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Journal of Applied Mathematics
Volume 2012, Article ID 597983, 14 pages
http://dx.doi.org/10.1155/2012/597983
Research Article

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

Mathematics Department, Faculty of Science, Taif University, Saudi Arabia

Received 2 April 2012; Revised 15 July 2012; Accepted 31 July 2012

Academic Editor: Renat Zhdanov

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.

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], Bäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], variable separation approach [5], various tanh methods [69], homogeneous balance method [10], similarity reductions method [11, 12], (𝐺/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 Painlevé 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: 𝐹𝑢,𝑢𝑡,𝑢𝑥,𝑢𝑡𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,=0,(2.1) where 𝑢=𝑢(𝑥,𝑡) is an unknown function, 𝐹 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 we give the main steps of a deformation method.

Step 1. The traveling wave variable 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑘(𝑥𝜔𝑡),(2.2) where 𝑘 and 𝜔 are the wave number and the wave speed, respectively. Under the transformation (2.2), (2.1) becomes an ordinary differential equation (ODE) as 𝑃𝑢,𝑢,𝑢,𝑢,=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: 𝑢(𝜉)=𝑎0+𝑛𝑖=1𝑎𝑖𝑓𝑖(𝜉)+𝑏𝑖𝑓𝑖+(𝜉)𝑛𝑖=2𝑐𝑖𝑓𝑖2(𝜉)𝑓(𝜉)+𝑛𝑖=1𝑑𝑖𝑓𝑖(𝜉)𝑓(𝜉),(2.4) where 𝑎0, 𝑎𝑖, 𝑏𝑖, 𝑐𝑖, and 𝑑𝑖 are constants to be determined later, while 𝑓(𝜉) satisfies the nonlinear ODE: 𝑓(𝜉)2=𝑝𝑓4(𝜉)+𝑞𝑓2(𝜉)+𝑟,(2.5) where 𝑝, 𝑞, and 𝑟 are constants.

Step 4. The positive integer “𝑛” 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(𝜉)[𝑓(𝜉)]𝑗, (𝑖=,2,1,0,1,2,;𝑗=0,1). Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for 𝑎0, 𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝜔, and 𝑘.

Step 6. Solving the algebraic equations by use of Maple or Mathematica, we have 𝑎0, 𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, and 𝑘 expressed by 𝑝, 𝑞, 𝑟.

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: 𝑣𝑡=13𝑢𝑥𝑥𝑥+83𝑢𝑢𝑥,𝑢𝑡=𝑣𝑥.(3.1) The traveling wave variable (2.2) permits us converting (3.1) into the following ODE: 𝜔𝑣+13𝑘2𝑢+83𝑢𝑢=0,𝜔𝑢+𝑣=0.(3.2) Integrating (3.2) with respect to 𝜉 once and taking the constant of integration to be zero, we obtain 1𝜔𝑣+3𝑘2𝑢+43𝑢2=0,(3.3)𝜔𝑢+𝑣=0.(3.4) Suppose that the solutions of (3.3) and (3.4) can be expressed by 𝑢(𝜉)=𝑎0+𝑛𝑖=1𝑎𝑖𝑓𝑖(𝜉)+𝑏𝑖𝑓𝑖+(𝜉)𝑛𝑖=2𝑐𝑖𝑓𝑖2(𝜉)𝑓(𝜉)+𝑛𝑖=1𝑑𝑖𝑓𝑖(𝜉)𝑓(𝜉),𝑣(𝜉)=𝐴0+𝑚𝑖=1𝐴𝑖𝑓𝑖(𝜉)+𝐵𝑖𝑓𝑖+(𝜉)𝑚𝑖=2𝐿𝑖𝑓𝑖2(𝜉)𝑓(𝜉)+𝑚𝑖=1𝐻𝑖𝑓𝑖(𝜉)𝑓(𝜉),(3.5) where𝑎0, 𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝐴𝑖, 𝐵𝑖, 𝐿𝑖, and 𝐻𝑖 are constants to be determined later.

