Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 91–96c© International Academic Publishers Vol. 46, No. 1, July 15, 2006

New Types of Exact Solutions for (N+1)-Dimensional φ4 Model∗

JIA Man1 and LOU Sen-Yue1,2

1Center of Nonlinear Science and Department of Physics, Ningbo University, Ningbo 315211, China2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China

(Received October 9, 2005)

Abstract New types of exact solutions of the (N + 1)-dimensional φ4-model are studied in detail. Some types ofinteraction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutionsare found.

PACS numbers: 11.10.Lm, 03.50.Kk, 03.65.Ge

Key words: φ4-model, periodic-periodic interaction waves

1 Introduction

In the classical and quantum field theory, the φ4 fieldmodel is one of the most important examples. The φ4 fieldis significant not only because of its basic status in fieldtheory but also because of its wide applications in variousbranches of physics and other scientific fields, such as insolid state physics, condensed matter physics, quantumfield theory, cosmology, etc.[1−3]

Owing to the wide applications of the φ4-model, tofind exact solutions of the (n + 1)-dimensional φ4-modelis of great importance and interest. Some properties andexact solutions for the φ4 field are known in Refs. [4] and[5]. However, there are still many open important prob-lems. For example, the periodic-periodic travelling wavesolutions and the periodic-solitary wave interaction solu-tions are still not found. In this paper, we are mainlyconcentrated on the exact solutions by seeking some newtype solutions for the φ4-model.

The (n+1)-dimensional φ4 equation reads

¤φ − λφ − µφ3 ≡

n∑

i=1

φxixi− φtt − λφ − µφ3 = 0 , (1)

which has been applied in almost all the branches ofphysics especially for n = 2 and 3 cases.

Though many special relations of the φ4 model havebeen given in Ref. [5], many kinds of exact solutions suchas the periodic-periodic interaction solutions have not yetbeen found. Here we give out some new exact solutionsof the φ4-model by extending the mapping relations ofRef. [4]. To find some special types of exact solutions ofthe φ4-model equation, many interesting results have beengiven by various authors. In order to find more exact so-lutions of Eq. (1), we try to extend the known solutionsto more general forms.

2 Kink-Like Solitons and Solitoffs and TheirConoid Wave Extentions

For two-dimensional integrable models such asKadomtsev–Petviashvili (KP) equation and Davey–Stewarzon (DS) equation,[6] there exist both the line soli-tons and solitoffs. A line soliton is defined as such a local-ized excitation that apart away from a line, rapidly tends

to constants. If the constants for both sides are the same,the excitation is called a bell-shape line soliton. Other-wise, it is called a kink-like line soliton. A solitoff is justa half line soliton. Three solitoffs can be connected to aY-shape soliton.

It is found that the line solitons and solitoffs may alsobe found for nonintegrable models such as the (2+1)-dimensional sine-Gordon and double sine-Gordon equa-tions. It is reasonable that there should also be solitoffsolutions for (2+1)-dimensional φ4-model though it hasnot yet been reported before.

It is clear that

φ = sn(V ) (2)

is a solution of the φ4-model equation, where V is a func-tion of g(x1, x2, . . . , xn, t), which satisfies

¤g(x1, x2, . . . , xn, t) = A(g) , (3)

and

(∇̃g(x1, x2, . . . , xn, t))2 ≡

n∑

i=1

g2xi

− g2t = B(g) , (4)

where A(g) and B(g) are functions of g = g(x1, x2, . . .,xn, t), and satisfy

A(g) =1

2

dB(g)

dg, (5)

V =

∫

B−1/2dg , (6)

while the constant n is the modulus of the Jacobi ellipticfunction sn(V ) ≡ sn(V, n).

Though the solution is known in Ref. [5], one can stillfind more general solution by including an arbitrary func-

tion to the “independent” variable V , say,

V = f

(

n∑

i=1

kixi + ω1t

)

+

n∑

i=1

mixi + ω0t

≡ f(ξ) + ξ0 , (7)

where f(ξ) is an arbitrary function of ξ and the constantski, mi, ω0, and ω1 satisfy

n∑

i=1

k2i − ω2

0 = 0 ,

∗The project supported by National Natural Science Foundations of China under Grant Nos. 90203001, 10475055, and 90503006

92 JIA Man and LOU Sen-Yue Vol. 46

n∑

i=1

kimi − ω1ω0 = 0 . (8)

It is easy to find that the solution (2) with Eq. (7) pos-sesses the model parameters,

λ = −δ(n2 + 1) , (9)

µ = 2δn2 , (10)

and

δ =n

∑

i=1

m2i − ω2

0 . (11)

The special solution (2) with (7) denotes some partic-

ular types of resonant solutions of two travelling wavesmoving in the directions perpendicular to the planes (orlines for n = 2):

ξ = 0, ξ0 = 0 .

