Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

Transcription

1 Huang and Xie Advances in Difference Equations (208) 208:29 R E S E A R C H Open Access Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics Bo Huang,2* and Shaofen Xie,2 * Correspondence: huangbo0407@26.com LMIB-School of Mathematics and Systems Science, Beihang University, Beijing, 009, China 2 Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Nanning, , China Abstract This paper deals with the analytical solutions for two models of special interest in mathematical physics, namely the (2 + )-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation and the (3 + )-dimensional generalized Boiti-Leon-Manna-Pempinelli equation. Using a modified version of the Fan sub-equation method, more new exact traveling wave solutions including triangular solutions, hyperbolic function solutions, Jacobi and Weierstrass elliptic function solutions have been obtained by taking full advantage of the extended solutions of the general elliptic equation, showing that the modified Fan sub-equation method is an effective and useful tool to search for analytical solutions of high-dimensional nonlinear partial differential equations. Keywords: mathematical physics; traveling wave solutions; Fan sub-equation method; evolution equations Introduction Nonlinear partial differential equations (NLPDEs) are important mathematical models to describe physical phenomena. They are also an important field in the contemporary study of nonlinear physics, especially in soliton theory. The research on the explicit solution and integrability is helpful in clarifying the movement of the matter under nonlinear interaction and plays an important role in scientifically explaining the physical phenomena. See, for example, fluid mechanics, plasma physics, optical fibers, solid state physics, chemical kinematic, chemical physics and geochemistry. In the present paper, we will consider the following two high-dimensional nonlinear equations: u xt + βu x u xy + δu y u xx + u xxxy =0 (.) and u xxxy 3(u x u y ) x 2u yt +3u yz = 0. (.2) Equation (.) [] is the (2 + )-dimensional generalized Calogero-Bogoyavlenskii-Schiff (GCBS) equation, where the parameter,β, δ are arbitrary nonzero constants. It contains the following famous NLPDEs: The Author(s) 208. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Huangand XieAdvances in Difference Equations (208) 208:29 Page 2 of 5 (i) Bogoyavlenskii-Schiff equation 4u xt +4u x u xy +2u y u xx + u xxxy = 0. (.3) (ii) Breaking soliton equation u xt 4u x u xy 2u y u xx + u xxxy = 0. (.4) (iii) Calogero-Bogoyavlenskii-Schiff equation 4u xt +8u x u xy +4u y u xx + u xxxy = 0. (.5) So it is very meaningful to further study equation (.). In particular, we can also get the solutions of (.3)-(.5). Equation (.2) is a generalization of theboiti-leon-manna-pempinelli (GBLMP) equation. If we take z = t,equation(.2) is reduced to the following BLMP equation [2, 3]: u xxxy + u yt 3u xx u y 3u x u xy =0. Searching for traveling wave solutions of NLPDEs acts a pivotal part in the study of nonlinear physical phenomena. In the past decades, much effort has been spent on the construction of exact solutions of NLPDEs, and many powerful methods have been developed such as inverse scattering transform [4], Bäcklund and Darboux transform [5 7], Hirota bilinear method [8], tanh-function method and extended tanh-function method [9 ], Fansub-equationmethod[2, 3], G /G method [4, 5], F-expansion method [6]andso on. Among these methods, the Fan sub-equation method not only gives a unified formation to construct various traveling wave solutions, but also provides a guideline to classify the various types of traveling wave solutions according to five parameters. Yomba [7] and Soliman and Abdou [8] extended the Fan sub-equation method to show that the general elliptic equation can be degenerated in some special conditions to the Riccati equation, first-kind elliptic equation, and generalized Riccati equation. Recently, Zhang and Peng [9] proposed and used a modification of the Fan sub-equation method with symbolic computation to construct a series of traveling wave solutions for the (3 + )-dimensional potential YTSF equation. These methods show the effectiveness of the modified Fan sub-equation method in handling the solution process of NLPDEs. Motivated by the work described in the above three papers, in this paper, using Zhang and Peng s modification of the Fan sub-equation method, we construct the exact traveling wave solutions for the above mentioned two high-dimensional nonlinear equations by taking full advantage of the extended solutions of the following general elliptic equation: ( ) dϕ(ξ) 2 = h 0 + h ϕ(ξ)+h 2 ϕ 2 (ξ)+h 3 ϕ 3 (ξ)+h 4 ϕ 4 (ξ). (.6) dξ In some special cases, when h 0 0,h 0,h 2 0,h 3 0andh 4 0, there may exist three parameters r, p and q such that ( ) dϕ(ξ) 2 = h 0 + h ϕ(ξ)+h 2 ϕ 2 (ξ)+h 3 ϕ 3 (ξ)+h 4 ϕ 4 (ξ)= ( r + pϕ(ξ)+qϕ 2 (ξ) ) 2. (.7) dξ

