PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of springeropenLink to Publisher's site
Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 314.
Published online 2017 December 21. doi:  10.1186/s13660-017-1587-5
PMCID: PMC5740214

Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations

Abstract

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the exp(−ϕ(z))-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

Keywords: nonlinear evolution equations, symmetry, exp(−ϕ(z))-expansion method, complex method, exact solutions, meromorphic function

Introduction

In 1998, Yu et al. [1] extended the Bogoyavlenskii Schiff equation

ut+Φ(u)us=0,Φ(u)=x2+4u+2uxx1,
1

to the (3 + 1)-dimensional NLEE

(4ut+Φ(u)us)x+3uyy=0,Φ(u)=x2+4u+2uxx1.
2

Setting u: = ux, equation (2) is changed into the (3 + 1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation

uxxxs − 4uxt + 4uxuxs + 2uxxus + 3uyy = 0.
3

The generalized (3 + 1)-dimensional Zakharov-Kuznetsov (gZK) equation is given by

a1u2uxa2uxxxa3uxyya4uxssa5uuxa6uxxtut = 0.
4

Here ai (i = 1, 2, …, 6) are arbitrary constants.

We note that equation (4) includes many famous NLEEs as its special cases. For instance, if a1a3a4a6 = 0, then equation (4) is the Korteweg-de Vries equation [2, 3]. If a2a4a5 = 0, then equation (4) is the (2 + 1) dimensional ZK-MEW equation [4]. If a3a4a6 = 0, then equation (4) is the Gardner equation [5]. If a4a5a6 = 0, then equation (4) is the modified Zakharov-Kuznetsov equation [6].

In recent years, it has aroused widespread interest in the study of NLEEs [713]. Equations (3) and (4) are very meaningful higher-dimensional NLEEs which can describe many dynamic processes and important phenomena in engineering and physics. The YTSF equation is a mostly used model for investigating the dynamics of solitons and nonlinear waves in fluid dynamics, plasma physics and weakly dispersive media [13]. Zakharov and Kuznetsov [14] proposed the ZK equation to describe nonlinear ion-acoustic waves in a plasma comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Many physical phenomena, in the purely dispersive limit, are governed by this type of equation, such as the long waves on a thin liquid film [15], the Rossby waves in a rotating atmosphere [16], and the isolated vortex of drift waves in a three-dimensional plasma [17]. The gZK equation is of a generalized setting of ZK equation. Seeking exact solutions of NLEEs is an interesting and significant subject. Over the past few years, many powerful methods for constructing the solutions of NLEEs have been used, for instance, the Bäcklund transform method [18], direct algebraic method [19], modified simple equation method [20], Lie group method [21, 22], exp(−ϕ(z))-expansion method [8, 9, 23, 24], and so on. Recently, Yuan et al. [2527] introduced the complex method to find the exact solutions of NLEEs in mathematical physics. In this paper, we study symmetries, symmetry reduction of the two higher-dimensional NLEEs, and then we obtain their exact solutions via the exp(−ϕ(z))-expansion method and complex method.

Description of the methods

Description of the exp(−ϕ(z))-expansion method

Suppose that a nonlinear partial differential equation (PDE) is given by

P(uuxuyutuxxuyyutt, …) = 0, 
5

where P is a polynomial of an unknown function u(xyt) and its derivatives in which nonlinear terms and highest order derivatives are involved. The main steps of this method are given in the following.

Step 1. Substituting the traveling wave transform

u(xyt) = w(z),  zkxlyrt

into equation (5) converts it to the following ordinary differential equation (ODE):

F(wwww, …) = 0, 
6

in which F is a polynomial of w(z) and its derivatives, while :=ddz.

Step 2. Assume that equation (6) has the following traveling wave solution:

w(z)=j=0nCj(exp(ϕ(z)))j,
7

where Cj (0 ≤ j ≤ n) are constants to be determined, such that Cj ≠ 0 and ϕϕ(z) satisfies the ODE as follows:

ϕ(z) = exp(−ϕ(z)) + μexp(ϕ(z)) + δ.
8

Equation (8) has the following solutions.

When δ2 − 4μ > 0, μ ≠ 0,

ϕ(z)=ln((δ24μ)tanh(δ24μ2(z+c)δ)2μ),
9

ϕ(z)=ln((δ24μ)coth(δ24μ2(z+c)δ)2μ).
10

When δ2 − 4μ < 0, μ ≠ 0,

ϕ(z)=ln((4μδ2)tan((4μδ2)2(z+c)δ)2μ),
11

ϕ(z)=ln((4μδ2)cot((4μδ2)2(z+c)δ)2μ).
12

When δ2 − 4μ > 0, μ = 0, δ ≠ 0,

ϕ(z)=ln(δexp(δ(z+c))1).
13

When δ2 − 4μ = 0, μ ≠ 0, δ ≠ 0,

ϕ(z)=ln(2(δ(z+c)+2)δ2(z+c)).
14

When δ2 − 4μ = 0, μ = 0, δ = 0,

ϕ(z) = ln(zc).
15

Here Cn ≠ 0, δ, μ are constants that will be determined later and c is an arbitrary constant. We take the homogeneous balance between nonlinear terms and highest order derivatives of equation (6) to determine the positive integer n.