Considering the homogeneous balance between the highest order derivative 𝑢 and the nonlinear term 𝑢2 in (3.3), the order of 𝑢 and 𝑣 in (3.4), then we can obtain 𝑛=𝑚=2, hence the exact solutions of (3.5) can be rewritten as, 𝑢(𝜉)=𝑎0+𝑎1𝑓(𝜉)+𝑏11𝑓(𝜉)+𝑎2𝑓2(𝜉)+𝑏21𝑓2(𝜉)+𝑐2𝑓(𝜉)+𝑑1𝑓(𝜉)𝑓(𝜉)+𝑑2𝑓(𝜉)𝑓2,(𝜉)𝑣(𝜉)=𝐴0+𝐴1𝑓(𝜉)+𝐵11𝑓(𝜉)+𝐴2𝑓2(𝜉)+𝐵21𝑓2(𝜉)+𝐿2𝑓(𝜉)+𝐻1𝑓(𝜉)𝑓(𝜉)+𝐻2𝑓(𝜉)𝑓2,(𝜉)(3.6) where 𝑎0, 𝑎1, 𝑎2, 𝑏1, 𝑏2, 𝑐2, 𝑑1, 𝑑2, 𝐴0, 𝐴1, 𝐵1, 𝐵2, 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(𝜉)[𝑓(𝜉)]𝑗, (𝑖=,2,1,0,1,2,;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 𝑎0=𝑎1=𝑎2=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐴2=𝐵1=𝐿1=𝐻1=𝐻2𝐴=0,0=arbitraryconstant,𝑏2=9𝜔2𝑟8𝑞,𝐵2=9𝜔3𝑟𝜔8𝑞,𝑘=±32𝑞.(3.7)

Case 2. Consider 𝑎0=𝑎1=𝑏2=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐵2=𝐵1=𝐿1=𝐻1=𝐻2𝐴=0,0=arbitraryconstant,𝑎2=9𝜔2𝑝8𝑞,𝐴2=9𝜔3𝑝𝜔8𝑞,𝑘=±32𝑞.(3.8)

Case 3. Consider 𝑎0=𝑎1=𝑎2=𝑏1=𝑐2=𝑑1=𝐴2=𝐴1=𝐵1=𝐿1=𝐻1𝑏=0,2=9𝜔2𝑟4𝑞,𝑑2=9𝜔2𝑟4𝑞,𝐵2=9𝜔3𝑟4𝑞,𝐻2=±9𝜔3𝑟,𝐴4𝑞0𝜔=arbitraryconstant,𝑘=3𝑞.(3.9)

Case 4. Consider 𝑎0=𝑎1=𝑏1=𝑐2=𝑑1=𝑑2=𝐴1=𝐵1=𝐿1=𝐻1=𝐻2𝑎=0,2=9𝜔2𝑝8𝑞,𝑏2=9𝜔2𝑟8𝑞,𝐴2=9𝜔3𝑝8𝑞,𝐵2=9𝜔3𝑟,𝐴8𝑞0𝜔=arbitraryconstant,𝑘=±32𝑞.(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 𝑢(𝜉)=9𝜔2𝑝𝑓8𝑞2(𝜉)9𝜔2𝑟18𝑞𝑓2,(𝜉)𝑣(𝜉)=9𝜔3𝑝𝑓8𝑞2(𝜉)+9𝜔3𝑟18𝑞𝑓2,(𝜉)(3.11) where 𝜔𝜉=±32𝑞(𝑥𝜔𝑡).(3.12)

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

Family 1. If𝑟=1,𝑞=(1+𝑚2),𝑝=𝑚2,𝑓(𝜉)=𝑠𝑛(𝜉), then we get 𝑢(𝜉)=9𝜔281+𝑚2𝑚2sn2(𝜉)+ns2,(𝜉)𝑣(𝜉)=9𝜔381+𝑚2𝑚2sn2(𝜉)+ns2,(𝜉)(3.13) where 𝜔𝜉=±32𝑖1+𝑚2(𝑥𝜔𝑡).(3.14)