3 Huangand XieAdvances in Difference Equations (208) 208:29 Page 3 of 5 Equation (.7) is satisfied only if the following relations hold: h 0 = r 2, h =2rp, h 2 =2rq + p 2, h 3 =2pq, h 4 = q 2. (.8) When h 2 =0andh 0 0,h 0,h 3 0andh 4 0, there may exist three parameters r, p and q such that ( ) dϕ(ξ) 2 = h 0 + h ϕ(ξ)+h 3 ϕ 3 (ξ)+h 4 ϕ 4 (ξ)= ( r + pϕ(ξ)+qϕ 2 (ξ) ) 2. (.9) dξ Equation (.9) requiresforits existencethe followingrelations: h 0 = r 2, h =2rp, h 3 =2pq, h 4 = q 2, p 2 = 2rq, rq < 0. (.0) Thus, for (.7)and(.9), the general elliptic equation is reduced to the generalized Riccati equation [20]. When h 0 = h = 0, the general elliptic equation is reduced to the auxiliary ordinary equation [2] ( ) dϕ(ξ) 2 = h 2 ϕ 2 (ξ)+h 3 ϕ 3 (ξ)+h 4 ϕ 4 (ξ). (.) dξ When h = h 3 = 0, the general elliptic equation is reduced to the elliptic equation ( ) dϕ(ξ) 2 = h 0 + h 2 ϕ 2 (ξ)+h 4 ϕ 4 (ξ). (.2) dξ Equation (.2) includes thericcati equation ( ) dϕ(ξ) 2 = ( A + ϕ 2 (ξ) ) 2, (.3) dξ when h 0 = A 2, h 2 =2A, h 4 =, and solutions of (.3) can be deduced from those of (.2) inthespecificcasewherethemodulusm of the Jacobi elliptic functions (solutions of (.2)) is driven to and 0. When h 2 = h 4 = 0, the general elliptic equation is reduced to the following: ( ) dϕ(ξ) 2 = h 0 + h ϕ(ξ)+h 3 ϕ 3 (ξ). (.4) dξ The rest of this paper is organized as follows. In Section 2, we describe and further develop Zhang and Peng s modification of the Fan sub-equation method by making good use of the extended solutions of the general elliptic equation. In Section 3, we apply the modified Fan sub-equation method to solve the above mentioned two high-dimensional nonlinear equations, and more new exact traveling wave solutions are explicitly obtained. In Section 4, some conclusions are given.

4 Huangand XieAdvances in Difference Equations (208) 208:29 Page 4 of 5 2 Modification of the Fan sub-equation method For a given NLPDE with independent variables x =(x 0 = t, x, x 2,...,x m ) and dependent variable u, F(u, u t, u xi, u xi x j,...), (2.) where F is in general a polynomial function of its argument, and the subscripts denote the partial derivatives. By using the traveling wave transformation, (2.)possesses the following ansatz: u = u(ξ), ξ = m k i x i, (2.2) i=0 where k i (i =0,...,m) are undetermined constants. Substituting (2.2) into(2.) yieldsan ODE Q ( u, u (r), u (r+),... ) = 0, (2.3) where u (r) = dr u dξ r, u (r+) = dr+ u dξ r+, r, and r is the least order of derivatives in the equation. To keep the solution process as simple as possible, the function Q should not be a total ξ-derivative of another function. Otherwise, taking integration with respect to ξ further reduces the transformed equation. Further introduce u (r) (ξ)=υ(ξ)= n i ϕ i + 0, (2.4) i= where ϕ = ϕ(ξ) satisfies(.6), while 0, i (i =,2,...,n) are constants to be determined later. To determine u(ξ) explicitly, onemaytakethefollowingfoursteps: Step. Determine the value of n by balancing the highest-order nonlinear term(s) and the highest-order partial derivative in (2.3). Step 2. With the aid of Maple, substitute (2.4) alongwith(.6) into(2.3) toderivea polynomial in ϕ. Set all the coefficients of the polynomial to zero to derive a set of algebraic equations for k i (i =0,...,m), 0 and i (i =,...,n). Step 3. Apply Wu-elimination method [22] to solve the above algebraic equations derived in Step 2, which yields the values of k i (i =0,...,m), 0 and i (i =,...,n). Step 4. Use the results obtained in the above steps to derive a series of fundamental solutions ϕl I, ϕii l, ϕiii l, ϕl IV and ϕl V to (.6) depending on the different values chosen for h 0, h, h 2, h 3 and h 4 [7, 2]. The superscripts I, II, III, IV and V determine the group of solutions, while the subscript l determines the rank of the solution. Then obtain exact solutions of (2.) by integrating each of the obtained fundamental solutions υ(ξ) with respect to ξ, r times: u(ξ)= ξ ξr ξ2 υ(ξ ) dξ dξ r dξ r + r d j ξ r j, (2.5) j= where d j (j =,2,...,r) are arbitrary constants.

