New exact solutions for the (3 + 1)-dimensional potential-YTSF equation by symbolic calculation

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New exact solutions for the ( 3 + 1 ) -dimensional potential-YTSF equation by symbolic calculation

XIURONG GUO^1,2^,∗, JIANGEN LIU³, YUFENG ZHANG²and QINGBIAO WANG⁴

1Basic Courses, Shandong University of Science and Technology, Tai’an, 271019, Shandong, People’s Republic of China

2School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

3State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

4National Engineering Laboratory for Coalmine Backfilling Mining, Shandong University of Science and Technology, Tai’an, 271019, Shandong, People’s Republic of China

∗Corresponding author. E-mail: guoxiurong007@163.com

MS received 2 May 2018; revised 19 June 2018; accepted 26 June 2018; published online 2 January 2019 Abstract. In this paper, we employ the improved homoclinic test technique and the extended homoclinic test technique. With the help of the symbolic calculation and applying the improved methods, we solve the(3+1)- dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation to obtain some new periodic kink-wave, periodic soliton and periodic wave solutions.

Keywords. (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation; homoclinic test technique; periodic kink-wave solutions; periodic soliton solutions.

PACS Nos 02.30.Jr; 02.60.Cb; 04.20.Jb; 52.35.Mw

1. Introduction

In this paper, we consider the following (3+1)- dimensional dynamical model called the potential-Yu–

Toda–Sasa–Fukuyama (YTSF) equation, first intro- duced by Yuet al[1], that is

∂⁴

∂x³∂z+4∂

∂x

∂²

∂x∂z−4∂²

∂x∂t+2∂²

∂x²

∂

∂z +3∂²

∂y² =0, (1) where = (x,y,z,t).

The (3 + 1)-dimensional potential-YTSF equation has been well studied by many scholars. For exam- ple, some exact solutions including the exact solitary wave and periodic solutions, the non-travelling wave solutions, the new periodic soliton solutions, the conser- vation laws etc. were obtained and presented in [2–9].

Meanwhile, there are many other related works on the (3+1)-dimensional potential-YTSF equation and other research related to this topic (see [10–36]).

In this paper, we employ the improved homoclinic test technique and the extended homoclinic test

technique. With the help of the symbolic calculation and applying the improved methods, we want to solve the (3+1)-dimensional potential-YTSF equation. As a result, we obtain some new periodic kink-wave, periodic soliton and periodic wave solutions. To the best of our knowledge, these solutions have not been reported anywhere else. At the same time, this paper can also be regarded as a supplement to other associated works.

2. The improved extended homoclinic test technique

In this section, we search for the periodic kink-wave solutions to eq. (1) by using the improved extended homoclinic test technique.

Assume that

ξ =x +cz, (2)

then eq. (1) can be rewritten as c∂⁴

∂ξ⁴ +6c∂

∂ξ

∂²

∂ξ² −4∂²

∂ξ∂t +3∂²

∂y² =0. (3)

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23 Page 2 of 7 Pramana – J. Phys.(2019) 92:23 Equation (3) reduces to the following Hirota bilinear

form:

(c D_ξ⁴−4D_ξDt +3D²_y)f · f =0, (4) under the dependent variable transformation = 2(ln f)ξ.

Equation (4) leads to c(f_ξξξξ f −4f_ξξξ f_ξ +3f_ξξ²)

−4(f_ξt f − f_ξ ft)+3(fyyf − f_y²)=0, (5) where Dis a Hirota bilinear operator [37,38]

D^m_x Dⁿ_yf ·g= ∂

∂x − ∂

∂x m

∂

∂y − ∂

∂y n

× f(x,y,t)·g(x,y,t)|₍x,y,t)=(x,y,t). (6) In what follows, we apply the improved extended homoclinic test technique to find solutions to the following form:

f =exp(−p(ξ +αt+βy))+b0cos(p(ξ −αt)) +b1exp(p(ξ+αt+βy)), (7) where p, α and β are free values which are to be determined later. b0 = b0(t) and b1 = b1(t) are unknown functions in the time variablet.

Substituting eq. (7) into eq. (5) and then collecting the coefficients in front of the different exponential functions, we get

Case1

β²= 4p(α−c p²)+2c1

4p ,b₀ =0,b₁=c₂e^c¹^t

,

where p, α,c1,c2andcare arbitrary constants.

Case2

β² = 8α+4c p²

3 ,b₀²= 4α+8c p² α−c p² b1

,

where p, α,candb1are free parameters.