Family 2. If 𝑟=1𝑚2, 𝑞=2𝑚21, 𝑝=𝑚2, 𝑓(𝜉)=cn(𝜉), then we get 𝑢(𝜉)=9𝜔282𝑚2𝑚12cn2(𝜉)1𝑚2nc2,(𝜉)𝑣(𝜉)=9𝜔3𝑚282𝑚21cn2(𝜉)1𝑚2nc2,(𝜉)(3.15) where 𝜔𝜉=±322𝑚21(𝑥𝜔𝑡).(3.16)

Family 3. If 𝑟=𝑚21, 𝑞=2𝑚2, 𝑝=1, 𝑓(𝜉)=dn(𝜉), then we get 𝑢(𝜉)=9𝜔282𝑚2dn2𝑚(𝜉)21nd2,(𝜉)𝑣(𝜉)=9𝜔382𝑚2dn2𝑚(𝜉)21nd2,(𝜉)(3.17) where 𝜔𝜉=±32𝑚21(𝑥𝜔𝑡).(3.18)

Family 4. If 𝑟=𝑚2, 𝑞=(1+𝑚2), 𝑝=1, 𝑓(𝜉)=dc(𝜉), then we get 𝑢(𝜉)=9𝜔281+𝑚2dc2(𝜉)+𝑚2cd2,(𝜉)𝑣(𝜉)=9𝜔381+𝑚2dc2(𝜉)+𝑚2cd2,(𝜉)(3.19) where 𝜔𝜉=±32𝑖1+𝑚2(𝑥𝜔𝑡).(3.20)

Family 5. If 𝑟=1, 𝑞=2𝑚2, 𝑝=1𝑚2, 𝑓(𝜉)=sc(𝜉), then we get 𝑢(𝜉)=9𝜔282𝑚21𝑚2sc2(𝜉)+cs2,(𝜉)𝑣(𝜉)=9𝜔382𝑚21𝑚2sc2(𝜉)+cs2,(𝜉)(3.21) where 𝜔𝜉=±322𝑚2(𝑥𝜔𝑡).(3.22)

Family 6. If 𝑟=1/4, 𝑞=(1/2)(12𝑚2), 𝑝=1/4, 𝑓(𝜉)=ns(𝜉)±cs(𝜉), then we get 𝑢(𝜉)=9𝜔212ns2(𝜉)812𝑚2,𝑣(𝜉)=9𝜔312ns2(𝜉)812𝑚2,(3.23) where 𝜔𝜉=±32(1/2)12𝑚2(𝑥𝜔𝑡).(3.24)