Step 3. Substituting equation (7) into equation (6) and accounting the function exp(−ϕ(z)), we obtain a polynomial of exp(−ϕ(z)). Equating all the coefficients of the same power of exp(−ϕ(z)) to zero yields a set of algebraic equations. By solving the algebraic equations, we get the values of Cn ≠ 0, δ, μ, and then we substitute them into equation (7) along with equations (9)-(15) to complete the determination of the solutions of equation (5).

Description of the complex method

Let m ∈ ℕ: = {1, 2, 3, …}, rj ∈ ℕ = ℕ ∪ {0}, j = 0, 1, …, m, r = (r0r1, …, rm), and

Kr[w](z):=j=0m[w(j)(z)]rj,

then d(r):=j=0mrj is the degree of Kr[w]. Let the differential polynomial be defined by

F(w,w,,w(m)):=rJarKr[w],

where J is a finite index set, and ar are constants, then  deg F(ww, …, w(m)): = maxrJ{d(r)} is the degree of F(ww, …, w(m)).

Consider the following differential equation:

F(ww, …, w(m)) = cwnd
16

where n ∈ ℕ, c ≠ 0, d are constants.

Set pq ∈ ℕ, and the meromorphic solutions w of equation (16) have at least one pole. If equation (16) has exactly p distinct meromorphic solutions, and their multiplicity of the pole at z = 0 is q, then equation (16) is said to satisfy the pq condition. It might not be easy to show that the pq condition of equation (16) holds, so we need the weak pq condition as follows.

Inserting the Laurent series

w(z)=λ=qβλzλ,βq0,q>0,
17

into equation (16), we can determine exactly p different Laurent singular parts:

λ=q1βλzλ,

then equation (16) is said to satisfy the weak pq condition.

Given two complex numbers ν1, ν2 such that Imν1ν2>0, and let L be the discrete subset L[2ν1, 2ν2]: = {νν = 2a1ν1 + 2a2ν2a1a2 ∈ ℤ}, and L is isomorphic to ℤ × ℤ. Let the discriminant Δ=Δ(b1,b2):=b1327b22 and

ln=ln(L):=νL{0}1νn.

A meromorphic function ℘(z): = ℘(zg2g3) with double periods 2ν1, 2ν2, which satisfies the following equation:

(℘(z))2 = 4℘(z)3 − g2℘(z)−g3

in which g2 = 60l4, g3 = 140l6, and Δ(g2g3) ≠ 0, is called the Weierstrass elliptic function and satisfies an addition formula [28] as follows:

(zz0)=14[(z)+(z0)(z)(z0)]2(z)(z0).

If a meromorphic function g is a rational function of z, or a rational function of eαz, α ∈ ℂ, or an elliptic function, then we say that g belongs to the class W.

In 2009, Eremenko et al. [29] studied the mth-order Briot-Bouquet equation (BBEq)

F(w,w(m))=j=0nFj(w)(w(m))j=0,

where Fj(w) are constant coefficients polynomials, m ∈ ℕ. For the mth-order BBEq, we have the following lemma.

Lemma 2.1

([28, 30, 31])

Let mnph ∈ ℕ,  deg F(ww(m)) < n, and a mth-order BBEq

F(ww(m)) = cwnd

satisfies the weak pq condition, then the meromorphic solutions w ∈ W. Supposing for some values of the parameters the solution w exists, then any other meromorphic solutions will be one parameter family w(zz0), z0 ∈ ℂ. In addition, every elliptic solution w with a pole at z = 0 is expressed as

w(z)=i=1h1j=2q(1)jβij(j1)!dj2dzj2(14[(z)+Di(z)Bi]2(z))+i=1h1βi12(z)+Di(z)Bi+j=2q(1)jβhj(j1)!dj2dzj2(z)+β0,
18

where βij are determined by (17), i=1hβi1=0 and Di2=4Bi3g2Big3.

Every rational function solution w: = R(z) is expressed as

R(z)=i=1hj=1qβij(zzi)j+β0,
19

which has h (≤p) distinct poles of multiplicity q.

Every simply periodic solution w: = R(ϑ) is a rational function of ϑeαz (α ∈ ℂ), and is expressed as

R(ϑ)=i=1hj=1qβij(ϑϑi)j+β0,
20

which has h (≤p) distinct poles of multiplicity q.

By the above definitions and lemma, we now present the complex method.

Step 1. Insert the transformation T:u(xyt) → w(z) defined by (xyt) → z into a given PDE to yield a nonlinear ODE.

Step 2. Insert (17) into the ODE to determine whether the weak pq condition holds.

Step 3. Insert the indeterminate solutions introduced in Lemma 2.1 into the ODE, and then get meromorphic solutions of the ODE with a pole at z = 0.

Step 4. Obtain meromorphic solutions w(z − z0) by Lemma 2.1 and the addition formula.

