Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

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 [6–9], 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: 𝐹𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,…=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(𝑥−𝜔𝑡),(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 𝑃𝑢,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+^𝑛_𝑖=1𝑎_𝑖𝑓^𝑖(𝜉)+𝑏_𝑖𝑓^−𝑖(𝜉)+^𝑛_𝑖=2𝑐_𝑖𝑓^𝑖−2(𝜉)𝑓^(𝜉)+^𝑛_𝑖₌₋₁𝑑_𝑖𝑓^𝑖(𝜉)𝑓^(𝜉),(2.4) where 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, and 𝑑_𝑖 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 𝑓^𝑖(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, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, 𝑑_𝑖, 𝜔, and 𝑘.

Step 6. Solving the algebraic equations by use of Maple or Mathematica, we have 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, 𝑑_𝑖, 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: 𝑣_𝑡=13𝑢^𝑥𝑥𝑥+83𝑢𝑢^𝑥𝑢_𝑡,=𝑣_𝑥.(3.1) The traveling wave variable (2.2) permits us converting (3.1) into the following ODE: 𝜔𝑣^+13𝑘²𝑢^+83𝑢𝑢^𝜔𝑢_=0,+𝑣^=0.(3.2) Integrating (3.2) with respect to 𝜉 once and taking the constant of integration to be zero, we obtain 𝜔𝑣+13𝑘²𝑢^+43𝑢²𝜔𝑢+𝑣=0.=0,((33..34)) Suppose that the solutions of (3.3) and (3.4) can be expressed by 𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝑓^𝑖(𝜉)+𝑏_𝑖𝑓^−𝑖(𝜉)+^𝑛_𝑖=2𝑐_𝑖𝑓^𝑖−2(𝜉)𝑓^(𝜉)+^−𝑛_𝑖₌₋₁𝑣(𝜉)=𝐴₀𝑑_𝑖𝑓^𝑖(𝜉)𝑓^(𝜉),+^𝑚_𝑖=1𝐴_𝑖𝑓^𝑖(𝜉)+𝐵_𝑖𝑓^−𝑖(𝜉)+^𝑚_𝑖=2𝐿_𝑖𝑓^𝑖−2(𝜉)𝑓^(𝜉)+^−𝑚_𝑖₌₋₁𝐻_𝑖𝑓^𝑖(𝜉)𝑓^(𝜉),(3.5) where𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, 𝑑_𝑖, 𝐴_𝑖, 𝐵_𝑖, 𝐿_𝑖, and 𝐻_𝑖 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)𝑣(𝜉)=𝐴₀𝑓²(𝜉),+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 𝑓^𝑖(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 𝐴𝑎₀=𝑎₁=𝑎₂=𝑏₁=𝑐₂=𝑑₁=𝑑₂=𝐴₁=𝐴₂=𝐵₁=𝐿₁=𝐻₁=𝐻₂=0,₀=arbitraryconstant,𝑏₂=−9𝜔²𝑟8𝑞,𝐵²=9𝜔³𝑟8𝑞,𝑘=±𝜔√32√𝑞.(3.7)

Case 2. Consider 𝐴𝑎₀=𝑎₁=𝑏₂=𝑏₁=𝑐₂=𝑑₁=𝑑₂=𝐴₁=𝐵₂=𝐵₁=𝐿₁=𝐻₁=𝐻₂=0,₀=arbitraryconstant,𝑎₂=−9𝜔²𝑝8𝑞,𝐴²=9𝜔³𝑝8𝑞,𝑘=±𝜔√32√𝑞.(3.8)

Case 3. Consider 𝑏𝑎₀=𝑎₁=𝑎₂=𝑏₁=𝑐₂=𝑑₁=𝐴₂=𝐴₁=𝐵₁=𝐿₁=𝐻₁=0,₂=−9𝜔²𝑟4𝑞,𝑑²=∓9𝜔²√𝑟4𝑞,𝐵²=9𝜔³𝑟4𝑞,𝐻²=±9𝜔³√𝑟𝐴4𝑞₀,=arbitraryconstant,𝑘=𝜔√3√𝑞.(3.9)