5 Huangand XieAdvances in Difference Equations (208) 208:29 Page 5 of 5 3 Application to evolution equations from mathematical physics 3. GCBS equation In thissubsection we consider the (2+)-dimensional GCBSequation (.). Using transformation (2.2) (herewedenoteξ = ax + by ωt), we reduce (.) intoanodeequation in the form aωu +()a 2 bu u + a 3 bu (4) = 0. (3.) Integrating (3.)once with respect to ξ and setting the integration constant to zero yields aωu + 2 ()a2 b ( u ) 2 + a 3 bu (3) = 0. (3.2) Further setting r =andu = υ,wehave aωυ + 2 ()a2 b(υ) 2 + a 3 bυ = 0. (3.3) According to Step, we get 2n = n +2;hencen =2.Wethensupposethat(3.3) hasthe formal solution: υ = 2 ϕ 2 + ϕ + 0. (3.4) Substituting (3.4) along with(.6) into(3.3) and collecting all terms with the same order of ϕ together, the left-hand side of (3.3) is converted into a polynomial in ϕ. Setting each coefficient of the polynomial to zero, we derive a set of algebraic equations for a, b, ω, 0,, 2 as follows: ϕ 0 : a 3 b h +4a 3 b 2 h 0 + a 2 bβ a 2 bδ 0 2 2aω 0 =0, ϕ : a 3 b h 2 +3a 3 b 2 h + a 2 bβ 0 + a 2 bδ 0 aω =0, ϕ 2 :3a 3 b h 3 +8a 3 b 2 h 2 +2a 2 bβ a 2 bβ 2 +2a 2 bδ a 2 bδ 2 2aω 2 =0, (3.5) ϕ 3 :2a 3 b h 4 +5a 3 b 2 h 3 + a 2 bβ 2 + a 2 bδ 2 =0, ϕ 4 :2a 3 b 2 h 4 + a 2 bβ a 2 bδ 2 2 =0. Solving the set of algebraic equations by using Maple, we can obtain many kinds of solutions depending on the special values chosen for h i (i =0,...,4). Case I. When h 0 = r 2, h =2rp, h 2 =2rq + p 2, h 3 =2pq, h 4 = q 2,from(3.5)wehave 2 = 2aq2, = 2apq, 0 = 2aqr, ω = (p2 4qr)a 2 b (3.6) and 2 = 2aq2, 0 = 2a(p2 +2qr) = 2apq,, ω = (p2 4qr)a 2 b. (3.7)

6 Huangand XieAdvances in Difference Equations (208) 208:29 Page 6 of 5 We, therefore, have υ = 2aq2 ϕ2 2apq ϕ 2aqr, ω = (p2 4qr)a 2 b (3.8) and υ = 2aq2 ϕ2 2apq ϕ 2a(p2 +2qr) For simplification, in the rest of this paper, we introduce that, ω = (p2 4qr)a 2 b. (3.9) M = 2 p2 4qr, N = 2 4qr p2. Note that ϕ is one of the twenty four ϕ I l (l =,2,...,24). For example, if we select l =7, by using (2.5) we can obtain the following traveling wave solutions: u (ξ)= 6aN [ ( ) ( )] tan 2 Nξ cot 2 Nξ + d, (3.0) where ξ = ax + by + 4N2 a 2 b t, d is an arbitrary constant; u 2 (ξ)= 6aN [ ( ) ( )] tan 2 Nξ cot 2 Nξ + 8aN 2 ξ + d, (3.) where ξ = ax + by 4N2 a 2 b t, d is an arbitrary constant. More fundamental solutions Using ansatz (3.9), similarly, we canobtain aseries ofthe fundamentalsolutions for (.) as follows. Family : When p 2 4qr >0, u I 2aM,2 (ξ)= tanh(mξ) 8aM2 ξ + d ; u I 8a 2,2 (ξ)= M2 ξ + 2aM coth(mξ)+d ; u I 3,2 (ξ)= a [ ±2pi arctan ( e 2Mξ ) 6pi arctan ( sinh(2mξ) ) +8M 2 ξ 2Mi + cosh(2mξ) u I 4,2 (ξ)= a ] 2Mtanh(2Mξ) + d ; [ 2p arctan ( e 2Mξ ) 6p ln ( tanh(mξ) ) +8M 2 ξ 2Mcoth(2Mξ) 2M sinh(2mξ) [ ( ) ( u I 6aM 5,2 (ξ)= tanh 2 Mξ + coth 2 Mξ u I 6,2 (ξ)= ] + d ; )] 8aM2 + d ; a [ 6ABM 2 sinh(2mξ)ξ 2B()[A sinh(2mξ)+b]

