Symmetry reductions and similarity solutions of the (3 + 1)-dimensional breaking soliton equation

Available online at www.sciencedirect.com

Applied Mathematics and Computation 201 (2008) 333–339 www.elsevier.com/locate/amc

Symmetry reductions and similarity solutions of the (3 + 1)-dimensional breaking soliton equation Zhi-lian Yan a,*, Jian-ping Zhou b,d, Xi-qiang Liu c a

School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002, Anhui, PR China b School of Computer Science, Anhui University of Technology, Maanshan 243002, Anhui, PR China c School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, Shandong, PR China d Department of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China

Abstract Using the compatibility method, we obtain three types of symmetry of the (3 + 1)-dimensional breaking soliton equation. By solving the corresponding characteristic equations associated with symmetry equations, symmetry reductions of the (3 + 1)-dimensional breaking soliton equation are given first, then some special types of similarity solutions are constructed. Ó 2008 Published by Elsevier Inc. Keywords: Breaking soliton equation; Symmetry; Similarity solution

1. Introduction Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in several aspects of physics as well as other natural and applied sciences. One of the most important tasks in the study of NPDEs is to construct exact solutions. In the last few years, a large number of effective solving methods, such as the tanh method, homogeneous balance method and symmetry group methods have been developed, see e.g. [1–7]. It is well known that many NPDEs are formulated in (3 + 1)-dimensional or more, thus the question of finding exact solutions of such high dimensional equations is of great importance. The purpose of this paper is to seek similarity solutions of the following (3 + 1)-dimensional breaking soliton equation [8] uxxt þ auxxx uyz þ buxxy uxz þ cuxy uxxz þ duxx uxyz þ euxxxyz ¼ 0;

ð1Þ

where a, b, c, d and e are given constants. Eq. (1) was originally proposed by Lin through means of the realizations of the generalized centerless Virasoro type symmetry algebra [9]. In Ref. [6], we presented a compatibility method of finding symmetry reductions and similarity solutions of NPDEs. One of the advantages of *

Corresponding author. E-mail address: [email protected] (Z.-l. Yan).

0096-3003/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.amc.2007.12.027

334

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

our method is that it is capable of greatly reducing the complexity in the computational process. The basic idea of the compatibility method is to seek the non-classical symmetry of a given NPDE such as Eq. (1) in the form ut ¼ aðt; x; y; zÞux þ bðt; x; y; zÞuy þ cðt; x; y; zÞuz þ dðt; x; y; zÞu þ hðt; x; y; zÞ;

ð2Þ

where a, b, c, d and h are functions to be determined later by the compatibility of Eqs. (1) and (2). Firstly, we can obtain the highest-order derivative term uxxxyz of Eq. (1) by substituting Eq. (2) into Eq. (1). Then we compute the derivatives of Eq. (1) with respect to t and Eq. (2) with respect to x twice and t to find uxxtt. Requiring the equality of uxxtt will lead to a set of determining equations of a, b, c, d and h. Once the determining equations are solved, the non-classical symmetries of Eq. (1) can be obtained. By using the compatibility method, we derive symmetries and the corresponding reductions of Eq. (1). Furthermore some new similarity solutions of the breaking soliton equation are found. 2. Symmetries of the (3 + 1)-dimensional breaking soliton equation Substituting Eq. (2) into Eq. (1) yields the highest-order derivative term uxxxyz of Eq. (1) as follows: uxxxyz ¼ ðaxx ux þ 2ax uxx þ auxxx þ bxx uy þ 2bx uxy þ buxxy þ cxx uz þ 2cx uxz þ cuxxz þ dxx u þ 2dx ux þ duxx þ hxx þ auxxx uyz þ buxxy uxz þ cuxy uxxz þ duxx uxyz Þ=e;

ð3Þ

where e is a non-zero constant. In terms of the equality of uxxtt of the Eqs. (1) and (2), one can get ðauxxx uyz þ buxxy uxz þ cuxy uxxz þ duxx uxyz þ euxxxyz Þt ¼ ðaux þ buy þ cuz þ du þ hÞxxt :