Using the dependent variable transformation = 2(ln f)ξ and Cases 1 and 2, we obtain the solutions of eq. (1) as follows:

1(x,y,z,t)=2−pe⁽¹²^c¹⁻^{c p}³⁻³⁴^β²^p⁾^t^−β^py⁻⁽^x⁺^cz⁾^p+c2pe⁽

12c1+c p3+3

4β2p)t+βpy+(x+cz)p

e⁽¹²^c¹^{−c p}³⁻³⁴^β²^p)t^−β^py−(x+cz)p+c2e⁽¹²^c¹^{+c p}³⁺³⁴^β²^p)t+βpy+(x+cz)p. (8) When c₁ = 0,c₂ = 1, the solitary wave solution is given by

1−1(x,y,z,t)=2p

tanh c p³+3 4β²p

t +βpy+ px+ pcz

. (9)

Whenc1 =0,c2 = −1, we obtain the solitary wave solution as

1−2(x,y,z,t)=2p

coth

c p³+ 3 4β²p

t

+βpy+px +pcz , (10)

wherec,pandβare free constants.

2(x,y,z,t)=2−pe⁻^p⁽^x⁺^cz^+α^t^+β^y⁾− pb0sin[p(x +cz−αt)]+pb1e^p⁽^x⁺^cz^+α^t^+β^y⁾

e⁻^p⁽^x⁺^cz^+α^t^+β^y⁾+b0cos[p(x+cz−αt)]+b1e^p⁽^x⁺^cz^+α^t^+β^y⁾ . (11)

Inserting b₁ = 1 into eq. (11), we get a periodic kink-wave solution of eq. (1) as follows:

2−1(x,y,z,t)=2p2 sinh[p(x +cz+αt+βy)] −b0sin[p(x+cz−αt)]

2 cosh[p(x +cz+αt+βy)] +b₀cos[p(x+cz−αt)]. (12)

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Figure 1. Characteristic form of the solitary wave solution (10) with parameters p =1,c =1, α =5,z = y =0: (a) 3D-plot and (b) 2D-contour.

Inserting b1 = −1 into eq. (11), we get a periodic kink-wave solution of eq. (1) as follows:

2−2(x,y,z,t)=2p2 cosh[p(x+cz+αt+βy)] −b0sin[p(x+cz−αt)]

2 sinh[p(x+cz+αt+βy)] +b0cos[p(x+cz−αt)], (13) wherec,p, αandβ are arbitrary constants.

Now, solutions (10) and (13) are plotted in three-dimensional (3D) and two-dimensional (2D) forms (see figures1and2).

Here, we point out that solution (11) has been obtained in [4]. Solution (8) has not been reported in other places.

3. The improved homoclinic test technique

In this section, we search for the periodic soliton solutions to eq. (1) by using the improved homoclinic test technique.

Figure 2. Characteristic form of the periodic solution (13) with parameters p = 1,c = 1, α = 2,z = y = 0: (a) 3D-plot and (b) 2D-contour.

We suppose that

ϑ =x +cz−αt, (14)

then eq. (3) can be changed into the form

c∂⁴

∂ϑ⁴ +3c ∂

∂ϑ

∂

∂ϑ 2

+4α∂²

∂ϑ² +3∂²

∂y² =0. (15) Equation (15) reduces to the following Hirota bilinear form:

(c D⁴_ϑ +4αD²_ϑ +3D²_y)f · f =0, (16) under the dependent variable transformation = 2(ln f)ϑ.

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23 Page 4 of 7 Pramana – J. Phys.(2019) 92:23 Next, we apply the improved homoclinic test

technique to find the solutions of the following form:

f =1+b1[exp(−I pϑ)+exp(I pϑ)] ·exp(ϒy+λ) +b2exp(2ϒy+2λ), (17) where p, ϒ andλare free values to be determined later.

b1 = b1(y)andb2 =b2(y)are unknown functions in the space variable y.

Substituting eq. (17) into eq. (16) and then collecting the coefficients in front of the different exponential functions so that they are equal to zero, we get Case1

ϒ = −1

2c1,p= −

√cα

c ,b1 =e⁻

(ϒ√√c+c2α)(c3−y),

b₂ =c₂e⁻²^ϒ^y

,

wherec, α,c1,c2andc3are arbitrary constants.