7 Huangand XieAdvances in Difference Equations (208) 208:29 Page 7 of 5 +24AM ( A + A 2 + B 2) sinh(2mξ)+48abm cosh 2 (Mξ) 6B 2 M 2 ξ ] + d, u I 7,2 (ξ)= a [ 8(A B) 2 (A B)()[2A cosh 2 M 2 ξ (Mξ) A + B] +2AM(A B) sinh(2mξ)+24am B 2 A 2 cosh 2 (Mξ) 6AM 2 (A B) cosh 2 (Mξ)ξ ] + d, where A and B are two nonzero real constants satisfying B 2 A 2 >0; where u I 8,2 (ξ)= 2a[L (ξ) tanh 2 ( 2 Mξ)+L 2(ξ) tanh( 2 Mξ)+L (ξ)] p()[p tanh 2 ( 2 Mξ) 4M tanh( 2 Mξ)+p] + d, L (ξ)=4m 2 p 2 ξ, L 2 (ξ)= 6pM 3 ξ 48Mqr; u I 9,2 (ξ)= 8a[M3 tanh 2 ( 2 Mξ)ξ M2 p tanh( 2 Mξ)ξ +3qr tanh( 2 Mξ)+M3 ξ] ()[M tanh 2 ( 2 Mξ) p tanh( 2 Mξ)+M] + d ; u I 0,2 (ξ)= 96aqrM ()(2M ± pi)[(2m ± pi) tanh(mξ) 2Mi p] 8M2 aξ + d ; u I,2 (ξ) 48aqrM + = p()[ p tanh(mξ)+2m] 8M2 aξ + d ; u I,2 (ξ) 24aqr = ()[2M tanh(mξ) p] 8M2 aξ + d ; u I 2,2 (ξ)=ui 9,2 (ξ). Family 2: When p 2 4qr <0, u I 3,2 (ξ)= 2aN tan(nξ)+8an 2 ξ + d ; u I 2 4,2 (ξ)=2an 8aN cot(nξ)+ ξ + d ; u I 5,2 (ξ)= 2aN [ ] 8aN 2 tan(2nξ) ± sec(2nξ) + ξ + d ; u I 6,2 (ξ)=2an [ ] 8aN 2 cot(2nξ) ± csc(2nξ) + ξ + d ; [ ( ) ( )] u I 6aN 7,2 (ξ)= cot 2 Nξ tan 2 Nξ + 8aN 2 ξ + d ; u I 8,2 (ξ)= a [ 8ABN 2 sin(2nξ)ξ +24ABN cos 2 (Nξ) B()[A sin(2nξ)+b] +2AN ( A ± A 2 B 2) sin(2nξ)+8b 2 N 2 ξ ] + d, u I 9,2 (ξ)= a [ ±24AN A2 B (A B)()[A cos(2nξ)+b] 2 cos 2 (Nξ) 2AN(A B) sin(2nξ)+6an 2 (A B) cos 2 (Nξ)ξ 8N 2 (A B) 2 ξ ] + d,

