšπ‘ž2,𝛼+12π‘π‘Ÿ0πœ”=βˆ’ξƒ©π‘ž121βˆ’βˆšπ‘ž2ξƒͺ+12π‘π‘Ÿ,𝛼2=πœ”π‘4βˆšπ‘ž2+12π‘π‘Ÿ,𝛽2=πœ”π‘Ÿ4βˆšπ‘ž2,√+12π‘π‘Ÿπ‘˜=Β±πœ”24βˆšπ‘ž2.+12π‘π‘Ÿ(3.35)

Case 4. Consider π‘Ž1=𝑏1=𝑏2=𝑑1=𝑑2=𝛼1=𝛽1=𝛽2=𝑒1=𝑒2π‘Ž=0,0πœ”=βˆ’ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺ+12π‘π‘Ÿ,π‘Ž2=βˆ’πœ”π‘2βˆšπ‘ž2+12π‘π‘Ÿ,𝑐2πœ”βˆš=βˆ’π‘2βˆšπ‘ž2,𝛼+12π‘π‘Ÿ0=πœ”ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺ+12π‘π‘Ÿ,𝛼2=πœ”π‘2βˆšπ‘ž2+12π‘π‘Ÿ,𝛾2=πœ”βˆšπ‘2βˆšπ‘ž2,√+12π‘π‘Ÿπ‘˜=Β±πœ”4βˆšπ‘ž2.+12π‘π‘Ÿ(3.36) Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case 4 to illustrate the effectiveness of the extended mapping method.

Substituting (3.36) into (3.32) yields πœ”π‘’(πœ‰)=βˆ’ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺβˆ’+12π‘π‘Ÿπœ”π‘2βˆšπ‘ž2𝑓+12π‘π‘Ÿ2πœ”βˆš(πœ‰)βˆ’π‘2βˆšπ‘ž2𝑓+12π‘π‘Ÿξ…žπœ”(πœ‰),𝑣(πœ‰)=ξƒ©π‘ž121+βˆšπ‘ž2ξƒͺ++12π‘π‘Ÿπœ”π‘2βˆšπ‘ž2𝑓+12π‘π‘Ÿ2πœ”βˆš(πœ‰)+𝑝2βˆšπ‘ž2𝑓+12π‘π‘Ÿξ…ž(πœ‰),(3.37) where βˆšπœ‰=Β±πœ”4βˆšπ‘ž2+12π‘π‘Ÿ(π‘₯βˆ’πœ”π‘‘).(3.38)

According to Appendix A, we have the following families of exact solutions.

Family 1. If π‘Ÿ=1, π‘ž=2π‘š2βˆ’1, 𝑝=π‘š2(π‘š2βˆ’1), 𝑓(πœ‰)=sd(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺβˆ’+1πœ”π‘š2ξ€·π‘š2ξ€Έβˆ’1sd2(πœ‰)2√16π‘š4βˆ’16π‘š2βˆ’βˆš+1πœ”π‘šπ‘š2βˆ’1nd(πœ‰)cd(πœ‰)2√16π‘š4βˆ’16π‘š2,πœ”+1𝑣(πœ‰)=121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺ++1πœ”π‘š2ξ€·π‘š2ξ€Έβˆ’1sd2(πœ‰)2√16π‘š4βˆ’16π‘š2+√+1πœ”π‘šπ‘š2βˆ’1nd(πœ‰)cd(πœ‰)2√16π‘š4βˆ’16π‘š2,+1(3.39) where βˆšπœ‰=Β±πœ”4√16π‘š4βˆ’16π‘š2+1(π‘₯βˆ’πœ”π‘‘).(3.40)

