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Soliton solution to BKP equation in Wronskian form
Applied Mathematics and Computation 224 (2013) 250–258
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Soliton solution to BKP equation in Wronskian form Yingli Kang, Yi Zhang, Ligang Jin ⇑ Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
a r t i c l e
i n f o
a b s t r a c t We find out a group of identities, named Wronskian identities of bilinear KP hierarchy here, as well as two useful properties of D-operator. This makes it possible to easily search for some new Wronskian solutions for PDE which owns bilinear form, and to simplify the process of the proof. As an application of this new method, we propose the first Wronskian condition for the BKP I equation and for the BKP II equation, respectively. In addition, we also discuss several generalized BKP equations and obtain their Wronskian conditions. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: D-operator KP hierarchy BKP equation Wronskian identities Wronskian solutions
1. Introduction For the nonlinear evolution equations that exhibit multisoliton solutions it has been usual to express the N-soliton solution in one of two ways. First, using inverse scattering or Hirota’s direct method, the N-soliton solution may be written in terms of Wronskian determinant. It has been revealed that a lot of soliton equations possess the N-soliton solution in terms of Wronskian determinant, such as KdV equation [1–6], KP equation [3], Boussinesq equation [7–10], Jimbo–Miwa equation [11–13] and so on. Second, for the BKP-type equation the N-soliton solution can be expressed as pfaffian [14–17]. Hirota [18] said that ‘‘in order to find soliton solutions for BKP-type equations by the inverse method, it is necessary to change the structure of the Gel’fand–Levitan integral equation because the solutions are expressed not as determinants but as pfaffians’’. However, in this letter we will construct Wronskian condition both for the first member and for the second member of the BKP hierarchy and express the N-soliton solution in terms of Wronskian determinant for the first time by an improved Wronskian technique. We express a Wronskian determinant by the compact Freeman and Nimmo’s notion [3]:
ð0Þ / 1 ð0Þ / 2 Wrð/1 ; /2 ; . . . ; /N Þ ¼ . . . ð0Þ / N
ð1Þ
/1
ð1Þ
/2 .. . ð1Þ /N
ðN1Þ /2 d .. .. ¼ j N 1j; . . ðN1Þ /N ðN1Þ
/1
ðN P 1Þ;
ð1:1Þ
where ð0Þ
/j
¼ /j ;
ðmÞ
/j
¼ @m x /j ;
ðm P 1; 1 6 j 6 NÞ:
ð1:2Þ
Next we sometimes denote jN 1j by s or j/ð0Þ ; /ð1Þ ; . . . ; /ðN1Þ j for convenience. 0 Let C ¼ jC 1 ; C 2 ; . . . ; C N j be a determinant with columns C 1 ; C 2 ; . . . ; C N where C i ¼ ðfi1 ; fi2 ; . . . ; fiN Þ , and every element in C, denoted fij ðxÞ, is a function with respect to variable x. Therefore, C is also a function with respect to x. Define that ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (L. Jin). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.085
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
@ ðmÞ fij ¼ @ m x fij ;
1 6 i;
j 6 N;
ð1:3aÞ 0
@ ðmÞ C i ¼ ð@ ðmÞ fi1 ; @ ðmÞ fi2 ; . . . ; @ ðmÞ fiN Þ ; @ ðmÞ C ¼
1 6 i 6 N;
ð1:3bÞ
N X jC 1 ; . . . ; C i1 ; @ ðmÞ C i ; C iþ1 ; . . . ; C N j;
ð1:3cÞ
i¼1
DpðmÞ DqðnÞ C C ¼ ð@ ðmÞ @ ðmÞ0 Þp ð@ ðnÞ @ ðnÞ0 Þq CðxÞCðx0 Þjx0 ¼x ;
ð1:3dÞ
where we similarly use notion @ ðmÞ0 when every element in C is a function with respect to variable x0 . In what follows, some examples, in which the determinant C is taken as a Wronskian determinant, are provided for a good understanding of the above definition (1.3).