8 Huangand XieAdvances in Difference Equations (208) 208:29 Page 8 of 5 where A and B are two nonzero real constants satisfying A 2 B 2 >0; u I 20,2 (ξ)= u I 2,2 (ξ)= u I 22,2 (ξ)= u I 23,2 (ξ) + = 24aqr ()[2N tan(nξ)+p] + 8aN 2 ξ + d ; 48aqrN p()[p tan(nξ) 2N] + 8aN 2 ξ + d ; 96aqrN ()(2N p)[(2n p) tan(nξ) ± 2N + p] + 8aN 2 ξ + d ; 48aqrN p()[p tan(nξ) 2N] + 8aN 2 ξ + d ; u I 23,2 (ξ) 24aqr = ()[2N tan(nξ)+p] + 8aN 2 ξ + d ; u I 24,2 (ξ)=ui 2,2 (ξ); where ξ = ax + by 4N2 a 2 b t, d is an arbitrary constant. Case II. When h 0 = r 2, h =2rp, h 2 =0,h 3 =2pq, h 4 = q 2 and p 2 = 2rq, from(3.5) we have 2 = 2aq2, = 2apq, 0 =0, ω = 3a2 bp 2 (3.2) and 2 = 2aq2, = 2apq, 0 = 6ap2, ω = 3a2 bp 2. (3.3) We, therefore, have υ = 2aq2 ϕ2 2apq ϕ, ω bp 2 = 3a2 (3.4) and υ = 2aq2 ϕ2 2apq ϕ + 6ap2, ω = 3a2 bp 2. (3.5) Note that ϕ is one of the twelve ϕl II (l =,2,...,2). For example, if we select l =2,by using (2.5), we can obtain the following traveling wave solutions: u 3 (ξ)= 8a 6qr ( ) ln coth 6qrξ a ( ) 6qr coth 6qrξ 2 ( ) ln sinh 6qrξ 2 4a(2 6qr ± 3 2qr 3p) ± 6ap 2qr ξ + d, (3.6)

9 Huangand XieAdvances in Difference Equations (208) 208:29 Page 9 of 5 where ξ = ax + by + 3a2 bp 2 t, d is an arbitrary constant; u 4 (ξ)= 8a 6qr ( ) ln coth 6qrξ a ( ) 6qr coth 6qrξ 2 ( ) ln sinh 6qrξ 2 4a(2 6qr ± 3 2qr 3p) + 6ap(± 2qr + p) ξ + d, (3.7) where ξ = ax + by 3a2 bp 2 t, d is an arbitrary constant. More fundamental solutions Using ansatz (3.5), if we select l = 5, 8, similarly, we can obtain more fundamental solutions for (.) as follows. u II 5,2 (ξ)= 2ap ln + 3a 6qr 2a 2qr ( ) ( coth 6qrξ tanh 6qrξ) 4 4 [ ( ) ( )] tanh 6qrξ + coth 6qrξ 4 4 [ ( )] ln sinh 6qrξ + 6a(4qr ± p 2qr + p 2 ) ξ + d ; 2 u II 8,2 (ξ)= 2aM(ξ) qr()(tanh 2 Y +2 3 tanh Y +) + d, where M(ξ)= 6 qr(p + 2qr) ( +tanh 2 Y ) ln [ ( qr tanh 2 Y +2 3 tanh Y + )] 2 3qr(p + 2qr) tanh Y ln [ ( qr tanh 2 Y +2 3 tanh Y + )] 24 3qr(p + 2qr) tanh Y ln(cosh Y ) 3ln 2 qr(p + 2qr) ( tanh 2 Y + ) 2 6 ( p 2 p 2qr +4qr )( tanh 2 Y + ) Y 2 2 ( p 2 p 2qr +4qr ) tanh Y Y 6 3 ln 2 qr(p + 2qr) tanh Y 2 6qr tanh Y 2 qr(p + 2qr) ( tanh 2 Y + ) ln(cosh Y ), with Y = 4 6qrξ, ξ = ax + by 3a 2 bp 2 t, d is an arbitrary constant. Case III. If h 0 = h =0,h 2, h 3, h 4 are arbitrary constants, the system does not admit any solution of this group. Case IV. If h = h 3 =0,whenh 2 2 3h 0h 4 >0,from(3.5)wehave 2 = 2ah 4, =0, 0 = 4a( h 2 ± h 2 2 3h 0h 4 ), ω = ± 4a2 b h 2 2 3h 0h 4. (3.8)