Family 2. If π‘Ÿ=π‘š2(π‘š2βˆ’1), π‘ž=2π‘š2βˆ’1,𝑝=1, 𝑓(πœ‰)=𝑑𝑠(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺβˆ’+1πœ”ds2(πœ‰)2√16π‘š4βˆ’16π‘š2++1πœ”cs(πœ‰)ns(πœ‰)2√16π‘š4βˆ’16π‘š2,πœ”+1𝑣(πœ‰)=121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺ++1πœ”ds2(πœ‰)2√16π‘š4βˆ’16π‘š2βˆ’+1πœ”cs(πœ‰)ns(πœ‰)2√16π‘š4βˆ’16π‘š2,+1(3.41) where βˆšπœ‰=Β±πœ”4√16π‘š4βˆ’16π‘š2+1(π‘₯βˆ’πœ”π‘‘).(3.42)

Family 3. If π‘Ÿ=π‘š2/4, π‘ž=(1/2)(π‘š2βˆ’2), 𝑝=π‘š2/4, 𝑓(πœ‰)=sn(πœ‰)±𝑖cn(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©π‘š121+2βˆ’22βˆšπ‘š4βˆ’π‘š2ξƒͺβˆ’+1πœ”π‘š2(sn(πœ‰)±𝑖cn(πœ‰))28βˆšπ‘š4βˆ’π‘š2βˆ’+1πœ”π‘š(𝑐𝑛(πœ‰)dn(πœ‰)βˆ“π‘–sn(πœ‰)dn(πœ‰))4βˆšπ‘š4βˆ’π‘š2,πœ”+1𝑣(πœ‰)=ξƒ©π‘š121+2βˆ’22βˆšπ‘š4βˆ’π‘š2ξƒͺ++1πœ”π‘š2(sn(πœ‰)±𝑖cn(πœ‰))28βˆšπ‘š4βˆ’π‘š2++1πœ”π‘š(cn(πœ‰)dn(πœ‰)βˆ“π‘–sn(πœ‰)dn(πœ‰))4βˆšπ‘š4βˆ’π‘š2,+1(3.43) where βˆšπœ‰=Β±πœ”4βˆšπ‘š4βˆ’π‘š2+1(π‘₯βˆ’πœ”π‘‘).(3.44)

Family 4. If π‘Ÿ=1,π‘ž=βˆ’(1+π‘š2), 𝑝=π‘š2, 𝑓(πœ‰)=sn(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121βˆ’1+π‘š2βˆšπ‘š4+14π‘š2ξƒͺβˆ’+1πœ”π‘š2sn2(πœ‰)2βˆšπ‘š4+14π‘š2βˆ’+1πœ”π‘šcn(πœ‰)dn(πœ‰)2βˆšπ‘š4+14π‘š2,πœ”+1𝑣(πœ‰)=121βˆ’1+π‘š2βˆšπ‘š4+14π‘š2ξƒͺ++1πœ”π‘š2sn2(πœ‰)2βˆšπ‘š4+14π‘š2++1πœ”π‘šcn(πœ‰)dn(πœ‰)2βˆšπ‘š4+14π‘š2,+1(3.45) where βˆšπœ‰=Β±πœ”4βˆšπ‘š4+14π‘š2+1(π‘₯βˆ’πœ”π‘‘).(3.46)

Family 5. If π‘Ÿ=1βˆ’π‘š2, π‘ž=2π‘š2βˆ’1, 𝑝=βˆ’π‘š2, 𝑓(πœ‰)=cn(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺ++1πœ”π‘š2cn2(πœ‰)2√16π‘š4βˆ’16π‘š2++1π‘–πœ”π‘šsn(πœ‰)dn(πœ‰)2√16π‘š4βˆ’16π‘š2,πœ”+1𝑣(πœ‰)=121+2π‘š2βˆ’1√16π‘š4βˆ’16π‘š2ξƒͺβˆ’+1πœ”π‘š2cn2(πœ‰)2√16π‘š4βˆ’16π‘š2βˆ’+1π‘–πœ”π‘šsn(πœ‰)dn(πœ‰)2√16π‘š4βˆ’16π‘š2,+1(3.47) where βˆšπœ‰=Β±πœ”4√16π‘š4βˆ’16π‘š2+1(π‘₯βˆ’πœ”π‘‘).(3.48)