1j ¼ @ x j Nd 1j ¼ j Nd 2; Nj; @ ð1Þ j Nd
ð1:4Þ
@ ð2Þ j Nd 1j ¼ j Nd 3; N 1; Nj þ j Nd 2; N þ 1j;
ð1:5Þ
1j ¼ j Nd 3; N 1; Nj þ j Nd 2; N þ 1j; @ 2x j Nd
ð1:6Þ
D2ð2Þ s s ¼ 2ð@ 2ð2Þ sÞs 2ð@ ð2Þ sÞ2 4; N 2; N 1; N þ 1j j Nd 3; N 1; N 5; N 3; N 2; N 1; Nj j Nd ¼ 2ðj Nd 2
þ 2j þ 2j Nd 3; N; N þ 1j þ j Nd 2; N þ 3jÞðj Nd 1jÞ 2ðj Nd 3; N 1; Nj þ j Nd 2; N þ 1jÞ :
ð1:7Þ
Notice that @ ðiÞ j Nd 1j does not equal to @ ix j Nd 1j for all integer i > 1. 2. Wronskian identities of bilinear KP hierarchy The KP hierarchy [19] is listed as follows
ðD41 4D1 D3 þ 3D22 Þf f ¼ 0; ½ðD31 þ 2D3 ÞD2 3D1 D4 f f ¼ 0; ðD61 20D31 D3 80D23 þ 144D1 D5 45D21 D22 Þf f ¼ 0;
ð2:1Þ
ðD61 þ 4D31 D3 32D23 þ 36D2 D4 9D21 D22 Þf f ¼ 0: .. . If Di is substituted for DðiÞ and let f ¼ j Nd 1j, then every equation in (2.1) is transformed into an identity. Especially, for the equations of degree four, degree five and degree six in (2.1), we have the following Lemma. Lemma 1. Using the above notions, it holds that
ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j 0; ½ðD3ð1Þ ðD6ð1Þ ðD6ð1Þ
ð2:2Þ
þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j 0;
20D3ð1Þ Dð3Þ
þ
4D3ð1Þ Dð3Þ
80D2ð3Þ
32D2ð3Þ
þ 144Dð1Þ Dð5Þ
þ 36Dð2Þ Dð4Þ
ð2:3Þ
1j 45D2ð1Þ D2ð2Þ Þj Nd
1j 9D2ð1Þ D2ð2Þ Þj Nd
j Nd 1j 0;
j Nd 1j 0:
ð2:4Þ ð2:5Þ
Proof of Lemma 1. After calculation , the identity (2.2) can be transformed into a Plücker relation, and the identity (2.3) can be transformed into a linear combination of two Plücker relations, and either of identities (2.3) and (2.4) can be transformed into a linear combination of five Plücker relations , that is,
ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j ¼ 24ðj Nd 1jj Nd 3; N; N þ 1j j Nd 2; Njj Nd 3; N 1; N þ 1j 2; N þ 1jj Nd 3; N 1; NjÞ 0; þ j Nd ½ðD3ð1Þ þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j ¼ 12ðj Nd 3; N; N þ 2jj Nd 1j j Nd 3; N 1; N þ 2jj Nd 2; Nj 3; N 1; Njj Nd 2; N þ 2jÞ 12ðj Nd 4; N 2; N; N þ 1jj Nd 1j j Nd 4; N 2; N 1; N þ 1jj Nd 2; Nj þ j Nd þ j Nd 4; N 2; N 1; Njj Nd 2; N þ 1jÞ 0;
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