Step 5. Inserting the inverse transformation T−1 into the meromorphic solutions, we get the exact solutions for the original PDE.

Symmetries and symmetry reduction

Symmetries

In order to find the symmetry σσ(xystu) of equation (4), we set

σauxbuycusduteuf
21

where u is the solution of equation (4), a, b, c, d, e, f are unknown functions of real variables x, y, s, t. According to Lie group analysis [21, 22], σ satisfies

σta1σ2uxa1u2σxa2σxxxa3σxyya4σxssa5σuxa5uσxa6σxxt = 0.
22

Substituting equation (21) into equation (22), we have a new differential equation, where

a2uxxx = −a1u2ux − a3uxyy − a4uxss − a5uux − a6uxxt − ut.
23

By equation (21), equation (22) and equation (23), we have

a=c5,b=(c2s+c3),c=(c4a4a3c2y),d=c1,e=0,f=0,
24

where ci (i = 2, 3, 4, 5) are real constants. Substituting equations (24) into equation (21), we achieve the symmetry of the gZK equation,

σ=c5ux+(c2s+c3)uy+(c4a4a3c2y)us+c1ut.
25

In order to find the symmetry σσ(xystu) of equation (3), we set

σauxbuycusduteuf.
26

Here u is the solution of equation (3), a, b, c, d, e, f are unknown functions of real variables x, y, s, t. According to Lie group analysis, σ satisfies

σxxxs − 4σxt + 4uxσxs + 4uxsσx + 2uxxσs + 2usσxx + 3σyy = 0.
27

Substituting equation (26) into equation (27), we have a new differential equation, where

uxxxs = 4uxt − 4uxuxs − 2uxxus − 3uyy.
28

By equation (26), equation (27) and equation (28), we have

a=c1x+c2,b=c3y+c4,c=(2c33c1)s+ρ(t),d=(2c3c1)t+c5,e=c1,f=ρ(t)x+23ρ(t)y2+τ(t)y+ψ(t),
29

where ci (i = 1, 2, …, 5) are real constants, ρ(t), τ(t), ψ(t) are arbitrary real functions of t. Substituting equations (29) into equation (26), we achieved the symmetry of YTSF equation

σ=(c1x+c2)ux+(c3y+c4)uy+((2c33c1)s+ρ(t))us+((2c3c1)t+c5)ut+c1u+ρ(t)x+23ρ(t)y2+τ(t)y+ψ(t).
30

Symmetry reduction

By solving the characteristic equation (25) of σ

dxc5=dyc2s+c3=dsc4a4a3c2y=dtc1=du0,
31

we find different symmetry reduced equations. Without loss of generality, we have two reduced equations as follows.

Setting c1c3c4c5 = 0, c2 = 1, we have the first similarity solution of equation (4)

uφ(ξη), 
32

where ξxt, η=y22a3+s22a4. Substituting equation (32) into equation (4), we have the first symmetry reduced equation of equation (4)

φξa1φ2φξ + (a2a3)φξξξ + 2φξηηa5φφξ = 0.
33

Setting c1c2 = 0, c3c4c5 = 1, solving σ = 0, we have the second similarity solution of equation (4)

uφ(ξη), 
34

where ξxy, ηs. Substituting equation (34) into equation (4), we have the second symmetry reduced equation of equation (4)

a1φ2φξ + (a2a3)φξξξa4φξηηa5φφξ = 0.

By solving the characteristic equation (30) of σ

dxc1x+c2=dyc3y+c4=ds(2c33c1)s+ρ(t)=dt(2c3c1)t+c5=duc1u+ρ(t)x+23ρ(t)y2+τ(t)y+ψ(t),
35

we obtain symmetry reduction of equation (3). Without loss of generality, we have two reduced equations as follows.

Setting c1c3c4 = 0, c2c5 = 1, ρ(t) = 1, solving σ = 0, we have the first similarity solution of equation (3)

uφ(ξηy)−∫(τ(t)yψ(t)) dt
36

where ξx − t, ηs − t. Substituting equation (36) into equation (3), we have the first symmetry reduced equation of equation (3)

φξξξη + 4φξξ + 4φξη + 4φξφξη + 2φξξφη + 3φyy = 0.
37

Setting c1c2c3c5 = 0, c4 = 1, ρ(t) = τ(t) = 0, solving σ = 0, we have the second similarity solution of equation (3)

uφ(xst)−ψ(t)y.
38

Substituting equation (38) into equation (3), we have the second symmetry reduced equation of equation (3)

φxxxs + 4φxφxs + 2φxxφs − 4φxt = 0.

Exact solutions

Exact solutions of gZK equation via the exp(−ϕ(z))-expansion method

Substituting the traveling wave transform

φ(ξη) = w(z),  zkξlη

into equation (33), then integrating it with respect to z, we obtain

((a2+a3)k2+2l2)w+w+a52w2+a13w3γ=0,
39

where γ is the integration constant which can be determined later.

Taking the homogeneous balance between w and w3 in equation (39) yields

w(z) = C0C1exp(−ϕ(z)), 
40

where C1 ≠ 0, C0 are constants to be determined, whereas δ and μ are arbitrary constants.