Because of the existence of the arbitrary function f(ξ),the structure of the periodic travelling waves shown byEq. (2) with (7) may be quite rich. For example, if wetake

f(ξ) =√

ξ2 + 1 , (12)

then the solution (2) denotes a special periodic-periodicwave interaction solution for the modulus n < 1.

Fig. 1 (a) A typical periodic solitoff structure of the (2+1)-dimensional equation expressed by Eq. (2) with Eqs. (7),(12), and (13) at t = 0. (b) A special two solitoff solution which is a limit case of (a) for the modulus n of the Jacobielliptic function n → 1.

Fig. 2 (a) A periodic-periodic wave interaction structure of the (2+1)-dimensional equation expressed by Eq. (7) withEqs. (15) and (13) at t = 0. (b) A periodic line kink soliton structure which is a limit case of (a) for the modulus n ofthe Jacobi elliptic function n → 1.

Figure 1(a) is a (2 + 1)-dimensional special structureof this type of solutions with the parameter selections,

k1 = 3, k2 = 4, ω1 = 5, m1 = 2 ,

m2 = 1, ω0 = 2, n = 910 , (13)

λ = − 181100 , µ = 81

50 (14)

at time t = 0.When n → 1, equation (2) with Eq. (7) and Eq. (12)

tends to a two kink-line solitoff solution. Figure 1(b)

shows the structure of a special two-solitoff solution ex-pressed by Eq. (2) with Eqs. (7) and (12), where the pa-rameter selections are taken as the same as in Eq. (13)except for n → 1.

Figure 2(a) shows another concrete example of the par-ticular periodic-periodic wave interaction structure by se-lecting the function f(ξ) as

f(ξ) = sin ξ (15)

with the same parameters in Fig. 1(a).

No. 1 New Types of Exact Solutions for (N+1)-Dimensional φ4 Model 93

Figure 2(b) is a plot of the straight-line kink solitonwith a periodic travelling wave deformation. The parame-ter and function selections of Fig. 2(a) except the modulusn is taken as the limiting value n = 1.

3 Bell Shape Line Solitons, Solitoffs and TheirConoid Wave Extentions

From Ref. [4] we know that

φ = cn(V ) (16)

is also a solution of the φ4-equation, where V satisfiesEqs. (7) and (8) while the corresponding parameter selec-tions are

λ = δ(2n2 − 1) , (17)

µ = −2δn2 (18)

with the free parameter δ the same as Eq. (11). The solu-

tions given by Eq. (16) with Eqs. (17) and (18) denote theinteraction solutions of two travelling periodic or solitarywaves. Here are two special explicit examples in (2 + 1)dimensions.

(i) If we take the function f(ξ) as

f(ξ) = 3

√

3ξ2 + 5 , (19)

then a special (2 + 1)-dimensional periodic-periodic wavesolution of Eq. (1) can be immediately read off from thecase of Eq. (16)

φ = cn(

3

√

3(k1x + k2y + ω1t)2 + 5 + m1x

+ m2y + ω0t, n) , (20)

where k1, k2, ω1, m1, m2, and ω0 satisfy the conditions

k21 + k2

2 − ω21 = 0 ,

k1m1 + k2m2 − ω1ω0 = 0 . (21)

Fig. 3 (a) A typical periodic solitoff structure of the (2+1)-dimensional equation expressed by Eq. (20) with Eqs. (19)and (22) at t = 0. (b) A special two solitoff solution which is a limit case of (a) for the modulus n of the Jacobi ellipticfunction n → 1.

Fig. 4 (a) A periodic-periodic wave interaction structure of the (2+1)-dimensional equation expressed by Eq. (20) withEqs. (24) and (22) at t = 0. (b) A special straight-line soliton interacting with a periodic travelling wave solution whichis a limit case of (a) for the modulus n of the Jacobi elliptic function n → 1.

Figure 3(a) shows a special (2 + 1)-dimensional structure of the two periodic waves expressed by Eq. (20) with theparameter selections

k1 = 3, k2 = 4, ω1 = 5, m1 = 2 ,

m2 = 1, ω0 = 2, n = 910 , (22)

and the corresponding parametersλ = 31

50 , µ = − 8150 , (23)

at t = 0. Figure 3(b) is a limit case of Fig. 3(a) for n → 1, which denotes the solution of φ4 tending to a two-curvedsolitoff solution. A curved line soliton[7] is defined as a localized excitation which is nonzero only at a curved line

94 JIA Man and LOU Sen-Yue Vol. 46

(its both ends extend to infinity) and decays exponentially apart from the curve. Then a curved-solitoff is just a halfcurved line solitoff, which means only one of its ends extends to infinity.