Family 7. If 𝑟=(1/4)(1𝑚2), 𝑞=(1/4)(1+𝑚2), 𝑝=(1/4)(1𝑚2), 𝑓(𝜉)=nc(𝜉)±sc(𝜉), then we get 𝑢(𝜉)=9𝜔21𝑚2sc2(𝜉)+nc2(𝜉)41+𝑚2,𝑣(𝜉)=9𝜔31𝑚2sc2(𝜉)+nc2(𝜉)41+𝑚2,(3.25) where 𝜔𝜉=±31+𝑚2(𝑥𝜔𝑡).(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]: 𝑢𝑡=𝑢𝑥𝑥𝑥+6𝑢𝑢𝑥+6𝑣𝑣𝑥,𝑣𝑡=𝑣𝑥𝑥𝑥+6𝑢𝑣𝑥+6𝑣𝑢𝑥.(3.27) Substituting (2.2) into (3.27) yields 𝜔𝑢+𝑘2𝑢+3(𝑢2+𝑣2)=0,𝜔𝑣+𝑘2𝑣+6(𝑢𝑣)=0.(3.28) Integrating (3.2) with respect to 𝜉 once and taking the constant of integration to be zero, we obtain 𝜔𝑢+𝑘2𝑢𝑢+32+𝑣2=0,(3.29)𝜔𝑣+𝑘2𝑣+6(𝑢𝑣)=0.(3.30) Suppose that the solutions of (3.27) can be expressed by 𝑢(𝜉)=𝑎0+𝑛𝑖=1𝑎𝑖𝑓𝑖(𝜉)+𝑏𝑖𝑓𝑖+(𝜉)𝑛𝑖=2𝑐𝑖𝑓𝑖2(𝜉)𝑓(𝜉)+𝑛𝑖=1𝑑𝑖𝑓𝑖(𝜉)𝑓(𝜉),𝑣(𝜉)=𝛼0+𝑚𝑖=1𝛼𝑖𝑓𝑖(𝜉)+𝛽𝑖𝑓𝑖+(𝜉)𝑚𝑖=2𝛾𝑖𝑓𝑖2(𝜉)𝑓(𝜉)+𝑚𝑖=1𝑒𝑖𝑓𝑖(𝜉)𝑓(𝜉),(3.31) where 𝑎0, 𝑎𝑖, 𝑏𝑖, 𝑐𝑖, 𝑑𝑖, 𝛼𝑖, 𝛽𝑖, 𝛾𝑖, and 𝑒𝑖 are constants to be determined later.
Balancing the order of 𝑢 and 𝑣2 in (3.29), the order of 𝑣 and 𝑢𝑣 in (3.30), then we can obtain 𝑛=𝑚=2, so (3.31) can be rewritten as 𝑢(𝜉)=𝑎0+𝑎1𝑓(𝜉)+𝑏11𝑓(𝜉)+𝑎2𝑓2(𝜉)+𝑏21𝑓2(𝜉)+𝑐2𝑓(𝜉)+𝑑1𝑓(𝜉)𝑓(𝜉)+𝑑2𝑓(𝜉)𝑓2,(𝜉)𝑣(𝜉)=𝛼0+𝛼1𝑓(𝜉)+𝛽11𝑓(𝜉)+𝛼2𝑓2(𝜉)+𝛽21𝑓2(𝜉)+𝛾2𝑓(𝜉)+𝑒1𝑓(𝜉)𝑓(𝜉)+𝑒2𝑓(𝜉)𝑓2,(𝜉)(3.32) where 𝑎0, 𝑎1, 𝑎2, 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(𝜉)[𝑓(𝜉)]𝑗, (𝑖=,2,1,0,1,2,;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 𝑎1=𝑎2=𝑏1=𝑐2=𝑑1=𝑑2=𝛼1=𝛼2=𝛽1=𝑒1=𝑒2=𝛾2𝑎=0,0𝜔=𝑞121+𝑞23𝑝𝑟,𝑏2=𝜔𝑟4𝑞2,𝛼3𝑝𝑟0𝜔=𝑞121+𝑞23𝑝𝑟,𝛽2=𝜔𝑟4𝑞23𝑝𝑟,𝑘=±𝜔24𝑞2.3𝑝𝑟(3.33)

Case 2. Consider 𝑎1=𝑏1=𝑏2=𝑐2=𝑑1=𝑑2=𝛼1=𝛽1=𝛽2=𝑒1=𝑒2=𝛾2𝑎=0,0𝜔=𝑞121+𝑞23𝑝𝑟,𝑎2=𝜔𝑝4𝑞2,𝛼3𝑝𝑟0=𝜔𝑞121+𝑞23𝑝𝑟,𝛼2=𝜔𝑝4𝑞23𝑝𝑟,𝑘=±𝜔24𝑞2.3𝑝𝑟(3.34)

Case 3. Consider 𝑎1=𝑏1=𝑐2=𝑑1=𝑑2=𝛼1=𝛽2=𝑒1=𝑒2=𝛾2𝑎=0,0𝜔=𝑞123+𝑞2+12𝑝𝑟,𝑎2=𝜔𝑝4𝑞2+12𝑝𝑟,𝑏2=𝜔𝑟4𝑞2,𝛼+12𝑝𝑟0𝜔=𝑞121𝑞2+12𝑝𝑟,𝛼2=𝜔𝑝4𝑞2+12𝑝𝑟,𝛽2=𝜔𝑟4𝑞2,+12𝑝𝑟𝑘=±𝜔24𝑞2.+12𝑝𝑟(3.35)