Substitute w, w2, w3, w into equation (39) and equate the coefficients of exp(−ϕ(z)) to zero, then

13a1C03+12a5C02+C0+2C1l2δμ+C1k2a2δμ+C1k2a3δμγ=0,C1a2k2δ2+C1a3k2δ2+2C1l2δ2+2C1a2k2μ+2C1a3k2μ+C02C1a1+4C1l2μ+C0C1a5+C1=0,12a5C12+a1C0C12+6C1l2δ+3C1k2a2δ+3C1k2a3δ=0,4C1l2+13a1C13+2C1k2a2+2C1k2a3=0.

Solving the above algebraic equations, we obtain

γ=2a1((δ24μ)(a2k2+a3k2+2l2)2)((δ24μ)(a2k2+a3k2+2l2)+1)6a1,C1=6(a2k2+a3k2+2l2)a1,C0=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a1,
41

where μ and δ are arbitrary constants.

Substituting equations (41) into equation (40) yields

w(z)=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a1+6(a2k2+a3k2+2l2)a1exp(ϕ(z)).
42

We apply equation (9) to equation (15) into equation (42), respectively, to get traveling wave solutions of the gZK equation as follows.

When δ2 − 4μ > 0, μ ≠ 0,

w11(z)=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a16(a2k2+a3k2+2l2)a12μ(δ24μ)tanh(δ24μ2(z+c)+δ),w12(z)=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a16(a2k2+a3k2+2l2)a12μ(δ24μ)coth(δ24μ2(z+c)+δ).

When δ2 − 4μ < 0, μ ≠ 0,

w13(z)=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a1+6(a2k2+a3k2+2l2)a12μ(4μδ2)tan(4μδ22(z+c)δ),w14(z)=6a1(a2k2+a3k2+2l2)δ2a1(2(δ24μ)(a2k2+a3k2+2l2))2a1+6(a2k2+a3k2+2l2)a12μ(4μδ2)cot(4μδ22(z+c)δ).

When δ2 − 4μ > 0, μ = 0, δ ≠ 0,

w15(z)=6a1(a2k2+a3k2+2l2)δ2a1(2δ2(a2k2+a3k2+2l2))2a1+6(a2k2+a3k2+2l2)a1δexp(δ(z+c))1.

When δ2 − 4μ = 0, μ ≠ 0, δ ≠ 0,

w16(z)=3(a2k2+a3k2+2l2)2a1δ1a16(a2k2+a3k2+2l2)a1δ2(z+c)2(δ(z+c)+2).

When δ2 − 4μ = 0, μ = 0, δ = 0,

w17(z)=1a1+6(a2k2+a3k2+2l2)a11z+c.

Exact solutions of gZK equation via the complex method

Inserting (17) into equation (39) we have p = 2, q = 1, β1=±6(a2k2+a3k2+2l2)a1, β0=a52a1, β1=a5224a126a1a2k2+a3k2+2l2, β2=12a12γa53+6a1a548a12(a2k2+a3k2+2l2) and β3 is an arbitrary constant.

Therefore, equation (39) is a second order BBEq and satisfies the weak 〈2, 1〉 condition. Hence, by Lemma 2.1, we see that meromorphic solutions of equation (39) belong to W. We will show meromorphic solutions of equation (39) in the following.

By (19), we infer that the indeterminate rational solutions of equation (39) are

R1(z)=β11z+β12zz1+β10,

with a pole at z = 0.

Substituting R1(z) into equation (39), we have

R1,1(z)=±6(a2k2+a3k2+2l2)a11za52a1,

where a52=4a1 and 9a1γ2 = 1;

R1,2(z)=±6(a2k2+a3k2+2l2)a1(1z1zz11z1)a52a1,

where k=4a1z12a52z1248l2a124a1(a2+a3) and γ=(a536a1a5+(a524a1)33)z13.

So the rational solutions of equation (39) are

wr,1(z)=±6(a2k2+a3k2+2l2)a11zz0a52a1

and

wr,2(z)=±6(a2k2+a3k2+2l2)a1(1zz01zz0z11z1)a52a1,

where z0 ∈ ℂ, z1 ≠ 0. a52=4a1, 9a1γ2 = 1 in the former case, or k=4a1z12a52z1248l2a124a1(a2+a3), γ=(a536a1a5+(a524a1)33)z13 in the latter case.

To obtain simply periodic solutions, let ϑeαz, and substitute wR(ϑ) into equation (39), then

((a2+a3)k2+2l2)α2(ϑR+ϑ2R)+R+a52R2+a13R3γ=0.
43

Substituting

R2(z)=β21ϑ1+β22(ϑϑ1)+β20

into equation (43), we obtain

R2,1(z)=±6(a2k2+a3k2+2l2)a1α(1ϑ1+12)a52a1
44

and

R2,2(z)=±6(a2k2+a3k2+2l2)a1α(1ϑ1ϑ1ϑϑ1ϑ1+12(ϑ11))a52a1,
45

where γ=a5(a526a1)12a12, l=12α4a1a522a1k2α2(a2+a3)a1 in the former case, or γ=3z1(z1+1)(4a1a52)32(z12+10z1+1)32a12+a5(a526a1)12a12, k=(4a1a524a1l2α2)(z12+1)+2(a524a120a1l2α2)z12a1(z12+10z1+1)(a2+a3)α2 in the latter case.

