New Exact Jacobi Elliptic Function Solutions for the Coupled Schrödinger-Boussinesq Equations

Abstract

A general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method, and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions for coupled Schrödinger-Boussinesq equations. As a result, several families of new generalized Jacobi elliptic function wave solutions are obtained by using this method, some of them are degenerated to solitary wave solutions and trigonometric function solutions in the limited cases, which shows that the general method is more powerful than plenty of traditional methods 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

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science. The investigation of exact solutions of NLPDEs plays an important role in the study of these phenomena such as the nonlinear dynamics and the mechanism behind the phenomena. With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as homotopy perturbation method [1], nonperturbative method [2], homogeneous balance method [3], Bäcklund transformation [4], Darboux transformation [5], extended tanh-function method [6], extended 𝐹-expansion method [7], 𝐺^′/𝐺 method [8], exp-function method [9], sine-cosine method [10], Jacobi elliptic function method [11], extended Riccati equation rational expansion method [12], extended auxiliary function method [13], and other methods [14, 15].

In [16, 17], Hong proposed a generalized Jacobi elliptic functions expansion method to obtain generalized exact solutions of NLPDEs. In [18], Hong and Lu proposed an improved general mapping deformation method which is more general than many other algebraic expansion methods [19, 20]. The solution procedure of this method, by the help of Matlab or Mathematica, is of utmost simplicity, and this method can be easily extended to all kinds of NLPDEs. In this work, we will propose the general algebraic method which contained the two methods [16–18] to obtain several new families of exact solutions for the coupled Schrödinger-Boussinesq equations.

2. Summary of the General Algebraic Method

Consider a given nonlinear evolution equation with one physical field 𝑢(𝑥,𝑡) in two variables 𝑥 and 𝑡𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,…=0. (1) We seek the following formal solutions of the given system by a new intermediate transformation: 𝑢(𝜉)=^𝑛_𝑖=0𝐴_𝑖𝜙^𝑖(𝜉), (2) where 𝐴_𝑖 are constants to be determined later. 𝜉=𝜉(𝑥,𝑡) are arbitrary functions with the variables 𝑥 and 𝑡. The parameter 𝑛 can be determined by balancing the highest order derivative terms with the nonlinear terms in (1). And 𝜙(𝜉) is a solution of the following ordinary differential equation: 𝜙^′2(𝜉)=⁴_𝑖=0𝑎_𝑖𝜙^𝑖(𝜉). (3) Substituting (3) and (2) into (1) and setting the coefficients of 𝜙^𝑖(𝜉)(𝑖=0,1,2,…) and 𝜙^𝑗(𝜉)∑⁴_𝑖=0𝑎_𝑖𝜙^𝑖(𝜉)(𝑗=…,−2,−1,0,1,2,…) to zero yield a set of algebraic equations for 𝐴_𝑖, 𝑎_𝑖, and 𝜉. Using the Mathematica to solve the algebraic equations and substituting each of the solutions of the set, that is, each of the expressions of 𝜙(𝜉) into (2), we can get the solutions of (1). In order to obtain some new general solutions of (3), we assume that (3) has the following solutions: 𝜙(𝜉)=𝑐₀+𝑐₁𝑒(𝜉)+𝑐₂𝑓(𝜉)+𝑐₃𝑔(𝜉)+𝑐₄ℎ(𝜉), (4) where 𝑐_𝑖=𝑐_𝑖(𝑡)(𝑖=0,…,4) are functions of 𝑡 to be determined later and the four functions 𝑒=𝑒(𝜉),𝑓=𝑓(𝜉),𝑔=𝑔(𝜉), and ℎ=ℎ(𝜉) are expressed as follows: 1𝑒=𝑝+𝑞𝐹+𝑟𝐹²+𝑙𝐹^′𝐹,𝑓=𝑝+𝑞𝐹+𝑟𝐹²+𝑙𝐹^′,𝐹𝑔=²𝑝+𝑞𝐹+𝑟𝐹²+𝑙𝐹^′𝐹,ℎ=^′𝑝+𝑞𝐹+𝑟𝐹²+𝑙𝐹^′, (5) where 𝑝,𝑞,𝑟, and 𝑙 are arbitrary constants which ensure the denominator unequal to zero, and 𝐹=𝐹(𝜉) is a solution of the following equations: 𝐹^′2=𝐴+𝐵𝐹²+𝐶𝐹⁴+2𝐷𝐹+2𝐸𝐹³,𝐹^′′=𝐵𝐹+2𝐶𝐹³+𝐷+3𝐸𝐹², (6) where “′” denotes 𝑑/𝑑𝜉, “′′” denotes 𝑑²/𝑑𝜉²,  𝐴,𝐵,𝐶,𝐷, and 𝐸 are arbitrary constants, and the four functions 𝑒,𝑓,𝑔, and ℎ satisfy the following relations: 𝑒^′=−𝑞𝑒ℎ−2𝑟𝑓ℎ−𝑙𝐷𝑒²+𝐵𝑒𝑓+2𝐶𝑓𝑔+3𝐸𝑓²,𝑓^′=𝑝𝑒ℎ−𝑟𝑔ℎ+𝑙𝐴𝑒²+𝐷𝑒𝑓−𝐶𝑔²,𝑔−𝐸𝑓𝑔_′=𝑞𝑔ℎ+2𝑝𝑓ℎ+𝑙2𝐴𝑒𝑓+3𝐷𝑓²+𝐵𝑓𝑔+𝐸𝑔²,ℎ^′=𝐷𝑝−𝐴𝑞𝑒²++𝑓𝐵𝑝−𝐷𝑞−2𝐴𝑟𝑒𝑓2𝐶𝑝+𝐸𝑞−𝐵𝑟𝑓𝑔+3𝐸𝑝−𝐷𝑟²+𝐶𝑞−𝐸𝑟𝑔²,𝑓²=𝑒𝑔,ℎ²=𝐴𝑒²+𝐵𝑓²+𝐶𝑔²+2𝐷𝑒𝑓+2𝐸𝑓𝑔,𝑝𝑒+𝑞𝑓+𝑟𝑔+𝑙ℎ=1. (7) And 𝑒,𝑓,𝑔, and ℎ satisfy one of the following relations at the same time.