(ii) Figure 4(a) is another concrete example of the particular periodic-periodic wave interaction structure by selectingthe function f(ξ) as

f(ξ) = cos2 ξ (24)

with the same parameters in Fig. 3(a). Figure 4(b) shows a straight-line soliton with a periodic travelling wavedeformation. The parameter and function selections of Fig. 4(b) are the same as those in Fig. 4(a) except that themodulus n is taken as the limiting value n = 1.

4 Mapping and Combining Two Solutions of φ4-model to a New Solution

In the last section, some new special types of solutions of φ4 have been found by adding an arbitrary function intothe expressions of quite free independent variable V of the known formal solutions proposed in Ref. [4].

In this section, we establish a new mapping relation among three special solutions of the φ4-model. Using themapping relation, we can obtain a new solution by combining two known special solutions of the φ4-model to a newone.

Superposition Theorem If φ1 and φ2 are solutions of the φ4-model equation

¤φ1 = λ1φ1 + µ1φ31, ¤φ2 = λ2φ2 + µ2φ

32 , (25)

under the constraint conditions

(∇̃φ1)2 = g1(φ1, φ2), (∇̃φ2)

2 = g2(φ1, φ2) , (26)

and(

∇̃φ1

)

·(

∇̃φ2

)

=n

∑

i=1

φ1xiφ2xi

− φ1tφ2t = g12(φ1, φ2) (27)

with g1(φ1, φ2) ≡ g1, g2(φ1, φ2) ≡ g2, and g12(φ1, φ2) ≡ g12 being functions of {φ1, φ2} and related by

µh3 + (λf2 − 4abg12)h + 2bφ2(−cφ22b + a)g1 + 2bφ1(−φ2

1b + ac)g2

− [(−cφ22b + a)(µ1φ

31 + λ1φ1) + (−bφ2

1 + ac)(µ2φ32 + λ2φ2)]f = 0 , (28)

where

f ≡ a + bφ1φ2, h ≡ φ1 + cφ2 , (29)

then

φ =h

f(30)

is also a solution of φ4 equation (1).Proof Substituting Eq. (30) into Eq. (1) yields

f(−cφ22b + a)¤φ1 + f(−φ2

1b + ac)¤φ2 − 4hab(∇̃φ1) · (∇̃φ2) − 2bφ2(−cφ22b + a)(∇̃φ1)

2

− 2bφ1(−φ21 + ac)(∇̃φ2)

2 + h(λf2 + µh2) = 0 . (31)

Now substituting the constraint conditions (25) ∼ (27) into Eq. (31) leads to the relation (28) and then the theorem 3is proved.

To obtain some explicit solutions from the superposition theorem, we have to solve the constraint system (25) ∼ (27).However, it is difficult to solve the constrained systems (25) ∼ (28) for arbitrary g1 and g2. So, in order to solve theproblem, here, we introduce some further restrictions

φ1 = φ1(V1(x1, . . . , xn, t)) ≡ φ1(V1), φ2 = φ2(V2(x1, . . . , xn, t)) ≡ φ2(V2) , (32)

and

φ21V1

= λ1φ21 +

µ1

2φ4

1 + C1, φ22V2

= λ2φ22 +

µ2

2φ4

2 + C2 (33)

with V1 and V2 being arbitrary solutions of the simple constraint equations

¤V1 = ¤V2 = (∇̃V1) · (∇̃V2) = 0, (∇̃V1)2 = G1, (∇̃V2)

2 = G2 , (34)

where G1 and G2 are constants.Under the constraints (32) ∼ (34), the superposition theorem is simplified to the following corollary.Corollary φ expressed by Eq. (30) with the constraints (32) ∼ (34) is a solution of Eq. (1) if and only if

c = −abλ

µ, a2 =

λ1µ

µ1λ, b2 =

µ2µ

λ2λ, C1 = G1 =

λ21

2µ1, C2 = G2 =

λ22

2µ2(35)

with

λ = λ1 + λ2 . (36)

No. 1 New Types of Exact Solutions for (N+1)-Dimensional φ4 Model 95

A special situation of Eq. (34) reads

V1 = f1

(

n∑

i=1

k1ixi + ω1t)

+

n∑

1=i

k0ixi + ω0t ≡ f1(ξ1) + ξ01 , (37)

V2 = f2

(

n∑

i=1

k2ixi + ω2t)

+

n∑

1=i

k3ixi + ω3t ≡ f2(ξ2) + ξ02 , (38)

n∑

i=1

kj1ikj2i − ωj1ωj2 = −G1δj10δj20 − G2δj13δj23, (j1, j2 = 0, 1, 2, 3) , (39)

where

δij =

{

0, i 6= j ,

1, i = j ,(40)

and f1(ξ1) and f2(ξ2) are arbitrary functions.The solutions given by Eq. (30) denotes the interaction solutions of three or four travelling periodic or kink waves.