Case 4. Consider 𝑎1=𝑏1=𝑏2=𝑑1=𝑑2=𝛼1=𝛽1=𝛽2=𝑒1=𝑒2𝑎=0,0𝜔=𝑞121+𝑞2+12𝑝𝑟,𝑎2=𝜔𝑝2𝑞2+12𝑝𝑟,𝑐2𝜔=𝑝2𝑞2,𝛼+12𝑝𝑟0=𝜔𝑞121+𝑞2+12𝑝𝑟,𝛼2=𝜔𝑝2𝑞2+12𝑝𝑟,𝛾2=𝜔𝑝2𝑞2,+12𝑝𝑟𝑘=±𝜔4𝑞2.+12𝑝𝑟(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 𝜔𝑢(𝜉)=𝑞121+𝑞2+12𝑝𝑟𝜔𝑝2𝑞2𝑓+12𝑝𝑟2𝜔(𝜉)𝑝2𝑞2𝑓+12𝑝𝑟𝜔(𝜉),𝑣(𝜉)=𝑞121+𝑞2++12𝑝𝑟𝜔𝑝2𝑞2𝑓+12𝑝𝑟2𝜔(𝜉)+𝑝2𝑞2𝑓+12𝑝𝑟(𝜉),(3.37) where 𝜉=±𝜔4𝑞2+12𝑝𝑟(𝑥𝜔𝑡).(3.38)

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

Family 1. If 𝑟=1, 𝑞=2𝑚21, 𝑝=𝑚2(𝑚21), 𝑓(𝜉)=sd(𝜉), then we get 𝜔𝑢(𝜉)=121+2𝑚2116𝑚416𝑚2+1𝜔𝑚2𝑚21sd2(𝜉)216𝑚416𝑚2+1𝜔𝑚𝑚21nd(𝜉)cd(𝜉)216𝑚416𝑚2,𝜔+1𝑣(𝜉)=121+2𝑚2116𝑚416𝑚2++1𝜔𝑚2𝑚21sd2(𝜉)216𝑚416𝑚2++1𝜔𝑚𝑚21nd(𝜉)cd(𝜉)216𝑚416𝑚2,+1(3.39) where 𝜉=±𝜔416𝑚416𝑚2+1(𝑥𝜔𝑡).(3.40)

Family 2. If 𝑟=𝑚2(𝑚21), 𝑞=2𝑚21,𝑝=1, 𝑓(𝜉)=𝑑𝑠(𝜉), then we get 𝜔𝑢(𝜉)=121+2𝑚2116𝑚416𝑚2+1𝜔ds2(𝜉)216𝑚416𝑚2++1𝜔cs(𝜉)ns(𝜉)216𝑚416𝑚2,𝜔+1𝑣(𝜉)=121+2𝑚2116𝑚416𝑚2++1𝜔ds2(𝜉)216𝑚416𝑚2+1𝜔cs(𝜉)ns(𝜉)216𝑚416𝑚2,+1(3.41) where 𝜉=±𝜔416𝑚416𝑚2+1(𝑥𝜔𝑡).(3.42)

Family 3. If 𝑟=𝑚2/4, 𝑞=(1/2)(𝑚22), 𝑝=𝑚2/4, 𝑓(𝜉)=sn(𝜉)±𝑖cn(𝜉), then we get 𝜔𝑢(𝜉)=𝑚121+222𝑚4𝑚2+1𝜔𝑚2(sn(𝜉)±𝑖cn(𝜉))28𝑚4𝑚2+1𝜔𝑚(𝑐𝑛(𝜉)dn(𝜉)𝑖sn(𝜉)dn(𝜉))4𝑚4𝑚2,𝜔+1𝑣(𝜉)=𝑚121+222𝑚4𝑚2++1𝜔𝑚2(sn(𝜉)±𝑖cn(𝜉))28𝑚4𝑚2++1𝜔𝑚(cn(𝜉)dn(𝜉)𝑖sn(𝜉)dn(𝜉))4𝑚4𝑚2,+1(3.43) where 𝜉=±𝜔4𝑚4𝑚2+1(𝑥𝜔𝑡).(3.44)