Family 1. When 𝑝=0𝐶𝑙²−𝑟²ℎ²=−𝐶+2𝐶𝑙ℎ−𝐵𝑟1−𝑙ℎ−𝑞𝑓𝑒−𝐴𝑒²𝑟²−2𝐷𝑟²𝑓+𝑒𝑓+2𝐶𝑞−2𝐸𝑟2𝑙𝐸𝑟−2𝐶𝑙𝑞𝑓ℎ+2𝐸𝑞𝑟−𝐶𝑞²𝑓². (8a)

Family 2. When 𝑞=0𝐶𝑙²−𝑟²ℎ²=2𝐶𝑙ℎ+𝑝𝑒−𝑝𝑙𝑒ℎ+2𝐸𝑟(𝑙ℎ−1)𝑓−𝐵𝑟1−𝑙ℎ−𝑝𝑒𝑒−𝐶𝑝²+𝐴𝑟²𝑒²+2𝐸𝑝𝑟−2𝐷𝑟²𝑒𝑓−𝐶. (8b)

Family 3. When 𝑟=0𝐶𝑙²𝑔²=1−2𝐸𝑙²𝑓𝑔−2𝑝𝑒+𝑝²−𝐴𝑙²𝑒²−2𝑞𝑓+2𝑝𝑞−𝐷𝑙²𝑒𝑓+𝑞²−𝐵𝑙²𝑒𝑔. (8c)

Family 4. When 𝑙=0𝑟²ℎ²=𝐶−2𝐶𝑝𝑒+2𝐸𝑟−2𝐶𝑞𝑓+𝐶𝑝²+𝐴𝑟²𝑒²+𝐶𝑞²−2𝐸𝑞𝑟+𝐵𝑟²+𝑒𝑔2𝐶𝑝𝑞−2𝐸𝑝𝑟+2𝐷𝑟²𝑒𝑓. (8d)