Inserting ϑeαz into equation (44) and equation (45), we can get simply periodic solutions to equation (39) with a pole at z = 0,

ws0,1(z)=±3(a2k2+a3k2+2l2)2a1αcothα2za52a1,ws0,2(z)=±3(a2k2+a3k2+2l2)2a1α(cothα2zcothα2(zz1)cothα2z1)a52a1,

where γ=a5(a526a1)12a12, l=12α4a1a522a1k2α2(a2+a3)a1 in the former case, or γ=3z1(z1+1)(4a1a52)32(z12+10z1+1)32a12+a5(a526a1)12a12, k=(4a1a524a1l2α2)(z12+1)+2(a524a120a1l2α2)z12a1(z12+10z1+1)(a2+a3)α2 in the latter case.

So simply periodic solutions of equation (39) are

ws,1(z)=±3(a2k2+a3k2+2l2)2a1αcothα2(zz0)a52a1

and

ws,2(z)=±3(a2k2+a3k2+2l2)2a1α(cothα2(zz0)cothα2(zz0z1)cothα2z1)a52a1,

where z0 ∈ ℂ, z1 ≠ 0. l=12α4a1a522a1k2α2(a2+a3)a1, γ=a5(a526a1)12a12 in the former case, or k=(4a1a524a1l2α2)(z12+1)+2(a524a120a1l2α2)z12a1(z12+10z1+1)(a2+a3)α2, γ=3z1(z1+1)(4a1a52)32(z12+10z1+1)32a12+a5(a526a1)12a12 in the latter case.

From (18), we have the indeterminate relations for the elliptic solutions of equation (39) with a pole at z = 0,

wd1(z)=β12(z)+D1(z)B1+β0,

where D12=4B13g2B1g3. Considering the results obtained above, we infer that β0=a52a1, g3 = 0, D1B1 = 0. So we obtain

wd1(z)=±3(a2k2+a3k2+2l2)2a1(z)(z)a52a1,

where g3 = 0.

Thus, the elliptic function solutions of equation (39) are

wd(z)=±3(a2k2+a3k2+2l2)2a1(zz0,g2,0)(zz0,g2,0)a52a1,

where z0 ∈ ℂ, g3 = 0, g2 is arbitrary. Applying the addition formula, we can rewrite it as

wd(z)=±3(a2k2+a3k2+2l2)2a1(+E)(4E2+(4E2g2)+2FEg2)((12E2g2)+4E33Eg2)+(43+12E23g2Eg2)Fa52a1,

where g3 = 0, F2 = 4E3 − g2E, E and g2 are arbitrary.

Exact solutions of YTSF equation via the exp(−ϕ(z))-expansion method

Substituting the traveling wave transform

φ(ξηy) = v(z),  zkξlηry

into equation (37), then integrating it with respect to z, we obtain

k3lv + (4k2 + 4kl + 3r2)v + 3k2l(v)2γ = 0, 
46

where γ is the integration constant which can be determine later.

Setting wv, equation (46) becomes

k3lw + (4k2 + 4kl + 3r2)w + 3k2lw2γ = 0.
47

Taking the homogeneous balance between w and w2 in equation (47) yields

w(z) = C0C1exp(−ϕ(z)) + C2(exp(ϕ(z)))2
48

where C2 ≠ 0, Ci (i = 0, 1, 2) are constants to be determined, whereas δ and μ are arbitrary constants.

Substitute w, w2, w into equation (47) and equate the coefficients of exp(−ϕ(z)) to zero, then

k3lC1δμ+2k3lC2μ2+3k2lC02+4C0k2+4C0kl+3C0r2+γ=0,C1lk3δ2+6C2lk3δμ+2C1lk3μ+6C0C1lk2+4C1k2+4C1lk+3C1r2=0,4C2lk3δ2+3C1lk3δ+8C2lk3μ+6C0C2lk2+3C12lk2+4C2k2+4C2lk+3C2r2=0,10C2lk3δ+6C1C2lk2+2C1lk3=0,3C22lk2+6C2lk3=0.

Solving the above algebraic equations, we obtain

γ=(δ24μ)2l2k6(4lk+4k2+3r2)212k2l,C2=2k,C1=2kδ,C0=lk3δ2+8lk3μ+4lk+4k2+3r26k2l,
49

where μ and δ are arbitrary constants.

Substituting equations (49) into equation (48), yields

w(z)=lk3δ2+8lk3μ+4lk+4k2+3r26k2l2kδexp(ϕ(z))2k(exp(ϕ(z)))2.
50

We apply equation (9) to equation (15) into equation (50), respectively, to get traveling wave solutions of the YTSF equation as follows.