10 Huangand XieAdvances in Difference Equations (208) 208:29 Page 0 of 5 We, therefore, have υ = 2ah 4 ϕ2 + 4a( h 2 ± h 2 2 3h 0 h 4 ), ω = ± 4a2 b h 2 2 3h 0h 4. (3.9) Note that ϕ is one of the sixteen ϕl IV (l =0,...,6).Forexample,ifweselect l =4,then h 0 = m 2,h 2 =2 m 2, h 4 =.Using(2.5), we can obtain the following traveling wave solutions: u 5 (ξ)= 2a E(φ, m)+ 4a(m2 2± m4 m 2 +) ξ + d, (3.20) where ξ = ax + by 4a2 b m 4 m 2 + t, E(φ, m) is the second kind elliptic integral, φ = arcsin(snξ) is the argument of ξ, d is an arbitrary constant. In the limit case when m, the solutions (3.20)become u 5 (ξ)= 2a 4a( ± ) tanh(ξ)+ ξ + d, (3.2) where ξ = ax + by 4a2 b t, d is an arbitrary constant. When m 0, the solutions (3.20) become a rational solution u 52 (ξ)= 4a( ± ) ξ + d, (3.22) where ξ = ax + by 4a2 b t, d is an arbitrary constant. When h 2 2 3h 0h 4 <0,from(3.5)wehave 2 = 2ah 4, =0, 0 = 4a(h 2 i 3h 0 h 4 h 2 2 ), ω = ± 4a2 b 3h 0 h 4 h 2 2 i. Note that in this case ϕ may be one of ϕ IV l (l =, 2, 3, 5). For example, if we select l =, i.e., h 4 = m 2, h 2 = ( m 2 ), h 0 =, which satisfy h 2 2 3h 0h 4 <0for < m. Using (2.5), we can obtain the following traveling wave solutions: u IV 2a (ξ)= E(φ, m) 4a(2 + m2 i 5m2 m 4 ) ξ + d, (3.23) where ξ = ax + by i 4a2 b 5m 2 m 4 t, E(φ, m) is the second kind elliptic integral, φ = arcsin(snξ) is the argument of ξ, d is an arbitrary constant. In the limit case when m, the solutions (3.23)become u IV 2a, (ξ)= tanh ξ 4a(3 i 3) ξ + d, (3.24) where ξ = ax + by i 4a2 b 3 t, d is an arbitrary constant.

11 Huangand XieAdvances in Difference Equations (208) 208:29 Page of 5 Case V. If h 2 = h 4 =0,h 3 >0,h 0, h are arbitrary constants, from (3.5)wehave 2 =0, = 3ah 3, 0 = ± 3h h 3 i, ω = ± a2 b 3h h 3 i. (3.25) We, therefore, have υ = 3ah 3 ϕ ± 3h h 3 i, ω = ± a2 b 3h h 3 i. (3.26) Substituting the corresponding solutions [7]of(.6)into(3.26)andusing(2.5), we can obtain a Weierstrass elliptic function solution u 6 (ξ)= 3ah ξ ( ) 3 h3 2 ξ, g 2, g 3 dξ ± 3h h 3 i ξ + d, (3.27) where ξ = ax + by a2 b 3h h 3 i t, g 2 = 4h h 3, g 3 = 4h 0 h 3, d is an arbitrary constant. Remark 3. In this case (in the condition of (.4)), the solution is a Weierstrass elliptic doubly periodic type solution only for h 3 > 0, so we just consider the case h 3 >0.Ifh <0, then imaginary number i appears again in (3.25), (3.26) and(3.27), and then other steps canbeconsideredinasimilarway. 3.2 GBLMP equation In this subsection we consider the (3 + )-dimensional GBLMP equation (.2). Note that since each term of equation (.2) contains a partial derivative with respect to y, forthe convenience of calculation, we denote transformation (2.2) byξ = ax+ y + cz ωt, then we reduce (.2) into an ODE equation in the form a 3 u (4) 6a 2 u u +2ωu +3cu = 0. (3.28) Integrating (3.28) oncewithrespecttoξ and setting the integration constant to zero yields a 3 u (3) 3a 2( u ) 2 +(2ω +3c)u = 0. (3.29) Further setting r =andu = υ,wehave a 3 υ 3a 2 (υ) 2 +(2ω +3c)υ = 0. (3.30) According to Step, we get 2n = n +2;hencen =2.Wethensupposethat(3.30) hasthe formal solution υ = 2 ϕ 2 + ϕ + 0. (3.3) Substituting (3.3) alongwith(.6) into(3.30) and collecting all terms with the same order of ϕ together, the left-hand side of (3.30)is convertedinto a polynomialin ϕ. Setting