Case 4. Consider 𝑎𝑎₀=𝑎₁=𝑏₁=𝑐₂=𝑑₁=𝑑₂=𝐴₁=𝐵₁=𝐿₁=𝐻₁=𝐻₂=0,₂=−9𝜔²𝑝8𝑞,𝑏²=−9𝜔²𝑟8𝑞,𝐴²=9𝜔³𝑝8𝑞,𝐵²=9𝜔³𝑟𝐴8𝑞₀,=arbitraryconstant,𝑘=±𝜔√32√𝑞.(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)=−9𝜔²𝑝8𝑞𝑓²(𝜉)−9𝜔²𝑟𝑣(𝜉)=9𝜔₃8𝑞𝑓²1(𝜉),𝑝8𝑞𝑓²(𝜉)+9𝜔³𝑟8𝑞𝑓²1(𝜉),(3.11) where 𝜉=±𝜔√32√𝑞(𝑥−𝜔𝑡).(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)=9𝜔²81+𝑚²𝑚²𝑣(𝜉)=−9𝜔₃sn²(𝜉)+ns²(𝜉),81+𝑚²𝑚²sn²(𝜉)+ns²(𝜉),(3.13) where 𝜉=±𝜔√32𝑖√1+𝑚²(𝑥−𝜔𝑡).(3.14)

Family 2. If 𝑟=1−𝑚², 𝑞=2𝑚²−1, 𝑝=−𝑚², 𝑓(𝜉)=cn(𝜉), then we get 𝑢(𝜉)=9𝜔²82𝑚²−1𝑚²𝑣(𝜉)=−9𝜔₃cn²(𝜉)−1−𝑚²nc²(𝜉),𝑚²82𝑚²−1cn²(𝜉)−1−𝑚²nc²(𝜉),(3.15) where 𝜉=±𝜔√32√2𝑚²−1(𝑥−𝜔𝑡).(3.16)

Family 3. If 𝑟=𝑚²−1, 𝑞=2−𝑚², 𝑝=−1, 𝑓(𝜉)=dn(𝜉), then we get 𝑢(𝜉)=9𝜔²82−𝑚²dn²𝑣(𝜉)=−9𝜔₃(𝜉)−𝑚²−1nd²(𝜉),82−𝑚²dn²(𝜉)−𝑚²−1nd²(𝜉),(3.17) where 𝜉=±𝜔√32√𝑚²−1(𝑥−𝜔𝑡).(3.18)

Family 4. If 𝑟=𝑚², 𝑞=−(1+𝑚²), 𝑝=1, 𝑓(𝜉)=dc(𝜉), then we get 𝑢(𝜉)=9𝜔²81+𝑚²dc²𝑣(𝜉)=−9𝜔₃(𝜉)+𝑚²cd²(𝜉),81+𝑚²dc²(𝜉)+𝑚²cd²(𝜉),(3.19) where 𝜉=±𝜔√32𝑖√1+𝑚²(𝑥−𝜔𝑡).(3.20)

Family 5. If 𝑟=1, 𝑞=2−𝑚², 𝑝=1−𝑚², 𝑓(𝜉)=sc(𝜉), then we get 𝑢(𝜉)=−9𝜔²82−𝑚²1−𝑚²𝑣(𝜉)=9𝜔₃sc²(𝜉)+cs²(𝜉),82−𝑚²1−𝑚²sc²(𝜉)+cs²(𝜉),(3.21) where 𝜉=±𝜔√32√2−𝑚²(𝑥−𝜔𝑡).(3.22)

Family 6. If 𝑟=1/4, 𝑞=(1/2)(1−2𝑚²), 𝑝=1/4, 𝑓(𝜉)=ns(𝜉)±cs(𝜉), then we get 𝑢(𝜉)=9𝜔²1−2ns²(𝜉)𝑣(𝜉)=−9𝜔8₃1−2𝑚²,1−2ns²(𝜉)81−2𝑚²,(3.23) where 𝜉=±𝜔√32(1/2)1−2𝑚²(𝑥−𝜔𝑡).(3.24)