When δ2 − 4μ > 0, μ ≠ 0,

w21(z)=lk3δ2+8lk3μ+4lk+4k2+3r26k2l+4kδμ(δ24μ)tanh(δ24μ2(z+c)+δ)8kμ2((δ24μ)tanh(δ24μ2(z+c)+δ))2,w22(z)=lk3δ2+8lk3μ+4lk+4k2+3r26k2l+4kδμ(δ24μ)coth(δ24μ2(z+c)+δ)8kμ2((δ24μ)coth(δ24μ2(z+c)+δ))2.

When δ2 − 4μ < 0, μ ≠ 0,

w23(z)=lk3δ2+8lk3μ+4lk+4k2+3r26k2l4kδμ(δ24μ)tan(δ24μ2(z+c)δ)8kμ2((δ24μ)tan(δ24μ2(z+c)δ))2,w24(z)=lk3δ2+8lk3μ+4lk+4k2+3r26k2l4kδμ(δ24μ)cot(δ24μ2(z+c)δ)8kμ2((δ24μ)cot(δ24μ2(z+c)δ))2.

When δ2 − 4μ > 0, μ = 0, δ ≠ 0,

w25(z)=lk3δ2+4lk+4k2+3r26k2l2kδ2exp(δ(z+c))12kδ2(exp(δ(z+c))1)2.

When δ2 − 4μ = 0, μ ≠ 0, δ ≠ 0,

w26(z)=12lk3μ+4lk+4k2+3r26k2l+kδ3(z+c)(δ(z+c)+2)kδ4(z+c)22((δ(z+c)+2))2.

When δ2 − 4μ = 0, μ = 0, δ = 0,

w27(z)=4lk+4k2+3r26k2l2k(z+c)2.

Exact solutions of YTSF equation via the complex method

Inserting (17) into equation (47) we have p = 1, q = 2, β−2 = −2k, β−1 = 0, β0=4lk+4k2+3r26k2l, β1 = 0, β2=16k4+32lk3+(16l212lγ+24r2)k2+24lkr2+9r4120k5l2, and β3 is an arbitrary constant.

Therefore, equation (47) is a second order BBEq and satisfies the weak 〈1, 2〉 condition. Hence, by Lemma 2.1, we see that meromorphic solutions of equation (47) belong to W. We will show meromorphic solutions of equation (47) in the following.

By (19), we deduce the indeterminacy rational solutions of equation (47) are

R1(z)=β32z2+β31z+β30,

with a pole at z = 0.

Substituting R1(z) into equation (47), we get the following form:

R1(z)=2kz24lk+4k2+3r26k2l,

where γ=16k4+32lk3+(16l2+24r2)k2+24lkr2+9r412k2l.

So the rational solutions of equation (47) are

wr(z)=2k(zz0)24lk+4k2+3r26k2l,

where γ=16k4+32lk3+(16l2+24r2)k2+24lkr2+9r412k2l, z0 ∈ ℂ.

To obtain simply periodic solutions, let ϑeαz, and substitute wR(ϑ) into equation (47), then we get

k3lα2(ϑRϑ2R) + (4k2 + 4kl + 3r2)R + 3k2lR2γ = 0.
51

Substituting

R2(z)=β42(ϑ1)2+β41(ϑ1)+β40,

into equation (51), we obtain

R2(z)=2kα2(ϑ1)22kα2(ϑ1)kα264lk+4k2+3r26k2l,
52

where γ=(4lk+4k2+3r2)2(lα2k3)212k2l. Substituting ϑeαz into equation (52), we can obtain simply periodic solutions of equation (47),

ws0(z)=2kα2(eαz1)22kα2(eαz1)kα264lk+4k2+3r26k2l=2kα2eαz(eαz1)2kα264lk+4k2+3r26k2l=kα22coth2αz2+kα234lk+4k2+3r26k2l,

with a pole at z = 0.

Therefore the simply periodic solutions of equation (47) are

ws(z)=kα22coth2α(zz0)2+kα234lk+4k2+3r26k2l,

where γ=(4lk+4k2+3r2)2(lα2k3)212k2l, z0 ∈ ℂ.

From (18), we can express the elliptic solutions of equation (47) as

wd0(z) = β−2℘(z) + β0

with a pole at z = 0.

Substituting wd0(z) into equation (47), we obtain

wd0(z)=2k(z)4lk+4k2+3r26k2l,

where g2=16k4+32lk3+(16l212lγ+24r2)k2+24lkr2+9r412k6l2, g3 is arbitrary.

Therefore, the elliptic solutions of equation (47) are

wd(z)=2k(zz0)4lk+4k2+3r26k2l,

in which z0 ∈ ℂ. Applying the addition formula, we can rewrite it as

wd(z)=2k((z)+14((z)+C(z)D)2)+2kD4lk+4k2+3r26k2l,

where g2=16k4+32lk3+(16l212lγ+24r2)k2+24lkr2+9r412k6l2, C2 = 4D3 − g2D − g3, g3 is arbitrary.