Case2

ϒ = −1

2c₁,p= −

√cα

c ,b₁ =e⁻

(ϒ√√c−2α) c (c₃−y), b2 =c2e⁻²^ϒ^y

,

wherec, α,c1,c2andc3are arbitrary constants.

Case3

ϒ = −1

2c1,p=

√cα

c ,b1 =e⁻

(ϒ√ c+2α)

√c (c3−y)

, b₂=c₂e⁻²^ϒ^y

,

wherec, α,c1,c2andc3are free parameters.

Case4

ϒ = −1

2c1,p=

√cα

c ,b1 =e⁻

(ϒ√√c−c2α)(c3−y)

, b₂=c₂e⁻²^ϒ^y

,

wherec, α,c1,c2andc3are free parameters.

Case5

b1 =c3e⁻⁽¹^/²⁾^c¹^y,b2 =c2e⁻⁽¹^/²⁾^c¹^y,

p² = 4α(b²₁−b2) c(4b₁²−b2)

,

where p, ϒ and c1 need to satisfy the constraint condition: 3c₁²+4(c p⁴−4αp²+3ϒ²+3c₁ϒ)=0.

Using the dependent variable transformation = 2(ln f)ϑ and Cases 1–5, we obtain solutions of eq. (1) as follows:

1(x,y,z,t)= − 4pe⁻⁽¹^/²⁾^c¹^c³^+λ+^ysin[p(x+cz−αt)]

2 e⁻^c¹^c³^+λ+^ycos[p(x +cz−αt)] +c2e²^λ+^c³+e^c³, (18) where p,c,c1,c2, =α/√

candc3are free parameters.

2(x,y,z,t)= − 4pe⁻⁽¹^/²⁾^c¹^c³^+λ+^c³sin[p(x+cz−αt)]

2 e^−(1/2)c¹^c³^+λ+c³cos[p(x+cz−αt)] +c2e^2λ+y+e^y, (19) where p,c,c₁,c₂, =α/√

candc₃are free parameters.

3(x,y,z,t)= − 4pe⁻⁽¹^/²⁾^c¹^c³^+λ+^ysin[p(x+cz−αt)]

2 e⁻⁽¹^/²⁾^c¹^c³^+λ+^ycos[p(x+cz−αt)] +c2e²^λ+^c³+e^c³, (20) where p,c,c1,c2, =α/√

4(x,y,z,t)= − 4pe^−(1/2)c¹^c³^+λ+c³sin[p(x+cz−αt)]

2 e⁻⁽¹^/²⁾^c¹^c³^+λ+^c³cos[p(x+cz−αt)] +c2e²^λ+^y+e^y, (21) where p,c,c1,c2, =α/√

5(x,y,z,t)= − 4c3psin[p(x+cz−αt)]e⁻⁽¹^/²⁾^c¹^y⁺^y^+λ

2c₃cos[p(x +cz−αt)]e⁽¹^/²⁾^c¹^y⁺^y^+λ+c₂e²^y⁺^c¹^y⁺²^γ +1, (22)

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Figure 3. Characteristic form of the periodic solution (21) with parameters p =1,c =1, α =1, λ=0,c1 =2, c2=c3=1, =1, ϒ = −1,z= y=0: (a) 3D-plot and (b) 2D-density.

where p, ϒ and c1 need to satisfy the constraint condition: 3c²₁+4(c p⁴−4αp²+3ϒ²+3c1ϒ)=0.

Now, solution (21) was plotted in 3D and 2D forms (see figures3and4).

Here, solutions (18)–(22) are new solutions.

If we make a variable transformation I y → y, eq. (15) can be represented as

c∂⁴

∂ϑ⁴ +3c ∂

∂ϑ

∂

∂ϑ 2

+4α∂²

∂ϑ² −3∂²

∂y² =0. (23) Equation (23) can be reduced to the following Hirota bilinear form:

(c D_ϑ⁴ +4αD_ϑ² −3D²_y)f · f =0, (24) under the dependent variable transformation = 2(ln f)ϑ.

Figure 4. Characteristic form of the periodic solution (21) with parameters p =1,c=1, α =1, λ=0,c1 =2, c2=c3 =1, =1, ϒ = −1,z =x =0: (a) 3D-plot and (b) 2D-density.

By the same manipulation as stated above, we have Case1

ϒ = −1

2c1,p= −

√cα

c ,b1=e⁻⁽¹^/²^)[^c¹⁽^c³⁻^y⁾⁺²^], b₂=c₂e⁻²^ϒ^y

,

where c, α,c1,c2, =α/√

−c and c3 are arbitrary constants.