Family 6. If π‘Ÿ=1βˆ’π‘š2, π‘ž=2βˆ’π‘š2, 𝑝=1, 𝑓(πœ‰)=cs(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121+2βˆ’π‘š2βˆšπ‘š4βˆ’16π‘š2ξƒͺβˆ’+16πœ”cs2(πœ‰)2βˆšπ‘š4βˆ’16π‘š2++16πœ”π‘›π‘ (πœ‰)ds(πœ‰)2βˆšπ‘š4βˆ’16π‘š2,πœ”+16𝑣(πœ‰)=121+2βˆ’π‘š2βˆšπ‘š4βˆ’16π‘š2ξƒͺ++16πœ”cs2(πœ‰)2βˆšπ‘š4βˆ’16π‘š2βˆ’+16πœ”ns(πœ‰)ds(πœ‰)2βˆšπ‘š4βˆ’16π‘š2,+16(3.49) where βˆšπœ‰=Β±πœ”4βˆšπ‘š4βˆ’16π‘š2+16(π‘₯βˆ’πœ”π‘‘).(3.50)

Family 7. If π‘Ÿ=βˆ’1, π‘ž=2βˆ’π‘š2,𝑝=π‘š2βˆ’1, 𝑓(πœ‰)=nd(πœ‰), then we get πœ”π‘’(πœ‰)=βˆ’ξƒ©121+2βˆ’π‘š2βˆšπ‘š4βˆ’16π‘š2ξƒͺβˆ’πœ”ξ€·π‘š+162ξ€Έβˆ’1nd2(πœ‰)2βˆšπ‘š4βˆ’16π‘š2βˆ’+16πœ”π‘š2βˆšπ‘š2βˆ’1sd(πœ‰)cd(πœ‰)2βˆšπ‘š4βˆ’16π‘š2,πœ”+16𝑣(πœ‰)=121+2βˆ’π‘š2βˆšπ‘š4βˆ’16π‘š2ξƒͺ+πœ”ξ€·π‘š+162ξ€Έβˆ’1nd2(πœ‰)2βˆšπ‘š4βˆ’16π‘š2++16πœ”π‘š2βˆšπ‘š2βˆ’1sd(πœ‰)cd(πœ‰)2βˆšπ‘š4βˆ’16π‘š2,+16(3.51) where βˆšπœ‰=Β±πœ”4βˆšπ‘š4βˆ’16π‘š2+16(π‘₯βˆ’πœ”π‘‘).(3.52)

4. Conclusion

The main objective of this paper is that we have found new exact solutions for the Boussinesq system and the coupled KdV equations by using the extended mapping method with the auxiliary equation method. Also, we conclude according to Appendix B that our results in terms of Jacobi elliptic functions generate into hyperbolic functions when π‘šβ†’1 and generate into trigonometric functions when π‘šβ†’0. This method provides a powerful mathematical tool to obtain more general exact solutions of a great many nonlinear PDEs in mathematical physics.

Appendices

A. The Jacobi Elliptic Functions

The general solutions to the Jacobi elliptic equation (2.3) and its derivatives [31] are listed in Table 1, where 0<π‘š<1 is the modulus of the Jacobi elliptic functions and βˆšπ‘–=βˆ’1.


𝑝 π‘ž π‘Ÿ 𝑓 ( πœ‰ ) 𝑓 ξ…ž ( πœ‰ )