12 Huangand XieAdvances in Difference Equations (208) 208:29 Page 2 of 5 each coefficient of the polynomial to zero, we derive a set of algebraic equations for a, c, ω, 0,, 2 as follows: ϕ 0 : a 3 h +4a 3 2 h 0 6a c 0 +4ω 0 =0, ϕ : a 3 h 2 +3a 3 2 h 6a c +2ω =0, ϕ 2 :3a 3 h 3 +8a 3 2 h 2 2a a c 2 +4ω 2 =0, (3.32) ϕ 3 :2a 3 h 4 +5a 3 2 h 3 6a 2 2 =0, ϕ 4 :2a 3 2 h 4 a =0. Solving the set of algebraic equations by using Maple, similarly, we can get the solutions 2,, 0 and ω depending on the special values chosen for h i (i = 0,...,4). Note that ϕ(ξ) intervening in (3.3) maybeoneofϕl I (ξ) orϕii l (ξ) orϕiv l (ξ) orϕl V (ξ), using (2.5) we can have many kinds of solutions. For the limit of length, we only consider Case I here. When h 0 = r 2, h =2rp, h 2 =2rq + p 2, h 3 =2pq, h 4 = q 2,from(3.32)wehave 2 =2aq 2, =2apq, 0 =2aqr, ω = 2 a3 p 2 +2a 3 qr 3 2 c (3.33) and 2 =2aq 2, =2apq, 0 = 3 ap aqr, ω = 2 a3 p 2 2a 3 qr 3 (3.34) 2 c. We, therefore, have υ =2aq 2 ϕ 2 +2apqϕ +2aqr, ω = 2 a3 p 2 +2a 3 qr 3 2 c (3.35) and υ =2aq 2 ϕ 2 +2apqϕ + 3 ap aqr, ω = 2 a3 p 2 2a 3 qr 3 c. (3.36) 2 Note that ϕ is one of the twenty four ϕ I l (l =,2,...,24).Forexample,ifweselect l =5, by using (2.5) we can obtain the following traveling wave solutions: [ ( ) ( )] u 7 (ξ)= am tanh 2 Mξ + coth 2 Mξ + d, (3.37) where ξ = ax + y + cz +( 2 a3 p 2 2a 3 qr c)t, d is an arbitrary constant; [ ( ) ( )] u 8 (ξ)= am tanh 2 Mξ + coth 2 Mξ + 4aM2 ξ + d, (3.38) 3 where ξ = ax + y + cz ( 2 a3 p 2 2a 3 qr 3 2 c)t, d is an arbitrary constant.

13 Huangand XieAdvances in Difference Equations (208) 208:29 Page 3 of 5 More fundamental solutions Using ansatz (3.36), if we select l = 2, 8, 20, similarly, we can obtain more fundamental solutions for (.2) as follows. uu I 2,2 (ξ)= 2aM coth(mξ)+ 3 ap2 ξ 4 3 aqrξ + d ; uu I 8,2 (ξ)= an(ξ) 6pM(p tanh 2 Y 4M tanh Y + p) + d, (3.39) where N(ξ)= 4p 2( p 2 +8qr ) tanh 2 Y Y +6pM ( p 2 +8qr ) tanh Y Y +96qrM 2 tanh Y 4p 2( p 2 +8qr ) Y, with Y = 2 Mξ; uu I 20,2 (ξ)= a [ 8N 3 tan(nξ)ξ 3(2N tan(nξ)+p) +4pN 2 ξ +2qr ] + d ; where ξ = ax + y + cz ( 2 a3 p 2 2a 3 qr 3 2 c)t, d is an arbitrary constant. Remark 3.2 All of the solutions presented in Section 3 have been checked with Maple 7 by putting them back into the original equations and the number of arbitrary constants has been reduced to a minimum. Remark 3.3 Most of the solutions presented in Section 3 (e.g., (3.6), (3.7), (3.20) and (3.27)) cannot be obtained by means of the Riccati equation expansion method [23] and its existing improvements like the one in [24].To thebestofourknowledge,theyhavenot been reported in literature. 4 Conclusions In this paper, we used and further developed Zhang and Peng s modification of the Fan sub-equation method, and many exact solutions of the (2+)-dimensional GCBS equation and the (3 + )-dimensional GBLMP equation were successfully found out. The obtained results show the effectiveness and advantages of the modified Fan sub-equation method in handling the solution process of high-dimensional NLPDEs. It is necessary to point out that in Step 3, to make the work feasible, how to choose appropriately variables in the ansatz would be the key step in the computation to find out a useful solution of the algebraic equations. It is interesting to note that research on solving fractional PDEs has attracted much attention recently, and there have been some new developments especially in solving time-fractional PDEs of the Fan sub-equations method (e.g., [25, 26]). However, searching for exact analytical solutions of nonlinear fractional PDEs is still on a preliminary stage. It is difficult to extend the methods for NLPDEs to fractional PDEs. Very recently, the known variable separation method has been extended to such fractional PDEs with initial