Family 7. If 𝑟=(1/4)(1−𝑚²), 𝑞=(1/4)(1+𝑚²), 𝑝=(1/4)(1−𝑚²), 𝑓(𝜉)=nc(𝜉)±sc(𝜉), then we get 𝑢(𝜉)=−9𝜔²1−𝑚²sc²(𝜉)+nc²(𝜉)𝑣(𝜉)=9𝜔₃41+𝑚²,1−𝑚²sc²(𝜉)+nc²(𝜉)41+𝑚²,(3.25) where 𝜉=±𝜔√3√1+𝑚²(𝑥−𝜔𝑡).(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]: 𝑣𝑢_𝑡𝑡=𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥+6𝑣𝑣_𝑥,=𝑣_𝑥𝑥𝑥+6𝑢𝑣_𝑥+6𝑣𝑢_𝑥.(3.27) Substituting (2.2) into (3.27) yields 𝜔𝑢𝜔^𝑣+_𝑘²𝑢^+3(𝑢²+𝑣²)^=0,+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 𝜔𝑢𝜔+𝑣𝑘+²𝑢𝑘^₂+3𝑢²+𝑣²=0,(3.29)𝑣^+6(𝑢𝑣)=0.(3.30) Suppose that the solutions of (3.27) can be expressed by 𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝑓^𝑖(𝜉)+𝑏_𝑖𝑓^−𝑖(𝜉)+^𝑛_𝑖=2𝑐_𝑖𝑓^𝑖−2(𝜉)𝑓^(𝜉)+^−𝑛_𝑖₌₋₁𝑣(𝜉)=𝛼₀𝑑_𝑖𝑓^𝑖(𝜉)𝑓^(𝜉),+^𝑚_𝑖=1𝛼_𝑖𝑓^𝑖(𝜉)+𝛽_𝑖𝑓^−𝑖(𝜉)+^𝑚_𝑖=2𝛾_𝑖𝑓^𝑖−2(𝜉)𝑓^(𝜉)+^−𝑚_𝑖₌₋₁𝑒_𝑖𝑓^𝑖(𝜉)𝑓^(𝜉),(3.31) where 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, 𝑑_𝑖, 𝛼_𝑖, 𝛽_𝑖, 𝛾_𝑖, and 𝑒_𝑖 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)𝑣(𝜉)=𝛼₀𝑓²(𝜉),+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 𝑓^𝑖(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=𝑎𝑎₂=𝑏₁=𝑐₂=𝑑₁=𝑑₂=𝛼₁=𝛼₂=𝛽₁=𝑒₁=𝑒₂=𝛾₂=0,₀=−𝜔121+√𝑞²𝑞−3𝑝𝑟,𝑏²=−𝜔𝑟𝛼4√𝑞²−3𝑝𝑟₀,=−𝜔121+√𝑞²𝑞−3𝑝𝑟,𝛽²=−𝜔𝑟4√𝑞²−3𝑝𝑟,𝑘=±√𝜔2√⁴𝑞²−3𝑝𝑟.(3.33)

Case 2. Consider 𝑎₁=𝑎𝑏₁=𝑏₂=𝑐₂=𝑑₁=𝑑₂=𝛼₁=𝛽₁=𝛽₂=𝑒₁=𝑒₂=𝛾₂=0,₀=−𝜔121+√𝑞²𝑞−3𝑝𝑟,𝑎²=−𝜔𝑝𝛼4√𝑞²−3𝑝𝑟₀,=𝜔121+√𝑞²𝑞−3𝑝𝑟,𝛼²=𝜔𝑝4√𝑞²−3𝑝𝑟,𝑘=±√𝜔2√⁴𝑞²−3𝑝𝑟.(3.34)

Case 3. Consider 𝑎𝑎₁=𝑏₁=𝑐₂=𝑑₁=𝑑₂=𝛼₁=𝛽₂=𝑒₁=𝑒₂=𝛾₂=0,₀=−𝜔123+√𝑞²+𝑞12𝑝𝑟,𝑎²=−𝜔𝑝4√𝑞²+12𝑝𝑟,𝑏²=−𝜔𝑟𝛼4√𝑞²+12𝑝𝑟₀,=−𝜔121−√𝑞²+𝑞12𝑝𝑟,𝛼²=𝜔𝑝4√𝑞²+12𝑝𝑟,𝛽²=𝜔𝑟𝑘=±√4√𝑞²+12𝑝𝑟,𝜔2√⁴𝑞²+12𝑝𝑟.(3.35)