Case2

ϒ = −1

2c1,p= −

√cα

c ,b1=e⁻⁽¹^/²^)[^c¹⁽^c³⁻^y⁾⁻²^], b2=c2e⁻²^ϒ^y

,

−c and c3 are arbitrary constants.

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23 Page 6 of 7 Pramana – J. Phys.(2019) 92:23 Case3

ϒ = −1

2c1,p=

√cα

c ,b1 =e⁻⁽¹^/²^)[^c¹⁽^c³⁻^y⁾⁺²^], b₂ =c₂e⁻²^ϒ^y

,

−c and c3 are free parameters.

Case4

ϒ = −1

2c₁,p=

√cα

c ,b₁ =e⁻⁽¹^/²^)[^c¹⁽^c³⁻^y⁾⁻²^], b2 =c2e⁻²^ϒ^y

,

where p, α,c1,c2andc3are free parameters.

Case5

b1=c3e⁽¹^/²⁾^c¹^y,b2=c2e^c¹^y,p² = 4α(b²₁−b2) c(4b₁²−b2)

, where p, ϒ and c1 need to satisfy the constraint condition: 3c₁²+4(−c p⁴+4αp²+3ϒ²+3ϒc₁)=0.

Using the dependent variable transformation = 2(ln f)ϑ and Cases 1–5, we can obtain the solutions of eq. (1) as follows:

1(x,y,z,t)= − 4pe⁻^c¹²^c³^+λ−^c³sin[p(x+cz−αt)]

2 e⁻^c²¹^c³^+λ−^c³cos[p(x+cz−αt)] +c2e²^λ−^y+e⁻^y

, (25)

wherec,c₁,c₂, =α/√

−candc₃are free parameters.

2(x,y,z,t)= −4pe^−c²¹^c³^+λ−^ysin[p(x+cz−αt)]

2 e^−c²¹^c³^+λ−^ycos[p(x +cz−αt)] +c₂e²^λ−^c³+e⁻^c³, (26) wherec,c1,c2, =α/√

−candc3are free parameters.

3(x,y,z,t)= − 4pe⁻^c¹²^c³^+λ−^c³sin[p(x+cz−αt)]

2 e⁻^c²¹^c³^+λ−^c³cos[p(x+cz−αt)] +c2e²^λ−^y+e⁻^y

, (27)

wherec,c1,c2, =α/√

4(x,y,z,t)= − 4pe^−c²¹^c³⁺²^λ−^ysin[p(x +cz−αt)]

2 e^−c²¹^c³⁺²^λ−^ycos[p(x +cz−αt)] +c2e²^λ−^c³+e⁻^c³, (28) wherec,c1,c2, =α/√

5(x,y,z,t)= − 4psin[p(x +cz−αt)]e⁽¹^/²⁾^c¹^y^+ϒ^y^+λ

2c3 cos[p(x+cz−αt)]e⁽¹^/²⁾^c¹^y^+ϒ^y^+λ+c2e²^ϒ^y⁺^c¹^y⁺²^λ+1, (29) where p, ϒ and c1 need to satisfy the constraint condition: 3c²₁+4(−c p⁴+4αp²+3ϒ²+3ϒc1)=0.

Solutions (25)–(29) were reported in other places.

Remark. The purpose of the paper is to seek different travelling-wave solutions of the (3 +1)-dimensional potential-YTSF equation by making use of various approaches. By taking various parameters, the exact solutions (10) and (13) have been manifested by fig- ures1and2. The periodic solution (21) has been shown by figures3and4when two sets of differential parameters p, c, α, λ, c1,c2,c3, are chosen. Actually, all solutions obtained in this paper can be described just like that of solution (21), and so here we do not want to discuss them again.

4. Conclusion

With the help of symbolic calculation and applying the improved homoclinic test technique and the extended homoclinic test technique, we found some new exact solutions to the (3 +1)-dimensional potential-YTSF equation. As a result, some new solutions, which include

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the periodic kink-wave, solitary wave and periodic wave solutions, were obtained.

Acknowledgements

This work was supported by the National Nature Science Foundation of China (Nos 11801323, 11705104), the Project of Shandong Province Higher Education Science and Technology Program (Nos J18KA227, J17KB130), the Shandong Provin- cial Natural Science Foundation (Nos. ZR2016AM31, ZR2016AQ19) and SDUST Research Fund (No. 2018TDJH101).

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