Here are two more special explicit examples in (2 + 1) dimensions.(i) The interaction solution of one kink and two periodic waves. If we take

φ1 = sn(V1, n1), φ2 = sn(V2, n2), a = b = c = 1 , (41)

V1 = sin(k11x + k12y + ω1t) + k01x + k02y + ω0t ,

V2 = cos2(k21x + k22y + ω2t) + k31x + k32y + ω3t , (42)

λ1 = −(n21 + 1)δ1, µ1 = 2n2

1δ1, G1 = k201 + k2

02 − ω20 ,

λ2 = −(n22 + 1)δ2, µ2 = 2n2

2δ2, G2 = k231 + k2

32 − ω23 , (43)

n1 = 1 , (44)

and k11, k12, ω1, k01, k02, ω0, k21, k22, ω2, k31, k32, and ω3 are determined by Eq. (39), then a special (2+1)-dimensionalperiodic-wave and a kink interaction solution can be immediately read off from the nonlinear superposition theorem:

φ =tanh(V1) + sn(V2, n2)

1 + tanh(V1)sn(V2, n2). (45)

Fig. 5 (a) A typical two-periodic-wave and a kink structure of the (2+1)-dimensional equation expressed by Eq. (45)with Eq. (46) at t = 0. (b) A special kink periodic wave interaction solution which is a limit case of (a) for the modulusn2 of the Jacobi elliptic function n2 → 1.

Figure 5(a) shows a special (2 + 1)-dimensional structure of the two-periodic-wave and a kink interaction solutionexpressed by Eq. (45) with Eqs. (41) ∼ (43) and the parameter selections

k11 = 3, k12 = 4, ω1 = 5, k01 = 2, k02 = 1, ω0 = 2, n1 = 1 ,

k21 = 95 , k22 = 12

5 , ω2 = 3, k31 = 32 , k32 = 2, ω3 = 5

2 , n2 = 710 , λ = 2, µ = −2 , (46)

at t = 0. Figure 5(b) is a plot of one periodic wave and one kink interaction solution. It is the limit case for Fig. 5(a)for n2 = 1 in φ2.

(ii) An interaction solution among solitoffs and periodic waves. A particular interaction solution for the (2 + 1)-dimensional φ4 equation possesses the form

φ =sn(V1, n1) + sn(V2, n2)

1 + sn(V1, n1)sn(V2, n2)(47)

with V1 and V2 selections

V1 =√

(k11x + k12y + ω1t)2 + 1 + k01x + k02y + ω0t ,

96 JIA Man and LOU Sen-Yue Vol. 46

V2 =√

(k21x + k22y + ω2t)2 + 5 + k31x + k32y + ω3t . (48)

Fig. 6 (a) A typical periodic-solitoff interaction structure of the (2+1)-dimensional equation expressed by Eq. (47)with Eqs. (48) and (46) at t = 0. (b) A special two-kink-like solitoff solution which is a limit case of (a) for the modulusn2 of the Jacobi elliptic function n2 → 1.

Figure 6(a) is a plot of a concrete example of Eq. (47) with the same parameter selections in Eq. (46) at t = 0.Figure 6(b) shows the structure of the limit case for n2 = 1 in φ2 which is similar to Fig. 1(b).

5 Summary and Discussion

In this paper, by adding an arbitrary function to the expressions of the known solutions of the (N +1)-dimensionalφ4-model, we obtained some new types of solutions including the periodic-periodic wave interaction solutions andthe periodic-solitoff interaction solutions etc. By seeking the possible nonlinear superpositions of two known specialsolutions, we have found more general solutions with two arbitrary functions. For instance, a line kink may be deformedby two periodic waves.

In summary, the existence of arbitrary functions can make the solutions of φ4 be much richer. A diversity ofexact explicit solutions of the φ4 can be obtained by selecting different functions. The abundant structure of the(2+1)-dimensional nonlinear systems is firstly found for many (2+1)-dimensional integrable systems.[8] The results ofthis paper show us that the rich structures of the nonlinear system exist also for nonintegrable systems.

Because of the existence of the deformation relations among the φ4-model and the other nonlinear Klein–Gordonsystems such as the φ6-equation, T.D. Lee model, multiple sine-Gordon systems, etc.,[4,5,9] the new type solutions ofφ4 proposed in this paper will lead to new types of other types of nonlinear Klein–Gordon equations.

Acknowledgments

The authors are indebt to Drs. X.Y. Tang, F. Huang, and H.C. Hu for their helpful discussions.

References

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