Substituting (4), (7) along with (8a)–(8d) into (3) separately yields four families of polynomial equations for 𝑒,𝑓,𝑔, and ℎ. Setting the coefficients of 𝑒^𝑖,𝑒^𝑖𝑓,𝑒^𝑖𝑔,𝑒^𝑖ℎ,𝑒^𝑖𝑓𝑔,𝑒^𝑖𝑓ℎ, and 𝑒^𝑖𝑔ℎ(𝑖=0,1,2,…) to zero yields a set of overdetermined differential equations (ODEs) in 𝑝,𝑞,𝑟,𝑙,𝑎_𝑖, and 𝑐_𝑖(𝑖=0,1,2,3,4),𝐴,𝐵,𝐶,𝐷,𝐸, and 𝜉(𝑥,𝑡), solving the ODEs by Mathematica and Wu elimination, we can obtain many exact solutions of (3) according to (4), (5), (6). If we let 𝑐₀=𝑐₁=𝑐₃=𝑐₄=0, 𝑐₂=1, 𝑝=1, 𝑞=𝑟=𝑙=0,  𝑎₀=𝐴, 𝑎₁=2𝐷, 𝑎₂=𝐵, 𝑎₃=2𝐸, and 𝑎₄=𝐶, we have 𝜑(𝜉)=𝐹(𝜉); our method contains the improved general mapping deformation method [18].

Remark 1. Our method proposed here is more general than the 𝐺^′/𝐺 method [8], the extended Riccati equation rational expansion method [12], the extended auxiliary function method [13], the generalized Jacobi elliptic functions expansion method [16, 17], and many other algebra expansion methods [6, 7, 11, 18–21].

Remark 2. Equations (2) and (3) can be extended to the following forms: 𝑢(𝜉)=^𝑛_𝑖=0𝐴_𝑖(𝑡)𝜙^𝑖(𝜉)+⁻¹_𝑖=−𝑛𝐴_𝑛−𝑖(𝑡)𝜙^𝑖(𝜉)+^𝑛_𝑖=−𝑛𝐵_𝑖(𝑡)𝜙^𝑖(𝜉)𝜙^′(𝜉),𝜙^′2(𝜉)=^𝑁_𝑖=0𝑎_𝑖(𝑡)𝜙^𝑖(𝜉), (9) where 𝑁 can be an arbitrary positive integer. 𝑛 is usually a positive integer. If 𝑛 is a fraction or a negative integer, we make the following transformation: (a)when 𝑛=𝑑/𝑐 is a fraction, we let 𝑢(𝜉)=𝑣^𝑑/𝑐(𝜉), then return to determine the balance constant 𝑛 again; (b)when 𝑛 is a negative integer, we suppose 𝑢(𝜉)=𝑣^𝑛(𝜉), then return to determine the balance constant 𝑛 again.

Remark 3. Notice that 𝐹₁(𝜉)_{(𝐴,𝐵,𝐶,𝐷,𝐸)1}⟶𝜑₁(𝜉)_{(𝑎0,𝑎1,𝑎2,𝑎3,𝑎4)1}=𝐹₂(𝜉)_{(𝐴,𝐵,𝐶,𝐷,𝐸)2}⟶𝜑₂(𝜉)_{(𝑎0,𝑎1,𝑎2,𝑎3,𝑎4)2}=𝐹₃(𝜉)_{(𝐴,𝐵,𝐶,𝐷,𝐸)3}⟶⋯. (10)
We find a meanful conclusion that this general method implies a BT of (1) with the compatible conditions (4), (5), (6), (7), and (8a)–(8d).

In the following, we will use this method to solve the Schrödinger-Boussinesq equations.

3. Exact Solutions to the Coupled Schrödinger-Boussinesq Equations

We consider the coupled Schrödinger-Boussinesq equations [21–28] 𝑖𝐸_𝑡+𝐸_𝑥𝑥+𝛼𝐸−𝑁𝐸=0,3𝑁_𝑡𝑡−𝑁_{𝑥𝑥𝑥𝑥}+3𝑁²_𝑥𝑥+𝛽𝑁_𝑥𝑥−|𝐸|²_𝑥𝑥=0. (11) These equations are known to describe various physical processes in laser and plasma, such as formation, Langmuir field amplitude, intense electromagnetic waves, and modulational instabilities [22–25]. The problem of the complete integrability of this system has been studied by Chowdhury et al. from the point of view of Painlevé analysis [25]. The solitary wave solutions for system (11) have been obtained in [26, 27]. The Jacobi doubly periodic wave solutions and a range of other solutions for this system have been investigated in [21, 28]. We are interested in searching new generalized Jacobi elliptic function solutions for (11) by using our method.