Family 4. If 𝑟=1,𝑞=(1+𝑚2), 𝑝=𝑚2, 𝑓(𝜉)=sn(𝜉), then we get 𝜔𝑢(𝜉)=1211+𝑚2𝑚4+14𝑚2+1𝜔𝑚2sn2(𝜉)2𝑚4+14𝑚2+1𝜔𝑚cn(𝜉)dn(𝜉)2𝑚4+14𝑚2,𝜔+1𝑣(𝜉)=1211+𝑚2𝑚4+14𝑚2++1𝜔𝑚2sn2(𝜉)2𝑚4+14𝑚2++1𝜔𝑚cn(𝜉)dn(𝜉)2𝑚4+14𝑚2,+1(3.45) where 𝜉=±𝜔4𝑚4+14𝑚2+1(𝑥𝜔𝑡).(3.46)

Family 5. If 𝑟=1𝑚2, 𝑞=2𝑚21, 𝑝=𝑚2, 𝑓(𝜉)=cn(𝜉), then we get 𝜔𝑢(𝜉)=121+2𝑚2116𝑚416𝑚2++1𝜔𝑚2cn2(𝜉)216𝑚416𝑚2++1𝑖𝜔𝑚sn(𝜉)dn(𝜉)216𝑚416𝑚2,𝜔+1𝑣(𝜉)=121+2𝑚2116𝑚416𝑚2+1𝜔𝑚2cn2(𝜉)216𝑚416𝑚2+1𝑖𝜔𝑚sn(𝜉)dn(𝜉)216𝑚416𝑚2,+1(3.47) where 𝜉=±𝜔416𝑚416𝑚2+1(𝑥𝜔𝑡).(3.48)

Family 6. If 𝑟=1𝑚2, 𝑞=2𝑚2, 𝑝=1, 𝑓(𝜉)=cs(𝜉), then we get 𝜔𝑢(𝜉)=121+2𝑚2𝑚416𝑚2+16𝜔cs2(𝜉)2𝑚416𝑚2++16𝜔𝑛𝑠(𝜉)ds(𝜉)2𝑚416𝑚2,𝜔+16𝑣(𝜉)=121+2𝑚2𝑚416𝑚2++16𝜔cs2(𝜉)2𝑚416𝑚2+16𝜔ns(𝜉)ds(𝜉)2𝑚416𝑚2,+16(3.49) where 𝜉=±𝜔4𝑚416𝑚2+16(𝑥𝜔𝑡).(3.50)

Family 7. If 𝑟=1, 𝑞=2𝑚2,𝑝=𝑚21, 𝑓(𝜉)=nd(𝜉), then we get 𝜔𝑢(𝜉)=121+2𝑚2𝑚416𝑚2𝜔𝑚+1621nd2(𝜉)2𝑚416𝑚2+16𝜔𝑚2𝑚21sd(𝜉)cd(𝜉)2𝑚416𝑚2,𝜔+16𝑣(𝜉)=121+2𝑚2𝑚416𝑚2+𝜔𝑚+1621nd2(𝜉)2𝑚416𝑚2++16𝜔𝑚2𝑚21sd(𝜉)cd(𝜉)2𝑚416𝑚2,+16(3.51) where 𝜉=±𝜔4𝑚416𝑚2+16(𝑥𝜔𝑡).(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 𝑚1 and generate into trigonometric functions when 𝑚0. 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<𝑚<1 is the modulus of the Jacobi elliptic functions and 𝑖=1.

tab1
Table 1

B. Hyperbolic Functions

The Jacobi elliptic functions sn(𝜉), cn(𝜉), dn(𝜉), ns(𝜉), cs(𝜉), ds(𝜉), sc(𝜉), sd(𝜉) generate into hyperbolic functions when 𝑚1 as in Table 2.

tab2
Table 2

C. Relations between the Jacobi Elliptic Functions

See Table 3.

tab3
Table 3

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