ð4Þ

Expanding the above equation and substituting Eqs. (2) and (3) into the expansion yield the following expression with the aid of computerized symbolic computation MAPLE eay uxxxxz þ eaz uxxxxy þ 3ebx uxxyyz þ ebz uxxxyy þ 3ecx uxxyzz þ ecy uxxxzz þ F 1 ðt; x; y; z; u; ux ; . . .Þ ¼ 0;

ð5Þ

where the function F1 is not dependent on uxxxxz, uxxxxy, uxxyyz, uxxxyy, uxxyzz and uxxxzz. To ensure Eq. (5) is true for an arbitrary solution u, it is necessary to take the coefficients of uxxxxz, . . ., uxxxzz to be zero, we have ay ¼ az ¼ bx ¼ bz ¼ cx ¼ cy ¼ 0;

i:e: a ¼ aðt; xÞ; b ¼ bðt; yÞ; c ¼ cðt; zÞ:

ð6Þ

Under Eq. (6), Eq. (5) can be further simplified to dduxx uxyz þ F 2 ðt; x; y; z; u; ux ; . . .Þ ¼ 0;

ð7Þ

where F2 is independent of uxxuxyz, which implies dðt; x; y; zÞ ¼ 0;

ðd 6¼ 0Þ:

ð8Þ

According to the same method step by step, it leads to a = a(t,x), b = b(t,y), c = c(t,z), d = 0, axx = 0 and hxx = 0. Then the following determining equations:  2by ax  2cz ax þ 2atx  2a2x þ dhxyz ¼ 0;

ð9Þ

 by a  cz a þ at  aax þ ahyz ¼ 0;

ð10Þ

 by b  cz b þ bt  bax þ bhxz ¼ 0;

ð11Þ

 by c  cz c þ ct  cax þ chxy ¼ 0;

ð12Þ

can be read off from Eq. (5). Solving Eqs. (9)–(12), various type solutions for a, b, c, d and h can be found. Here we will discuss the following concrete cases. Case 1 a is an arbitrary constant. From Eqs. (9)–(12), one can get

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

 a ¼ a;

Z

335



b ¼ y  bA2 ðtÞðc0 t þ c1 Þ dt ðc0 t þ c1 Þ;   Z ðc0 t þ c1 Þ; c ¼  ðc0 þ 1Þz þ cA1 ðtÞðc0 t þ c1 Þ dt d ¼ 0;

a ¼ a; d ¼ 0;

h ¼ ðA1 ðtÞy þ A2 ðtÞz þ A3 ðtÞÞx  ac0 yz=½aðc0 t þ c1 Þ þ A4 ðtÞ; ða 6¼ 0Þ Z Z b ¼ y=c1  bA2 ðtÞ dt; c ¼ z=c1  cA1 ðtÞ dt; h ¼ ðA1 ðtÞy þ A2 ðtÞz þ A3 ðtÞÞx þ A4 ðtÞ; ða ¼ 0Þ

ð13Þ

ð14Þ

where c0 and c1 are integral constants, A1(t), A2(t), A3(t) and A4(t) are arbitrary functions of t. Hence we obtain the non-classical symmetry of Eq. (1) expressed by r0 ¼ aux þ buy  ut þ cuz þ h;

ð15Þ

where a, b, c, d and h are determined by Eq. (13) or (14). Case 2 a = a(t) is the function of t. Solving Eqs. (9)–(12), we have   Z a ¼ 1=ðc2 t þ c3 Þ; b ¼ c4 y  bB2 ðtÞðc2 t þ c3 Þdt ðc2 t þ c3 Þ; h ¼ ðB1 ðtÞy þ B2 ðtÞz þ B3 ðtÞÞx þ B4 ðtÞ;   Z ðc2 t þ c3 Þ; d ¼ 0; ð16Þ c ¼  ðc2 þ c4 Þz þ cB1 ðtÞðc2 t þ c3 Þ dt where c2, c3 and c4 are integral constants, B1(t), B2(t), B3(t) and B4(t) are arbitrary functions of t. As a result we obtain the following form symmetry of Eq. (1) r1 ¼ aux þ buy  ut þ cuz þ h;