½D6ð1Þ 20D3ð1Þ Dð3Þ 80D2ð3Þ þ 144Dð1Þ Dð5Þ 45D2ð1Þ D2ð2Þ j Nd 1j j Nd 1j ¼ 720½j Nd 1jj Nd 4; N 2; N; N þ 2j 2; Njj Nd 4; N 2; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 4; N 2; N 1; Nj 360½j Nd 1jj Nd 5; N 3; N j Nd 5; N 3; N 2; N 1; N þ 1j þ j Nd 5; N 3; N 2; N 1; Nj 2; N; N þ 1j j Nd 2; Njj Nd 2; N þ 1jj Nd 360½j Nd 1jj Nd 4; N 1; N; N þ 1j j Nd 3; N 1; Njj Nd 4; N 2; N 1; N þ 1j þ j Nd 3; N 1; N þ 1jj Nd 4; N 2; N 1; Nj 360½j Nd 1jj Nd 3; N; N þ 3j j Nd 2; Njj Nd 3; N 1; N þ 3j þ j Nd 2; N þ 3jj Nd 3; N 1; Nj 1jj Nd 3; N þ 1; N þ 2j j Nd 2; N þ 1jj Nd 3; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 3; N 1; N þ 1j 0; 360½j Nd and
5; N 3; N 2; N; N þ 1j ½D6ð1Þ þ 4D3ð1Þ Dð3Þ 32D2ð3Þ þ 36Dð2Þ Dð4Þ 9D2ð1Þ D2ð2Þ j Nd 1j j Nd 1j ¼ 72½j Nd 1jj Nd j Nd 2; Njj Nd 2; N þ 1jj Nd 1jj Nd 4; N 5; N 3; N 2; N 1; N þ 1j þ j Nd 5; N 3; N 2; N 1; Nj þ 144½j Nd 2; N; N þ 2j j Nd 2; Njj Nd 4; N 2; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 4; N 2; N 1; Nj 216½j Nd 1jj Nd 4; N 3; N 1; Njj Nd 4; N 2; N 1; N þ 1j þ j Nd 3; N 1; N þ 1jj Nd 4; N 2; N 1; Nj 1; N; N þ 1j j Nd 216½j Nd 1jj Nd 3; N þ 1; N þ 2j j Nd 2; N þ 1jj Nd 3; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 3; N 1; N þ 1j þ 72½j Nd 1jj Nd 3; N; N þ 3j j Nd 2; Njj Nd 3; N 1; N þ 3j þ j Nd 2; N þ 3jj Nd 3; N 1; Nj 0: Therefore, all of the identities (2.2)–(2.5) hold. h
3. Two useful properties of D-operator Now we are ready to give two useful properties of D-operator which play an important role in our solving method: Lemma 2. If the functions /0i s with respect to variables {x, y, z} in Wronskian determinant (1.1) satisfy
@ y / ¼ a1 @ x þ a2 @ 2x þ þ am @ m x /; @ z / ¼ b1 @ x þ b2 @ 2x þ þ bn @ nx /; 0
where / ¼ ð/1 ; /2 ; . . . ; /N Þ and
1j Dpy Dqz j Nd
a0i s;
0 bi s
ð3:1Þ ð3:2Þ
are arbitrary functions independent on x, then
j Nd 1j ¼ ða1 Dð1Þ þ a2 Dð2Þ þ þ am DðmÞ Þp ðb1 Dð1Þ þ b2 Dð2Þ þ þ bn DðnÞ Þ j Nd 1j j Nd 1j: q
ð3:3Þ
Proof of Lemma 2. Without loss of generality, we can assume that m P n. By the definition (1.3) and the conditions (3.1) and (3.2), we have
Dpy Dqz s s ¼ ð@ y @ y0 Þp ð@ z @ z0 Þq sðx; y; zÞsðx; y0 ; z0 Þjy0 ¼y;z0 ¼z p ¼ a1 ð@ ð1Þ @ ð1Þ0 Þ þ a2 ð@ ð2Þ @ ð2Þ0 Þ þ am ð@ ðmÞ @ ðmÞ0 Þ q b1 ð@ ð1Þ @ ð1Þ0 Þ þ b2 ð@ ð2Þ @ ð2Þ0 Þ þ bn ð@ ðnÞ @ ðnÞ0 Þ sðx; y; zÞsðx0 ; y; zÞjx0 ¼x ( ! ) m n X Y Y x ¼ ½ai ð@ i @ i0 Þki ½bj ð@ j @ j0 Þ j sðx; y; zÞsðx0 ; y; zÞjx0 ¼x k1 þk2 þþkm ¼p x1 þx2 þþxn ¼q ki ;xi ¼0;1;2;...