π‘š 2 βˆ’ ( 1 + π‘š 2 ) 1 s n ( πœ‰ ) c n ( πœ‰ ) d n ( πœ‰ )
βˆ’ π‘š 2 2 π‘š 2 βˆ’ 1 1 βˆ’ π‘š 2 c n ( πœ‰ ) βˆ’ s n ( πœ‰ ) d n ( πœ‰ )
βˆ’1 2 βˆ’ π‘š 2 π‘š 2 βˆ’ 1 d n ( πœ‰ ) βˆ’ π‘š 2 s n ( πœ‰ ) c n ( πœ‰ )
1 βˆ’ ( 1 + π‘š 2 ) π‘š 2 n s ( πœ‰ ) βˆ’ d s ( πœ‰ ) c s ( πœ‰ )
π‘š 2 βˆ’ 1 2 βˆ’ π‘š 2 βˆ’1 n d ( πœ‰ ) π‘š 2 s d ( πœ‰ ) c d ( πœ‰ )
1 2 βˆ’ π‘š 2 1 βˆ’ π‘š 2 c s ( πœ‰ ) βˆ’ n s ( πœ‰ ) d s ( πœ‰ )
1 βˆ’ π‘š 2 2 βˆ’ π‘š 2 1 s c ( πœ‰ ) n c ( πœ‰ ) d c ( πœ‰ )
π‘š 2 ( π‘š 2 βˆ’ 1 ) 2 π‘š 2 βˆ’ 1 1 s d ( πœ‰ ) n d ( πœ‰ ) c d ( πœ‰ )
1 2 π‘š 2 βˆ’ 1 π‘š 2 ( π‘š 2 βˆ’ 1 ) d s ( πœ‰ ) βˆ’ c s ( πœ‰ ) n s ( πœ‰ )
1 4 1 2 ( 1 βˆ’ 2 π‘š 2 ) 1 4 n s ( πœ‰ ) Β± c s ( πœ‰ ) βˆ’ d s ( πœ‰ ) c s ( πœ‰ ) βˆ“ n s ( πœ‰ ) d s ( πœ‰ )
1 4 ( 1 βˆ’ π‘š 2 ) 1 4 ( 1 + π‘š 2 ) 1 4 ( 1 βˆ’ π‘š 2 ) n c ( πœ‰ ) Β± s c ( πœ‰ ) s c ( πœ‰ ) d c ( πœ‰ ) Β± n c ( πœ‰ ) d c ( πœ‰ )
π‘š 2 4 1 2 ( π‘š 2 βˆ’ 2 ) π‘š 2 4 s n ( πœ‰ ) Β± 𝑖 c n ( πœ‰ ) c n ( πœ‰ ) d n ( πœ‰ ) βˆ“ 𝑖 s n ( πœ‰ ) d n ( πœ‰ )

B. Hyperbolic Functions

The Jacobi elliptic functions sn(πœ‰), cn(πœ‰), dn(πœ‰), ns(πœ‰), cs(πœ‰), ds(πœ‰), sc(πœ‰), sd(πœ‰) generate into hyperbolic functions when π‘šβ†’1 as in Table 2.


s n ( πœ‰ ) β†’ t a n h ( πœ‰ ) c n ( πœ‰ ) β†’ s e c h ( πœ‰ ) d n ( πœ‰ ) β†’ s e c h ( πœ‰ ) n s ( πœ‰ ) β†’ c o t h ( πœ‰ )
c s ( πœ‰ ) β†’ c s c h ( πœ‰ ) d s ( πœ‰ ) β†’ c s c h ( πœ‰ ) s c ( πœ‰ ) β†’ s i n h ( πœ‰ ) s d ( πœ‰ ) β†’ s i n h ( πœ‰ )

C. Relations between the Jacobi Elliptic Functions

See Table 3.


1 n c ( πœ‰ ) = c n ( πœ‰ ) 1 n d ( πœ‰ ) = d n ( πœ‰ ) c d ( πœ‰ ) = c n ( πœ‰ ) d n ( πœ‰ ) d c ( πœ‰ ) = d n ( πœ‰ ) c n ( πœ‰ )
c s ( πœ‰ ) = c n ( πœ‰ ) s n ( πœ‰ ) s c ( πœ‰ ) = s n ( πœ‰ ) c n ( πœ‰ ) s d ( πœ‰ ) = s n ( πœ‰ ) d n ( πœ‰ ) d s ( πœ‰ ) = d n ( πœ‰ ) s n ( πœ‰ )

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Copyright © 2012 J. F. Alzaidy. 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.


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