We consider the following transformations: 𝐸(𝑥,𝑡)=𝐸(𝜉)=𝑢(𝜉)𝑒^{𝑖(𝑘𝑥+𝜔𝑡+𝜉00)},𝑁(𝑥,𝑡)=𝑁(𝜉)=𝑣(𝜉),𝜉=𝑎𝑥−2𝑎𝑘𝑡+𝜉₀, (12)(13) where 𝑎 and 𝑘 are constants to be determined later and 𝜉₀ and 𝜉₀₀ are arbitrary constants.

Substituting (12) and (13) into (11) and integrating the second equation of system (11) twice, we have 𝑎²𝑢^′′+𝛼−𝜔−𝑘²𝑎𝑢−𝑢𝑣=0,₂𝑣^′′+𝑢²−12𝑘²+𝛽𝑣−3𝑣²=0. (14) By balancing the highest derivative term with the nonlinear terms in (14), we obtain 𝑛=2. Therefore, we assume that (14) have the following solutions: 𝑢(𝜉)=𝐴₀+𝐴₁𝜑(𝜉)+𝐴₂𝜑²(𝜉),𝑣(𝜉)=𝐵₀+𝐵₁𝜑(𝜉)+𝐵₂𝜑²(𝜉). (15) Substituting (3), (13), and (15) into (14) and setting the coefficients of 𝜙^𝑖(𝜉)(𝑖=0,1,2,…) and 𝜑^𝑗(𝜉)∑⁴_𝑖=0𝑎_𝑖𝜑^𝑖(𝜉)(𝑗=…,−2,−1,0,1,2,…) to zero yield a set of over-determined equations (ODEs) for 𝐴_𝑖,𝑘,𝜔, and 𝑎_𝑖. After solving the ODEs by Mathematica, we could determine the following solutions.

Family 1. 𝐴₀=±𝑎²−𝑎₁𝑎₃−8𝑎₀𝑎₄2,𝐴₁=±2𝑎²𝑎₄𝑎₃−2𝑎₁𝑎₃−16𝑎₀𝑎₄,𝐴₂𝐵=0,₀=0,𝐵₁=𝑎²𝑎₃,𝐵₂=2𝑎²𝑎₄,𝑎𝜔=−²𝑎⁴₃−80𝑎₁𝑎₃𝑎²₄−128𝑎₀𝑎³₄48𝑎²₃𝑎₄𝛽+𝛼+,𝜀12𝑘=4𝑎²𝑎⁴₃+16𝑎₁𝑎₃𝑎²₄−128𝑎₀𝑎³₄−4𝑎²₃𝑎₄𝛽3𝑎²₃𝑎₄,𝜀²=1,𝑎₂=𝑎²₃4𝑎₄+2𝑎₁𝑎₄𝑎₃. (16)

Family 2. 𝐴₀=±√2𝐵₀,𝐴₁=0,𝐴₂=±6√2𝑎²𝑎₄,𝐵₀=const,𝐵₁=0,𝐵₂=6𝑎²𝑎₄,𝑎₁=𝑎₃1=0,𝜔=1244𝑎²𝑎₂−22𝐵₀+12𝛼+𝛽,𝑘=𝜀4𝑎²𝑎₂−2𝐵₀−𝛽12,𝜀²=1. (17)

Substituting (4), (5), (6), (7) along with (8a)–(8d) and (16) into (14) separately yields an ODEs; after solving the ODEs by Mathematica and Wu elimination, we can obtain the following solutions of (14) according to (4), (5), (6), (16).