¼
X k1 þk2 þþkm ¼p x1 þx2 þþxn ¼q ki ;xi ¼0;1;2;...
i¼1
(" m Y
ðai Di Þki
i¼1
j¼1
n Y
# ðbj Dj Þ
xj
)
ss
j¼1
¼ ða1 Dð1Þ þ a2 Dð2Þ þ þ am DðmÞ Þp ðb1 Dð1Þ þ b2 Dð2Þ þ þ bn DðnÞ Þ
q
s s:
It means that the conclusion (3.3) holds. h
Lemma 3. If the functions /0i s in Wronskian determinant (1.1) satisfy
A/ ¼ ðc1 @ x þ c2 @ 2x þ þ cm @ m x Þ/;
ð3:4Þ
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
where / ¼ ð/1 ; /2 ; . . . ; /N Þ0 ; c0i s are arbitrary functions independent on x and A ¼ ðaij ÞNN is an arbitrary real matrix, then for any non-negative integer n,
1j j Nd 1j ¼ 0: DðnÞ ðc1 Dð1Þ þ c2 Dð2Þ þ þ cm DðmÞ Þj Nd
ð3:5Þ
Proof of Lemma 3. Let us compute the value of TrðAÞs as follows:
ð0Þ / 1 . . . N X a /ð0Þ TrðAÞs ¼ ii i i¼1 .. . ð0Þ /N
ð1Þ /1
.. . ð1Þ
aii /i .. .
ð1Þ
/N
/ð0Þ 1 . . .. N . X . N X ð0Þ ðN1Þ aij /j aii /i ¼ i¼1 j¼1 .. . .. . ðN1Þ /N /ð0Þ ðN1Þ /1
N
ð1Þ
/1 .. . N X ð1Þ aij /j j¼1
.. . ð1Þ
/N
.. . N X ðN1Þ aij /j ; j¼1 .. . ðN1Þ /N
ðN1Þ
/1
ð3:6Þ
where we have used the elementary row operations of a determinant. Then under condition (3.3), the above expression (3.5) equals to
ð0Þ /1 . m .. N X X ð0Þ ck @ kx /i i¼1 k¼1 .. . ð0Þ /N
ð1Þ
m X
ð1Þ
ck @ kx /i
k¼1
.. .
ð1Þ /N
ð0Þ ð1Þ ðN1Þ / /1 /1 1 .. .. .. .. . m X m N . . . X X ðN1Þ k ð1Þ k ðN1Þ ck @ kx /i ck @ kx /ð0Þ ¼ @ / @ / x i x i i k¼1 .. .. k¼1 i¼1 .. .. . . . ð0Þ ð1Þ ðN1Þ . / / / ðN1Þ N N N /N /ð0Þ /ði1Þ @ k /ðiÞ /ðiþ1Þ 1 x 1 1 1 ði1Þ ðiþ1Þ k ðiÞ m N1 ð0Þ X X / / @ /2 x /2 2 2 ¼ ck . .. .. .. . i¼0 . k¼1 . . . ð0Þ ði1Þ ðiÞ ðiþ1Þ k / /N @ x /N /N N
/1 .. .