ð17Þ

where a, b, c, d and h are determined by Eq. (16). Case 3 a = a(t,x) is the function of t and x. It follows from Eqs. (9)–(12) that     Z Z a ¼ x  ak 4 ðtÞðc5 t þ c6 Þ dt ðc5 t þ c6 Þ; b ¼ c7 y  bk 2 ðtÞðc5 t þ c6 Þ dt ðc5 t þ c6 Þ;   Z ðc5 t þ c6 Þ; d ¼ 0; c ¼  ðc5 þ c7 þ 1Þz þ ck 1 ðtÞðc5 t þ c6 Þ dt h ¼ ðk 1 ðtÞy þ k 2 ðtÞz þ k 3 ðtÞÞx þ k 4 ðtÞyz þ k 5 ðtÞ;

ð18Þ

where c5, c6 and c7 are integral constants, ki(t) (i = 1, . . ., 5) are arbitrary functions of t. Thus we have the symmetry r2 of the (3 + 1)-dimensional breaking soliton equation r2 ¼ aux þ buy  ut þ cuz þ h;

ð19Þ

where a, b, c, d and h are determined by Eq. (18). It is not difficult to see that r0, r1 and r2 have not been reported in the existing literatures. 3. Symmetry reductions of the (3 + 1)-dimensional breaking soliton equation In this section we will obtain symmetry reductions of Eq. (1) by the compatibility of r = 0 and Eq. (1). One first solves the associated characteristic equations of r = 0 to derive similarity variables and then substitutes these results into Eq. (1) to determine the corresponding reduced equations.

336

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

3.1. r0 = aux + buy  ut + cuz + h The determining equations for similarity variables of r0 = 0 are dx dy dt dz du ¼ ¼ ¼ ¼ ; a b 1 c h

ð20Þ

where a, b, c, d and h are determined by Eq. (13) and c0 6¼ 0. Solving the above system, one can derive the following expression of u Z Z Z 1=c 1þ1=c0 1þ1=c0 u ¼ f ðn; g; sÞ þ ng A1 ðtÞðc0 t þ c1 Þ 0 dt þ ns A2 ðtÞðc0 t þ c1 Þ dt  s ½atA2 ðtÞðc0 t þ c1 Þ Z 1=c 1þ1=c0 þ D1 ac0 =a dt þ n ½D1 A1 ðtÞðc0 t þ c1 Þ 0 þ D2 A2 ðtÞðc0 t þ c1 Þ þ A3 ðtÞ dt Z Z 1=c 1=c  g ½atA1 ðtÞðc0 t þ c1 Þ 0 þ D2 ac0 =a dt  gsac0 t=a  ½A1 ðtÞD1 atðc0 t þ c1 Þ 0 þ A2 ðtÞD2 atðc0 t þ c1 Þ1þ1=c0 þ A3 ðtÞat þ D1 D2 ac0 =a  A4 ðtÞ dt; 1=c0

ð21Þ

ðc0 þ1Þ=c0

whereR n ¼ x þ at; g ¼ Ryðc0 t þ c1 Þ  D1 ; s ¼ zðc0 t þ cR1 Þ  D2 and f are similarity variables, R D1 ¼ ½ðc0 t þ c1 Þð1c0 Þ=c0 bA2 ðtÞðc0 t þ c1 Þ dt dt and D2 ¼ ½ðc0 t þ c1 Þ21=c0 cA1 ðtÞðc0 t þ c1 Þ dt dt. Substituting Eq. (21) into Eq. (1) yields the reduced equation of the (3 + 1)-dimensional breaking soliton equation ac1 fnnn þ gfnng  sfnns =ðc0 þ 1Þ þ afnnn fgs þ bfnng fns þ cfnns fng þ dfnn fngs þ efnnngs ¼ 0;