Case 1. 𝐴=𝑎₀₁=141−𝑚²,𝐵=𝑎₂₁=2−𝑚²,𝐶=𝑎₄₁1=1,𝐷=2𝑎₁₁=12𝑚²,1−1𝐸=2𝑎₃₁𝑐=−1,0≤𝑚≤1,𝑝=1,𝑞=𝑟=𝑙=0,₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₁𝜉₁=𝐹₁𝜉₁=cn𝜉₁±1^±_±sn𝜉₁+cn𝜉₁,𝜉₁=𝑎𝑥−2𝑎𝑘₁𝑡+𝜉₀. (18)

Case 2. 𝐴=𝑎₀₂=14,𝐵=𝑎₂₂=2−𝑚²,𝐶=𝑎₄₂=1−𝑚²1,𝐷=2𝑎₁₂=12,1𝐸=2𝑎₃₂=1−𝑚²𝑐,0≤𝑚≤1,𝑝=1,𝑞=𝑟=𝑙=0,₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₂𝜉₂=𝐹₂𝜉₂=sn𝜉₂±1−sn𝜉₂^±_±cn𝜉₂,𝜉₂=𝑎𝑥−2𝑎𝑘₂𝑡+𝜉₀. (19)

Case 3. 𝐴=𝑎₀₃=−𝑐²₀−11+𝑐²₀−1𝑚²,𝐵=𝑎₂₃=2−6𝑐²₀𝑚²−1,𝐶=𝑎₄₃=−𝑚²,1𝐷=2𝑎₁₃=𝑐₀−2𝑐₀𝑚²+2𝑐³₀𝑚²,1𝐸=2𝑎₃₃=2𝑐₀𝑚²𝑐,0≤𝑚≤1,𝑝=1,𝑞=𝑟=𝑙=0,₀=const,𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₃𝜉₃=𝐹₃𝜉₃=𝑐₀+cn²𝜉₃,𝜉₃=𝑎𝑥−2𝑎𝑘₃𝑡+𝜉₀. (20)

Case 4. 𝐴=1,𝐵=−𝑚²−1,𝐶=𝑚²,𝑎𝐷=𝐸=0,𝐹=sn𝜉,0≤𝑚≤1,₀₄=1,𝑎₁₄=∓42(1+𝑚)√𝑎𝑚,₃₄=±22(1+𝑚)√𝑚1+6𝑚+𝑚²−4(1+𝑚)√𝑚,𝑎₂₄=12(1+𝑚)√𝑚−6𝑚−𝑚²𝑎−1,₄₄=8𝑚(𝑚+1)²−2(1+𝑚)1+𝑚²+6𝑚√𝑚,𝑝=1,𝑞=±2(1+𝑚)√𝑚,𝑟=𝑚,𝑙=0,𝑐₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₄𝜉₄=sn𝜉₄1±2(1+𝑚)√𝑚sn𝜉₄+𝑚sn²𝜉₄,𝜉₄=𝑎𝑥−2𝑎𝑘₄𝑡+𝜉₀. (21)

Case 5. 𝐴=1−𝑚²,𝐵=2𝑚²−1,𝐶=−𝑚²,𝐷=𝐸=0,0≤𝑚≤1,𝐹=cn𝜉𝑎₀₅=1,𝑎₁₅𝑎=−4,₂₅=8−4𝑚²,𝑎₃₅=8𝑚²−8,𝑎₄₅=4−4𝑚²,𝑐𝑝=0,𝑞=1,𝑟=0,𝑙=±1,₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₅𝜉₅=cn𝜉₅cn𝜉₅∓sn𝜉₅dn𝜉₅,𝜉₅=𝑎𝑥−2𝑎𝑘₅𝑡+𝜉₀. (22)

Case 6. 𝐴=𝑚²−1,𝐵=2−𝑚²,𝐶=−1,𝐷=𝐸=0,0≤𝑚≤1,𝐹=dn𝜉𝑎₀₆=1,𝑎₁₆𝑎=−4𝑚,₂₆=8𝑚²−4,𝑎₃₆=8𝑚−8𝑚³,𝑎₄₆=4𝑚⁴−4𝑚²,𝑐𝑝=0,𝑞=𝑚,𝑟=0,𝑙=±1,₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₆𝜉₆=dn𝜉₆dn𝜉₆∓𝑚²sn𝜉₆cn𝜉₆,𝜉₆=𝑎𝑥−2𝑎𝑘₆𝑡+𝜉₀. (23)