Case 4. Consider 𝑎𝑎₁=𝑏₁=𝑏₂=𝑑₁=𝑑₂=𝛼₁=𝛽₁=𝛽₂=𝑒₁=𝑒₂=0,₀=−𝜔121+√𝑞²+𝑞12𝑝𝑟,𝑎²=−𝜔𝑝2√𝑞²+12𝑝𝑟,𝑐²=−𝜔√𝑝𝛼2√𝑞²+12𝑝𝑟₀,=𝜔121+√𝑞²+𝑞12𝑝𝑟,𝛼²=𝜔𝑝2√𝑞²+12𝑝𝑟,𝛾²=𝜔√𝑝𝑘=±√2√𝑞²+12𝑝𝑟,₄𝜔√𝑞²+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 𝑢(𝜉)=−𝜔121+√𝑞²+𝑞12𝑝𝑟−2√𝑞²𝜔+𝑝12𝑝𝑟𝑓²(𝜉)−𝜔√𝑝2√𝑞²+12𝑝𝑟𝑓^𝑣(𝜉)=𝜔(𝜉),121+√𝑞²+𝑞12𝑝𝑟+2√𝑞²𝜔+𝑝12𝑝𝑟𝑓²(𝜉)+𝜔√𝑝2√𝑞²+12𝑝𝑟𝑓^(𝜉),(3.37) where 𝜉=±√₄𝜔√𝑞²+12𝑝𝑟(𝑥−𝜔𝑡).(3.38)

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

Family 1. If 𝑟=1, 𝑞=2𝑚²−1, 𝑝=𝑚²(𝑚²−1), 𝑓(𝜉)=sd(𝜉), then we get 𝑢(𝜉)=−𝜔121+2𝑚²−1√16𝑚⁴−16𝑚²+1−𝜔𝑚²𝑚²−1sd²(𝜉)2√16𝑚⁴−16𝑚²+1−𝜔𝑚√𝑚²−1nd(𝜉)cd(𝜉)𝑣(𝜉)=𝜔2√16𝑚⁴−16𝑚²+1,121+2𝑚²−1√16𝑚⁴−16𝑚²+1+𝜔𝑚²𝑚²−1sd²(𝜉)2√16𝑚⁴−16𝑚²+1+𝜔𝑚√𝑚²−1nd(𝜉)cd(𝜉)2√16𝑚⁴−16𝑚²+1,(3.39) where 𝜉=±√₄𝜔√16𝑚⁴−16𝑚²+1(𝑥−𝜔𝑡).(3.40)

Family 2. If 𝑟=𝑚²(𝑚²−1), 𝑞=2𝑚²−1,𝑝=1, 𝑓(𝜉)=𝑑𝑠(𝜉), then we get 𝑢(𝜉)=−𝜔121+2𝑚²−1√16𝑚⁴−16𝑚²+1−𝜔ds²(𝜉)𝑣(𝜉)=𝜔2√16𝑚⁴−16𝑚²+1+2√1𝜔6c𝑚s⁴(𝜉−)n1s6(𝑚𝜉²)+1,121+2𝑚²−1√16𝑚⁴−16𝑚²+1+𝜔ds²(𝜉)2√16𝑚⁴−16𝑚²+1−2√1𝜔6c𝑚s⁴(𝜉−)n1s6(𝑚𝜉²)+1,(3.41) where 𝜉=±√₄𝜔√16𝑚⁴−16𝑚²+1(𝑥−𝜔𝑡).(3.42)

Family 3. If 𝑟=𝑚²/4, 𝑞=(1/2)(𝑚²−2), 𝑝=𝑚²/4, 𝑓(𝜉)=sn(𝜉)±𝑖cn(𝜉), then we get 𝑢(𝜉)=−𝜔121+𝑚²−22√𝑚⁴−𝑚²+1−𝜔𝑚²(sn(𝜉)±𝑖cn(𝜉))²𝑣(𝜉)=𝜔8√𝑚⁴−𝑚²+1−𝜔𝑚(𝑐𝑛(𝜉4)√d𝑚n(⁴𝜉−)∓𝑚²𝑖s+n(1𝜉)dn(𝜉)),121+𝑚²−22√𝑚⁴−𝑚²+1+𝜔𝑚²(sn(𝜉)±𝑖cn(𝜉))²8√𝑚⁴−𝑚²+1+𝜔𝑚(cn(𝜉4)√d𝑚n(⁴𝜉−)∓𝑚²𝑖s+n(1𝜉)dn(𝜉)),(3.43) where 𝜉=±√₄𝜔√𝑚⁴−𝑚²+1(𝑥−𝜔𝑡).(3.44)