ð22Þ

which is a (2 + 1)-dimensional variable coefficients partial differential equation. 3.2. r0 = aux + buy  ut + cuz + h In this case a = 0 and c1 6¼ 0, we obtain the following expression of u by solving the corresponding characteristic equations of r0 = 0 Z Z Z t=c1 t=c1 u ¼ f ðn; g; sÞ þ ng A1 ðtÞe dt þ ns A2 ðtÞe dt þ n ðD01 A1 ðtÞet=c1 þ D02 A2 ðtÞet=c1 þ A3 ðtÞÞ dt Z Z Z  g A1 ðtÞatet=c1 dt  s A2 ðtÞatet=c1 dt  ðatA1 ðtÞD01 et=c1 þ atA2 ðtÞD02 et=c1 þ A3 ðtÞat  A4 ðtÞÞ dt; ð23Þ R

R

where n ¼ xR þ at; gR ¼ yet=c1  D01 ; s ¼ zet=c1  D02 and f are similarity variables, D01 ¼ b ðet=c1 A2 ðtÞ dtÞ dt and D02 ¼ c ðet=c1 A1 ðtÞ dtÞ dt. Substituting Eq. (23) into Eq. (1), one can get the following NPDE afnnn þ gfnng =c1  sfnns =c1 þ bfnng fns þ cfnns fng þ dfnn fngs þ efnnngs ¼ 0:

ð24Þ

3.3. r1 = aux + buy  ut + cuz + h In this case c2 6¼ 0, it follows from r1 = 0 that Z Z c =c 1þc =c u ¼ f ðn; g; sÞ þ ng B1 ðtÞðc2 t þ c3 Þ 4 2 dt þ ns B2 ðtÞðc2 t þ c3 Þ 4 2 dt Z Z  s ½B2 ðtÞðc2 t þ c3 Þ1þc4 =c2 lnðc2 t þ c3 Þdt=c2 þ n ½D3 B1 ðtÞðc2 t þ c3 Þc4 =c2 þ D4 B2 ðtÞðc2 t þ c3 Þ1þc4 =c2 Z c =c þ B3 ðtÞdt  g ½B1 ðtÞðc2 t þ c3 Þ 4 2 lnðc2 t þ c3 Þdt=c2 Z  c =c 1þc =c  ½ðB1 ðtÞD3 ðc2 t þ c3 Þ 4 2 þ B2 ðtÞD4 ðc2 t þ c3 Þ 4 2 þ B3 ðtÞÞ lnðc2 t þ c3 Þ  c2 B4 ðtÞdt c2 ; ð25Þ

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

337

ðc þc Þ=c

c =c

where n ¼ xRþ lnðc2 t þ c3 Þ=c2 ; gR ¼ yðc2 t þ c3 Þ 4 2  D3 ; s ¼ zðc2 tR þ c3 Þ 2 4 2  DR4 and f are similarity variðc c Þ=c 2c4 =c2 ables, D3 ¼ ½ðc2 t þ c3 Þ 4 2 2 bB2 ðtÞðc2 t þ c3 Þ dt dt and D4 ¼ ½ðc2 t þ c3 Þ cB1 ðtÞðc2 t þ c3 Þ dt dt: Substituting Eq. (25) into Eq. (1), we have fnnn þ gfnng =c4  sfnns =ðc2 þ c4 Þ þ afnnn fgs þ bfnng fns þ cfnns fng þ dfnn fngs þ efnnngs ¼ 0: ð26Þ 3.4. r2 = aux + buy  ut + cuz + h In this case, the determining equations for similarity variables are dx dy dt dz du ¼ ¼ ¼ ¼ ; a b 1 c h