Case 7. 𝐶𝐴=¹𝐶₃𝑞−5𝐶²₃𝑞³+𝜀𝐶₁+3𝐶₃𝑞²𝐶₃3𝐶₃𝑞²−2𝐶₁4𝐶₃,𝐵=0,𝐶=0,2𝐷=𝐶₁,2𝐸=𝐶₃,𝐹=℘𝐶₃2𝜉,−4𝐶₁𝐶₃,−4𝑀𝐶₃,𝑎₀₇=0,𝑎₁₇=𝐶₃,𝑎₂₇=−3𝐶₃𝑎𝑞,₃₇=𝐶₁+3𝐶₃𝑞²,𝑎₄₇=−3𝐶₁𝐶₃𝑞−9𝐶²₃𝑞³+𝜀𝐶₁+3𝐶₃𝑞²𝐶₃3𝐶₃𝑞²−2𝐶₁×4𝐶₃⁻¹,𝜀=sgn𝐶₁+3𝐶₃𝑞²,𝑝=0,𝑞=const,𝑟=1,𝑙=0,𝑐₀=𝑐₁=𝑐₃=𝑐₄=0,𝑐₂𝜑=1,₇𝜉₇=1𝑞+℘𝐶₃/2𝜉₇,−4𝐶₁/𝐶₃,−4𝑀/𝐶₃,𝜉₇=𝑎𝑥−2𝑎𝑘₇𝑡+𝜉₀. (24)

We obtain the following solutions of (11) according to (4), (5), (12), (13), (15), and (16) and Cases 1–7: 𝐸_𝑖(𝑥,𝑡)=±𝑎²−𝑎_1𝑖𝑎_3𝑖−8𝑎_0𝑖𝑎_4𝑖2±2𝑎²𝑎_4𝑖𝑎_3𝑖−2𝑎_1𝑖𝑎_3𝑖−16𝑎_0𝑖𝑎_4𝑖𝜑_𝑖𝜉_𝑖×𝑒^{𝑖(𝑘𝑖𝑥+𝜔𝑖𝑡+𝜉00)},𝑁_𝑖(𝑥,𝑡)=𝑎²𝑎_3𝑖𝜑_𝑖𝜉_𝑖+2𝑎²𝑎_4𝑖𝜑²_𝑖𝜉_𝑖,𝜉_𝑖=𝑎𝑥−2𝑎𝑘_𝑖𝑡+𝜉₀,(𝑖=1,…,7), (25) where 𝑎 is an arbitrary constant,  𝑎_0𝑖,𝑎_1𝑖,𝑎_2𝑖,𝑎_3𝑖, and 𝑎_4𝑖 are defined as Cases 1–7, and 𝜔_𝑖 and 𝑘_𝑖 are defined as follows: 𝜔_𝑖𝑎=−²𝑎⁴_3𝑖−80𝑎_1𝑖𝑎_3𝑖𝑎²_4𝑖−128𝑎_0𝑖𝑎³_4𝑖48𝑎²_3𝑖𝑎_4𝑖𝛽+𝛼+,𝑘12_𝑖=𝜀4𝑎²𝑎⁴_3𝑖+16𝑎_1𝑖𝑎_3𝑖𝑎²_4𝑖−128𝑎_0𝑖𝑎³_4𝑖−4𝑎²_3𝑖𝑎_4𝑖𝛽3𝑎²_3𝑖𝑎_4𝑖,𝜀²=1,(𝑖=1,…,7). (26)

With the similar process,substituting (4), (5), (6), and (7) along with (8a)–(8d) and (17) into (14) separately yields ODEs; after solving the ODEs by Mathematica and Wu elimination, we can obtain the following solutions of (11) according to (4), (5), (12), (13), (15), and (17) and the solutions of (3) mentioned in [29]: 𝐸_𝑗(𝑥,𝑡)=±√2𝐵₀±6√2𝑎²𝑎_4𝑗𝜑²_𝑗𝜉_𝑗𝑒^{𝑖(𝑘𝑗𝑥+𝜔𝑗𝑡+𝜉00)},,𝑁𝑗=0,1,2,…_𝑗(𝑥,𝑡)=𝐵₀+6𝑎²𝑎_4𝑗𝜑²_𝑗𝜉_𝑗,𝜉_𝑗=𝑎𝑥−2𝑎𝑘_𝑗𝑡+𝜉₀,𝜔_𝑗=11244𝑎²𝑎_2𝑗−22𝐵₀,𝑘+12𝛼+𝛽_𝑗=𝜀4𝑎²𝑎_2𝑗−2𝐵₀−𝛽12,𝜀²=1. (27)