14 Huangand XieAdvances in Difference Equations (208) 208:29 Page 4 of 5 and boundary conditions [27]. So, extending the work in the present paper is worthy of study. Acknowledgements The first author wishes to thank Dongming Wang for his encouragement and profound concern. The authors also wish to thank the reviewers for their valuable comments that have helped to improve the presentation of the paper. Funding The project was supported by open fund of Guangxi Key laboratory of hybrid computation and IC design analysis (HCIC 20602). Competing interests The authors declare that they have no competing interests. Authors contributions The authors contributed equally to this paper. All authors read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 23 June 207 Accepted: 7 December 207 References. Toda, K, Yu, S, Fukuyama, F: The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2 + ) dimensions. Rep. Math. Phys. 44, (999) 2. Gilson, C, Nimmo, J-J: A (2 + )-dimensional generalization of the akns shallow-water wave-equation. Phys. Lett. A 80, (993) 3. Luo, L: New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation. Phys. Lett. A 375, (20) 4. Ablowitz, M, Clarkson, P: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (99) 5. Miurs, M: Bäcklund Transformation. Springer, Berlin (978) 6. Matveev, V: Darboux Transformation and Solitons. Springer, Berlin (99) 7. Fan, E: Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J. Phys. A, Math. Gen. 33, (2000) 8. Tam, H-W, Ma, W-X, Hu, X-B, Wang, D-L: The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited. J. Phys. Soc. Jpn. 69, (2000) 9. Parkes, E, Duffy, B: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 98, (996) 0. Fan, E: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, (2000). Lü, Z, Zhang, H: On a new modified extended tanh-function method. Commun. Theor. Phys. 39, (2003) 2. Fan, E, Dai, H: A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Comput. Phys. Commun. 53, 7-30 (2003) 3. Fan, E: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos Solitons Fractals 6, (2003) 4. Zayed, E, Gepreel, K: The (G /G)-expansion method for finding traveling wave solutions of nonlinear pdes in mathematical physics. J. Math. Phys. 50, (2009) 5. Zhang, Y: Solving STO and KD equations with modified Riemann-Liouville derivative using improved (G /G)-expansion function method. IAENG Int. J. Appl. Math. 45, 6-22 (205) 6. Wang, M, Li, X: Extended F-expansion and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A 343, (2005) 7. Yomba, E: The extended Fan s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations. Phys. Lett. A 336, (2005) 8. Soliman, A, Abdou, M: Exact travelling wave solutions of nonlinear partial differential equations. Chaos Solitons Fractals 32, (2007) 9. Zhang, S, Peng, A: A modification of fan sub-equation method for nonlinear partial differential equations. Int. J. Appl. Math. 44, 0-4 (204) 20. Xie, F, Zhang, Y, Lü, Z: Symbolic computation in non-linear evolution equation: application to (3 + )-dimensional Kadomtsev-Petviashvili equation. Chaos Solitons Fractals 24, (2005) 2. Sirendaoreji, Jiong, S: Auxiliary equation method for solving nonlinear partial differential equations. Phys. Lett. A 309, (2003) 22. Wu, W: Algorithms and Computation. Springer, Berlin (994) 23. Zhu, W, Ma, S, Fang, J, Ma, Z, Zhu, H: Fusion, fission, and annihilation of complex waves for the (2 + )-dimensional generalized Calogero-Bogoyavlenskii-Schiff system. Chin. Phys. B 23, (204) 24. Li, B, Chen, Y: Exact analytical solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation using symbolic computation.czechoslov.j.phys.54, (2004) 25. Zhang, S, Zhang, H-Q: Fractional sub-equation method and its applications to nonlinear fractional pdes. Phys. Lett. A 375, (20)

15 Huangand XieAdvances in Difference Equations (208) 208:29 Page 5 of Zhang, S, Cai, B, Xu, B: Variable separation method for nonlinear time fractional biological population model. Int. J. Numer. Methods Heat Fluid Flow 25, (205) 27. Zhang, S, Hong, S: Variable separation method for a nonlinear time fractional partial differential equation with forcing term. J. Comput. Appl. Math. (207).