ð27Þ

where a, b, c, d and h are determined by Eq. (18) and c5 6¼ 0. Solving the above system yields that Z Z Z ðc þ1Þ=c5 1þc =c 1þ1=c5 dt þ ns k 2 ðtÞðc5 t þ c6 Þ 7 5 dt þ gs k 4 ðtÞðc5 t þ c6 Þ dt u ¼ f ðn; g; sÞ þ ng k 1 ðtÞðc5 t þ c6 Þ 7 Z Z ðc þ1Þ=c5 1þ1=c5 1þc =c þ D7 k 4 ðtÞðc5 t þ c6 Þ  dt þ s ½D5 k 2 ðtÞðc5 t þ c6 Þ 7 5 þ g ½D5 k 1 ðtÞðc5 t þ c6 Þ 7 Z 1þ1=c5 ðc þ1Þ=c5 1þc =c þ D6 k 4 ðtÞðc5 t þ c6 Þ  dt þ ½k 1 ðtÞD5 D6 ðc5 t þ c6 Þ 7 þ k 2 ðtÞD5 D7 ðc5 t þ c6 Þ 7 5 Z 1=c 1þ1=c5 ðc þ1Þ=c5 þ k 3 ðtÞD5 ðc5 t þ c6 Þ 5 þ k 4 ðtÞD6 D7 ðc5 t þ c6 Þ þ k 5 ðtÞ dt þ n ½D6 k 1 ðtÞðc5 t þ c6 Þ 7 þ D7 k 2 ðtÞðc5 t þ c6 Þ

1þc7 =c5

þ k 3 ðtÞðc5 t þ c6 Þ

1=c5

 dt;

ð28Þ

where n ¼ xðc5 t Rþ c6 Þ1=c5  D5 ; g ¼ yðc5 t þ c6 Þc7 =c5  D6 ; s ¼ zðcR5 t þ c6 Þ1ðc7 þ1Þ=c5  D7 and f are similarity 1þ1=c5 R 1þc7 =c5 R variables, D ¼ ½ðc t þ c Þ ðtÞðc t þ c Þ dt dt, D ¼ ½ðc t þ c Þ ak bk 5 5 6 4 5 6 6 5 6 2 ðtÞðc5 t þ c6 Þ dt dt and R 2ðc7 þ1Þ=c5 R D7 ¼ ½ðc5 t þ c6 Þ ck 1 ðtÞðc5 t þ c6 Þ dt dt. Substituting Eq. (28) into Eq. (1), we have nfnnn þ gfnng =c7  sfnns =ðc5 þ c7 þ 1Þ þ afnnn fgs þ bfnng fns þ cfnns fng þ dfnn fngs þ efnnngs ¼ 0:

ð29Þ

In fact, the reduced Eqs. (22), (24), (26) and (29) of Eq. (1) can be written uniformly as ðp0 þ p1 nÞfnnn þ qgfnng þ rsfnns þ afnnn fgs þ bfnng fns þ cfnns fng þ dfnn fngs þ efnnngs ¼ 0;

ð30Þ

where p0 and p1 are arbitrary constants, q and r are non-zero constants. 4. Similarity solutions of (3 + 1)-dimensional breaking soliton equation It is easily shown that one must solve the (2 + 1)-dimensional variable coefficients Eq. (30) to obtain similarity solutions of Eq. (1). Herein we consider the following two special cases. 4.1. Supposing a special solution of Eq. (30) is of the form f ðn; g; sÞ ¼ s1 ðg; sÞn2 þ s2 ðg; sÞn þ s3 ðg; sÞ;

ð31Þ

where s1(g,s), s2(g,s) and s3(g,s) are functions to be determined later. Substituting Eq. (31) into Eq. (30) leads to the following system ðb þ cÞs1g s1s þ ds1 s1gs ¼ 0;

ð32Þ

qgs1g þ rss1s þ bs1g s2s þ cs1s s2g þ ds1 s2gs ¼ 0:

ð33Þ

Solving the above Eqs. (32) and (33), one can get the following special solutions of Eq. (30) f1 ðn; g; sÞ ¼ s2 ðg; sÞn þ s3 ðg; sÞ; f2 ðn; g; sÞ ¼ D1 ðgsðq=rÞ ÞD2 n2 þ s3 ðg; sÞ;

ð34Þ ð35Þ

ðb þ c þ d ¼ 0Þ

f3 ðn; g; sÞ ¼ ½ðb þ c þ dÞðD1 lnðgsðq=rÞ Þ þ D2 Þ=dd=ðbþcþdÞ n2 þ s3 ðg; sÞ;