Comparison

Implementing the exp(−ϕ(z))-expansion method, we found seven solutions for the gZK and YSFT equation, respectively. Using the complex method, we found five solutions for the gZK equation and three solutions for the YSFT equation. Rational solutions w17(z) and w27(z) are obtained via the exp(−ϕ(z))-expansion method, and Wr,1(z) and Wr(z) are obtained via the complex method. If we let c = −z0, then w17(z) is equivalent to Wr,1(z), and w27(z) is equivalent to Wr(z). For getting rational solutions, these two methods are in good agreement. Rational solutions Wr,2(z) and simply periodic solutions Ws,2(z) and Ws(z) are new and cannot be degenerated successively through elliptic function solutions. From the results, we can find more solutions by the exp(−ϕ(z))-expansion method, whereas we can obtain elliptic function solutions just by the complex method. These two methods are very useful tools in finding the exact solutions of NLEEs.

Computer simulations

In this section, we illustrate some results by the computer simulations. We carry out further analysis to the properties of simply periodic solutions Ws,2(z) and Ws(z) as in Figures 1 and and2.2.

  1. By employing the complex method, we are capable to obtain simply periodic solutions Ws,1(z) and Ws,2(z) of the gZK equation. The solutions Ws,1(z) and Ws,2(z) come from hyperbolic function. Figure 1 shows the shape of solutions Ws,2(z) for k = 1, l = 1, α = 1, a1 = −6, a2 = 1, a3 = 1, a5 = −24, and z1 = 1 within the interval −2π ≤ ξη ≤ 2π. Note that they have two distinct generation poles which are showed by Figure 1.
  2. By using the complex method, we achieve to obtain simply periodic solutions Ws(z) of the YSTF equation. The solutions Ws(z) are in terms of the hyperbolic function solution. The solutions Ws(z) in Figure 2 of the YSTF equation are represented the singular soliton solution for the parameters k = 1, l = 1, r = 1, α = 1 and y = 0 within the interval −2π ≤ ξη ≤ 2π.

Figure 1
The solution of the gZK equation corresponding to Ws,2(z) . (a) z0 = −8, (b) z0 = 0, (c) z0 = 8.
Figure 2
The solution of the YSTF equation corresponding to Ws(z) . (a) z0 = −8, (b) z0 = 0, (c) z0 = 8.

Conclusions

In this article, we utilize Lie group analysis to obtain symmetries and symmetry reduction for two higher-dimensional NLEEs. In this way, we can reduce the dimension of the NLEEs, which is relevant in the fields of mathematical physics and engineering. Five types of explicit function solutions are constructed by the exp(−ϕ(z))-expansion method and complex method. It demonstrates these methods are very efficient and powerful to seek the exact solutions of NLEEs. We can apply the idea of this study to other NLEEs.

Acknowledgements

This work was supported by the NSF of China (11271090, 11701111); the NSF of Guangdong Province (2016A030310257); the Foundation for Young Talents in Educational Commission of Guangdong Province (2015KQNCX116). Thanks to the Joint PHD Program of Guangzhou University and Curtin University. Thanks to the editors and referees with their very useful suggestions and helpful comments.

Authors’ contributions

Authors’ contributions

All authors typed, read and approved the final manuscript.

Notes

Competing interests

The authors declare that they have no competing interests.

Footnotes

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Yongyi Gu, moc.361@iygnoyugdg.

Jianming Qi, moc.361@ujdsgnimnaijiq.