Family 4. If 𝑟=1,𝑞=−(1+𝑚²), 𝑝=𝑚², 𝑓(𝜉)=sn(𝜉), then we get 𝑢(𝜉)=−𝜔121−√𝑚⁴1++14𝑚𝑚²²+1−𝜔𝑚²sn²(𝜉)𝑣(𝜉)=𝜔2√𝑚⁴+14𝑚²+1−2√𝜔𝑚𝑚c⁴n+(𝜉1)4d𝑚n²(𝜉+)1,121−√𝑚⁴1++14𝑚𝑚²²+1+𝜔𝑚²sn²(𝜉)2√𝑚⁴+14𝑚²+1+2√𝜔𝑚𝑚c⁴n+(𝜉1)4d𝑚n²(𝜉+)1,(3.45) where 𝜉=±√₄𝜔√𝑚⁴+14𝑚²+1(𝑥−𝜔𝑡).(3.46)

Family 5. If 𝑟=1−𝑚², 𝑞=2𝑚²−1, 𝑝=−𝑚², 𝑓(𝜉)=cn(𝜉), then we get 𝑢(𝜉)=−𝜔121+2𝑚²−1√16𝑚⁴−16𝑚²+1+𝜔𝑚²cn²(𝜉)𝑣(𝜉)=𝜔2√16𝑚⁴−16𝑚²+1+2√𝑖𝜔16𝑚𝑚sn⁴−(𝜉)1d6n𝑚(²𝜉+)1,121+2𝑚²−1√16𝑚⁴−16𝑚²+1−𝜔𝑚²cn²(𝜉)2√16𝑚⁴−16𝑚²+1−2√𝑖𝜔16𝑚𝑚sn⁴−(𝜉)1d6n𝑚(²𝜉+)1,(3.47) where 𝜉=±√₄𝜔√16𝑚⁴−16𝑚²+1(𝑥−𝜔𝑡).(3.48)

Family 6. If 𝑟=1−𝑚², 𝑞=2−𝑚², 𝑝=1, 𝑓(𝜉)=cs(𝜉), then we get 𝑢(𝜉)=−𝜔121+√𝑚⁴−2−16𝑚𝑚²²+16−𝜔cs²(𝜉)𝑣(𝜉)=𝜔2√𝑚⁴−16𝑚²+16+2√𝜔𝑚𝑛⁴𝑠−(𝜉1)6d𝑚s²(𝜉+)16,121+√𝑚⁴−2−16𝑚𝑚²²+16+𝜔cs²(𝜉)2√𝑚⁴−16𝑚²+16−2√𝜔𝑚n⁴s−(𝜉1)6d𝑚s²(𝜉+)16,(3.49) where 𝜉=±√₄𝜔√𝑚⁴−16𝑚²+16(𝑥−𝜔𝑡).(3.50)

Family 7. If 𝑟=−1, 𝑞=2−𝑚²,𝑝=𝑚²−1, 𝑓(𝜉)=nd(𝜉), then we get 𝑢(𝜉)=−𝜔121+√𝑚⁴−2−16𝑚𝑚²²+16−𝜔𝑚²−1nd²(𝜉)2√𝑚⁴−16𝑚²+16−𝜔𝑚²√𝑚²−1sd(𝜉)cd(𝜉)𝑣(𝜉)=𝜔2√𝑚⁴−16𝑚²+16,121+√𝑚⁴−2−16𝑚𝑚²²+16+𝜔𝑚²−1nd²(𝜉)2√𝑚⁴−16𝑚²+16+𝜔𝑚²√𝑚²−1sd(𝜉)cd(𝜉)2√𝑚⁴−16𝑚²+16,(3.51) where 𝜉=±√₄𝜔√𝑚⁴−16𝑚²+16(𝑥−𝜔𝑡).(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 𝑚→1 and generate into trigonometric functions when 𝑚→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<𝑚<1 is the modulus of the Jacobi elliptic functions and 𝑖=√−1.

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 𝑚→1 as in Table 2.

Table 2 

C. Relations between the Jacobi Elliptic Functions

See Table 3.

Table 3