ðdðb þ c þ dÞ 6¼ 0Þ

ð36Þ

338

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

where s2(g,s) and s3(g,s) are arbitrary functions of g and s, D1 and D2 are integral constants. Substituting f1, f2 and f3 into Eq. (21), we can obtain rational function solutions of the (3 + 1)-dimensional breaking soliton equation as follows: uðt; x; y; zÞ ¼ s2 ðg; sÞn þ s3 ðg; sÞ þ r1 ; uðt; x; y; zÞ ¼ D1 ðgs

ð37Þ

ðq=rÞ D2 2

Þ n þ s3 ðg; sÞ þ r1 ;

uðt; x; y; zÞ ¼ ½ðb þ c þ dÞðD1 lnðgs

ðq=rÞ

ðb þ c þ d ¼ 0Þ

Þ þ D2 Þ=d

d=ðbþcþdÞ 2

n þ s3 ðg; sÞ þ r1 ;

ð38Þ ðdðb þ c þ dÞ 6¼ 0Þ

ð39Þ

where n ¼ x þ at; g ¼ yðc0 t þ c1 Þ1=c0  D1 ; s ¼ zðc0 t þ c1 Þðc0 þ1Þ=c0  D2 and Z Z Z 1=c 1þ1=c0 1=c dt  g ½atA1 ðtÞðc0 t þ c1 Þ 0 þ D2 ac0 =a dt r1 ¼ ng A1 ðtÞðc0 t þ c1 Þ 0 dt þ ns A2 ðtÞðc0 t þ c1 Þ Z Z 1þ1=c0 1=c þ D1 ac0 =a dt  gsac0 t=a  ½A1 ðtÞD1 atðc0 t þ c1 Þ 0  s ½atA2 ðtÞðc0 t þ c1 Þ Z 1þ1=c0 1=c þ A2 ðtÞD2 atðc0 t þ c1 Þ þ A3 ðtÞat þ D1 D2 ac0 =a  A4 ðtÞ dt þ n ½D1 A1 ðtÞðc0 t þ c1 Þ 0 þ D2 A2 ðtÞðc0 t þ c1 Þ

1þ1=c0

þ A3 ðtÞ dt:

Similarly, substituting f1, f2 and f3 into Eqs. (23), (25) and (28) can obtain other nine type rational function solutions of Eq. (1), which is omitted here. 4.2. Assume Eq. (30) have the following form solution f ðn; g; sÞ ¼ w1 ðnÞw2 ðg; sÞ;

ð40Þ

where w1(n) and w2(g,s) are functions to be determined. Substituting Eq. (40) into Eq. (30) yields 00 000 0 00 ½ðp0 þ p1 nÞw2 þ ew2gs w000 1 þ ðqgw2g þ rsw2s Þw1 þ aw2 w2gs w1 w1 þ ½ðb þ cÞw2g w2s þ dw2 w2gs w1 w1

¼ 0:

ð41Þ

Let p1 ¼ 0;

qgw2g þ rsw2s ¼ 0;

ðb þ cÞw2g w2s þ dw2 w2gs ¼ D3 ðp0 w2 þ ew2gs Þ;

ð42Þ

where D3 is a non-zero constant. If q = r, p0 = (b + c + d)/D3, it follows from Eq. (42) that w2 ðg; sÞ ¼ gs  D3 e=ðb þ cÞ:

ð43Þ w000 1

D3 w01 w001

þ ¼ 0. Under Eq. (42), Eq. (41) can be further simplified to an ordinary differential equation Integrating the obtained equation with respect to n and supposing u ¼ w01 yield the desired Riccati equation u0 þ D3 u2 =2 þ D4 ¼ 0;