References

1. Yu S, Toda K, Sasa N, Fukuyama T. N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 + 1)(3+1) dimensions. J. Phys. A, Math. Gen. 1998;31(14):3337–3347. doi: 10.1088/0305-4470/31/14/018. [Cross Ref]
2. Mei JQ, Zhang HQ. New soliton-like and periodic-like solutions for the KdV equation. Appl. Math. Comput. 2005;169:589–599.
3. Cui AG, Li HY, Zhang CY. A splitting method for shifted skew-Hermitian linear system. J. Inequal. Appl. 2016;2016:160. doi: 10.1186/s13660-016-1105-1. [Cross Ref]
4. Khalique CM, Adem KR. Exact solutions of the (2 + 1)(2+1) dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis. Math. Comput. Model. 2011;54(1-2):184–189. doi: 10.1016/j.mcm.2011.01.049. [Cross Ref]
5. Wazwaz AM. New solitons and kink solutions for the Gardner equation. Commun. Nonlinear Sci. Numer. Simul. 2007;12:1395–1404. doi: 10.1016/j.cnsns.2005.11.007. [Cross Ref]
6. Tascan F, Bekir A, Koparan M. Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Numer. Simul. 2009;14(5):1810–1815. doi: 10.1016/j.cnsns.2008.07.009. [Cross Ref]
7. Roshid HO, Alam MN, Akbar MA. Traveling wave solutions for fifth order (1 + 1)(1+1)-dimensional Kaup-Keperschmidt equation with the help of Exp(−ϕη)Exp(ϕη)-expansion method. Walailak J. Sci. Technol. 2015;12(11):1063–1073.
8. Roshid HO, Alam MN, Akbar MA, Islam R. Traveling wave solutions of the simplified MCH equation via Exp(−Φ(ξ))Exp(Φ(ξ))-expansion method. Br. J. Math. Comput. Sci. 2015;5(5):595–605. doi: 10.9734/BJMCS/2015/10800. [Cross Ref]
9. Roshid HO, Kabir MR, Bhowmik RC, Datta BK. Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))Exp(ϕ(ξ))-expansion method. SpringerPlus. 2014;3:692. doi: 10.1186/2193-1801-3-692. [PMC free article] [PubMed] [Cross Ref]
10. Roshid HO, Roshid MM, Rahman N, Pervin MR. New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation. Propuls. Power Res. 2017;6(1):49–57. doi: 10.1016/j.jppr.2017.02.002. [Cross Ref]
11. Roshid HO. Novel solitary wave solution in shallow water and ion acoustic plasma waves in-terms of two nonlinear models via MSE method. J. Ocean Eng. Sci. 2017;2(3):196–202. doi: 10.1016/j.joes.2017.07.004. [Cross Ref]
12. Roshid HO, Rashidi MM. Multi-soliton fusion phenomenon of Burgers equation and fission, fusion phenomenon of Sharma-Tasso-Olver equation. J. Ocean Eng. Sci. 2017;2(2):120–126. doi: 10.1016/j.joes.2017.04.001. [Cross Ref]
13. Roshid HO. Lump solutions to a (3 + 1)(3+1)-dimensional potential-Yu-Toda-Sasa-Fukuyama (YTSF) like equation. Int. J. Appl. Comput. Math. 2017;3:1455–1461. doi: 10.1007/s40819-017-0430-5. [Cross Ref]
14. Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Sov. Phys. JETP. 1974;39:285–288.
15. Toh S, Iwasaki H, Kawahara T. Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion. Phys. Rev. A. 1989;40:5472–5475. doi: 10.1103/PhysRevA.40.5472. [PubMed] [Cross Ref]
16. Petviashvihi VI. Red spot of Jupiter and the drift soliton in plasma. JETP Lett. 1980;32:619–622.
17. Nozaki K. Vortex solutions of drift waves and anomalous diffusion. Phys. Rev. Lett. 1981;46:184–187. doi: 10.1103/PhysRevLett.46.184. [Cross Ref]
18. Li B, Chen Y, Zhang H. Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order. Phys. Lett. A. 2002;305(6):377–382. doi: 10.1016/S0375-9601(02)01515-3. [Cross Ref]
19. Taghizadeh N, Neirameh A. New complex solutions for some special nonlinear partial differential systems. Comput. Math. Appl. 2011;62(4):2037–2044. doi: 10.1016/j.camwa.2011.06.046. [Cross Ref]
20. Jawad AJM, Petkovic MD, Biswas A. Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 2010;217(2):869–877.
21. Tian C. Lie Group and Its Applications in Partial Differential Equations. Beijing: Higher Education Press; 2001.
22. Liu H, Li J, Zhang Q. Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. J. Comput. Appl. Math. 2009;228(1):1–9. doi: 10.1016/j.cam.2008.06.009. [Cross Ref]
23. Islam SMR, Khan K, Akbar MA. Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations. SpringerPlus. 2015;4:124. doi: 10.1186/s40064-015-0893-y. [PMC free article] [PubMed] [Cross Ref]
24. Khan K, Akbar MA. The exp(−ϕ(ξ))exp(ϕ(ξ))-expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation. Int. J. Dyn. Syst. Differ. Equ. 2014;5(1):72–83.
25. Yuan WJ, Xiao B, Wu YH, Qi JM. The general traveling wave solutions of the Fisher type equations and some related problems. J. Inequal. Appl. 2014;2014:500. doi: 10.1186/1029-242X-2014-500. [Cross Ref]
26. Yuan WJ, Huang ZF, Fu MZ, Lai JC. The general solutions of an auxiliary ordinary differential equation using complex method and its applications. Adv. Differ. Equ. 2014;2014:147. doi: 10.1186/1687-1847-2014-147. [Cross Ref]
27. Yuan WJ, Meng FN, Huang Y, Wu YH. All traveling wave exact solutions of the variant Boussinesq equations. Appl. Math. Comput. 2015;268:865–872.
28. Lang S. Elliptic Functions. 2. New York: Springer; 1987.
29. Eremenko A, Liao LW, Ng TW. Meromorphic solutions of higher order Briot-Bouquet differential equations. Math. Proc. Camb. Philos. Soc. 2009;146:197–206. doi: 10.1017/S030500410800176X. [Cross Ref]
30. Yuan WJ, Shang YD, Huang Y, Wang H. The representation of meromorphic solutions to certain ordinary differential equations and its applications. Sci. Sin., Math. 2013;43(6):563–575. doi: 10.1360/012012-159. [Cross Ref]
31. Kudryashov NA. Meromorphic solutions of nonlinear ordinary differential equations. Commun. Nonlinear Sci. Numer. Simul. 2010;15(10):2778–2790. doi: 10.1016/j.cnsns.2009.11.013. [Cross Ref]

Articles from Springer Open Choice are provided here courtesy of Springer