Remark 4. If we let 𝑎_01.1=1−𝑚²,  𝑎_21.1=2𝑚²−1,  𝑎_41.1=−𝑚²,  𝑎_11.1=𝑎_31.1=0, and 𝜑_1.1(𝜉_1.1)=cn𝜉_1.1, then we have 𝐸_1.1(𝑥,𝑡)=±√2𝐵₀∓6√2𝑎²𝑚²cn²𝜉_1.1𝑒^{𝑖(𝑘1𝑥+𝜔1𝑡+𝜉00)},𝑁_1.1(𝑥,𝑡)=𝐵₀−6𝑎²𝑚²cn²𝜉_1.1,𝜉_1.1=𝑎𝑥−2𝑎𝑘_1.1𝑡+𝜉₀,𝜔_1.1=11244𝑎²2𝑚²−1−22𝐵₀,𝑘+12𝛼+𝛽_1.1=𝜀4𝑎²2𝑚²−1−2𝐵₀−𝛽12,𝜀²=1. (28) If we let 𝑎=(1/2)𝑏₀,  𝑚=1, or 𝑎=(1/2)(1/2𝑚²−1)𝑏₀, then solution (𝐸_1.1(𝑥,𝑡),𝑁_1.1(𝑥,𝑡)) is in full agreement with the solution 𝐸₁,  𝑁₁ and 𝐸₂,  𝑁₂ mentioned in [21].

Remark 5. The seven types of explicit solutions we obtained here to (11) are not shown in the previous literature to our knowledge. They are new exact solutions of (11). Notice that sn𝜉→tanh𝜉,  cn𝜉→sech𝜉, and dn𝜉→sech𝜉 when the modulus 𝑚→1, and sn𝜉→sin𝜉,  cn𝜉→cos𝜉, and dn𝜉→1 when the modulus 𝑚→0. Solutions 𝐸_𝑖,𝑁_𝑖(𝑖=1,…,6) are degenerated to solitary wave solutions when the modulus 𝑚→1 and to triangular functions solutions when the modulus 𝑚→0.

We can give the numerical simulation of 𝐸₆ to show their physical properties (see Figures 1, 2, and 3).

(a)

(b)

Figure 1 

The real part of 𝐸6 when 𝑎=𝛼=1,𝛽=−1,𝜉0=0, and 𝑚=0.9 and a plane graph when 𝑡=0.

(a)

(b)

Figure 2 

The imaginary part of 𝐸6 when 𝑎=𝛼=1,𝛽=−1,𝜉0=0, and 𝑚=0.9 and a plane graph when 𝑡=0.

(a)

(b)

Figure 3 

The modulus of 𝐸6 when 𝑎=𝛼=1,  𝛽=−1,  𝜉0=0, and 𝑚=0.9 and a plane graph when 𝑡=0.

4. Conclusion

In this paper, we succeed to propose a general algebraic method approach for finding new exact solutions of the nonlinear evolution equations. By using this method and computerized symbolic computation, we have found abundant new exact solutions for the coupled Schrödinger-Boussinesq equations (11). More importantly, our method is much simpler and powerful to find new solutions to various kinds of nonlinear evolution equations, such as KdV equation, Boussinesq equation, and zakharov equation. We believe that this method should play an important role for finding exact solutions in the mathematical physics.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 61070231), the Outstanding Personal Program in Six Fields of Jiangsu (Grant no. 2009188), the Graduate Student Innovation Project of Jiangsu Province (Grant no. CXLX13_673), and the General Program of Innovation Foundation of Nanjing Institute of Technology (Grant no. CKJB201218).