ð44Þ

Table 1 Some solutions of Eq. (30) D3

D4

1 1 1 2 2 2 8 8 8

1/2 1/2 1/2 1 1 1 1 1 1

Solutions f(n,g,s) R R R R w2 R ðcoth n  csch nÞ dn; wR2 ðtanh n  isech nÞ dn; R w2 ½tanh n=ð1  sech nÞ dn; w2 ½coth n=ð1  icsch nÞ dn w2 R ðcsc n  cot nÞ dn; w2 R ðtan n  sec nÞ dn; w2 R ½tan n=ð1  sec nÞ dn w2 R ðcot n  csc nÞ Rdn; w2 ðsec n  tan nÞ dn; w2 ½cot n=ð1  csc nÞ dn w2 R tanh n dn; w2 coth n dn w2 R tan n dn w2 R cot n dn w2 R ½tanh n=ð1 þ tanh2 nÞ dn w2 R ½tan n=ð1  tan2 nÞ dn w2 ½cot n=ð1  cot2 nÞ dn

Z.-l. Yan et al. / Applied Mathematics and Computation 201 (2008) 333–339

339

where D4 is an integral constant. Considerable explicit solutions of Eq. (44) have been given in Ref. [10], so we derive the corresponding solutions of Eq. (30) while p1 = a = 0, q = r and p0 = (b + c + d)/D3 as follows in Table 1. Hence the nonlinear partial differential equation uxxt þ buxxy uxz þ cuxy uxxz þ duxx uxyz þ euxxxyz ¼ 0; possesses the following similarity solutions: Z Z Z u ¼ f ðn; g; sÞ þ ng A1 ðtÞet=c1 dt þ ns A2 ðtÞet=c1 dt þ n ðD01 A1 ðtÞet=c1 þ D02 A2 ðtÞet=c1 þ A3 ðtÞÞdt Z Z Z t=c1 t=c1 dt  s A2 ðtÞate dt  ðatA1 ðtÞD01 et=c1 þ atA2 ðtÞD02 et=c1 þ A3 ðtÞat  A4 ðtÞÞdt;  g A1 ðtÞate

ð45Þ

ð46Þ

where f is determined by Table 1, n ¼ x þ ðb þ c þ dÞt=D3 ; g ¼ yet=c1  D01 and s ¼ zet=c1  D02 . where w2 is determined by Eq. (43) and i2 = 1. 5. Conclusions We have discussed the symmetry reductions and similarity solutions of the (3 + 1)-dimensional breaking soliton equation in this paper. It is shown that Eq. (1) can be reduced to (2 + 1)-dimensional variable coefficients partial differential Eqs. (22), (24), (26) and (29) by the compatibility method. Abundant similarity solutions of Eq. (1) can be constructed by solving the obtained (2 + 1)-dimensional nonlinear partial differential equations, including rational function solutions, triangular function solutions and hyperbolic function solutions. Acknowledgement This work was supported by the Nature Science Foundation of Shandong Province in China (No. Q2005A01). References [1] L. Huibin, W. Kelin, Exact solutions for two nonlinear evolution equations, J. Phys. A 23 (1990) 4097–4105. [2] S.Y. Lou, G.X. Huang, H.Y. Ruan, Exact solitary waves in a convecting fluid, J. Phys. A 24 (1991) L 587–L 590. [3] M.L. Wang, Y.B. Zhou, Z.B. Li, Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [4] G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969) 1025–1042. [5] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin, 1986. [6] Z.L. Yan, X.Q. Liu, Symmetry and similarity solutions of variable coefficients generalized Zakharov-Kuznetsov equation, Appl. Math. Comput. 180 (2006) 288–294. [7] Z.L. Yan, X.Q. Liu, L. Wang, The direct symmetry method and its application in viable coefficients Schro¨dinger equation, Appl. Math. Comput. 187 (2007) 701–707. [8] H. Zhao, C.L. Bai, Abundant multisoliton structure of (3 + 1)-dimensional breaking soliton equation, Commun. Theor. Phys. 42 (2004) 561–564. [9] J. Lin, S.Y. Lou, K.L. Wang, High-dimensional Virasoro integral models and exact solutions, Phys. Lett. A 287 (2001) 257–267. [10] H.T. Chen, H.Q. Zhang, New multiple soliton-like solutions to (3 + 1)-dimensional Burgers equation with variable coefficients, Commun. Theor. Phys. 42 (2004) 497–500.