ðN1Þ
/1
ðN1Þ /1 ðN1Þ /2 .. ; . ðN1Þ /
ð3:7Þ
N
which gives that
TrðAÞs ¼ ðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þs:
ð3:8Þ
Through a similar calculation, we have
TrðAÞð@ ðnÞ sÞ ¼ ðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þð@ ðnÞ sÞ:
ð3:9Þ
Substituting (3.7) and (3.8) into an apparently established identity
ð@ ðnÞ sÞ½TrðAÞs ¼ s½TrðAÞð@ ðnÞ sÞ;
ð3:10Þ
we have
ð@ ðnÞ sÞðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þs ¼ sðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þð@ ðnÞ sÞ:
ð3:11Þ
So we can get
c1 ½ð@ ðnÞ @ ð1Þ sÞs ð@ ðnÞ sÞð@ ð1Þ sÞ þ c2 ½ð@ ðnÞ @ ð2Þ sÞs ð@ ðnÞ sÞð@ ð2Þ sÞ þ þ cm ½ð@ ðnÞ @ ðmÞ sÞs ð@ ðnÞ sÞð@ ðmÞ sÞ ¼ 0
ð3:12Þ
and then
Dn ðc1 Dð1Þ þ c2 Dð2Þ þ þ cm DðmÞ Þs s ¼ 0:
ð3:13Þ
The proof is completed. h
4. Application of the improved Wronskian technique In this section, the Wronskian condition will be constructed for the first member and the second member of BKP hierarchy and for the high-dimensional equations, respectively. Based on these Wronskian conditions, various types of Wronskian solutions, such as soliton, positon, negaton, complexiton, rational solutions and so on (see Refs. [6,13,11,10,2] for detail), can be considered. Some sample solutions will be provided for illumination.
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
4.1. Wronskian condition for BKP I equation The bilinear form of the first equation of BKP hierarchy reads
h i ðDt D3x ÞDy þ 3D2x f f ¼ 0:
ð4:1:1Þ
Next the first Wronskian condition for Eq. (4.1.1) will be constructed. Theorem 1. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ a/x ;
ð4:1:2Þ
3 /t ¼ 4/3x /x ; a
ð4:1:3Þ
A/ ¼ /2x ;
ð4:1:4Þ
where A ¼ ðaij ÞNN is an arbitrary real matrix and a is arbitrary real nonzero constant. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.1.1). Proof of Theorem 1. By Lemma 2 and conditions (4.1.2) and (4.1.3), we have
Dt Dy j Nd 1j j Nd 1j ¼ ð4aDð1Þ Dð3Þ 3D2ð1Þ Þj Nd 1j j Nd 1j;
ð4:1:5Þ
1j D3x Dy j Nd
ð4:1:6Þ
1j 3D2x j Nd
j Nd 1j ¼
j Nd 1j ¼
aD4ð1Þ j Nd 1j
3D2ð1Þ j Nd 1j
j Nd 1j;
j Nd 1j
ð4:1:7Þ
and using Lemma 3 and condition (4.1.4), we get
D2ð2Þ j Nd 1j j Nd 1j ¼ 0:
ð4:1:8Þ
Then substituting the results (4.1.5)–(4.1.7) into the left-side of Eq. (4.1.1), we obtain
½ðDt D3x ÞDy þ 3D2x j Nd 1j j Nd 1j ¼ ½aðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þ þ 3aDð2Þ2 j Nd 1j j Nd 1j ¼ 0 where we have made use of the identities (2.2) of Lemma 1 and result (4.1.8). The proof is completed. h Soliton Let
2 k 1 A¼
2
k2
: .. . 2 kN
ð4:1:9Þ
By solving the Wronskian conditions (4.1.2)–(4.1.4), N-soliton soluton to Eq. (4.1.1) is given by
f ¼ Wrð/1 ; /2 ; . . . ; /N Þ; ki xþaki yþð4k3i 3aki Þt
where /i ¼ e Positon Let
2 k 1 A¼
2
k2
ð4:1:10Þ ði ¼ 1; 2; . . . ; NÞ.
: .. . 2 kN
ð4:1:11Þ
By solving the Wronskian conditions (4.1.2)–(4.1.4), positon to Eq. (4.1.1) is given by
f ¼ Wrð/1 ; /2 ; . . . ; /N Þ;
3 3 with /i ¼ ci;1 cos ki x þ aki y 4ki þ 3ka i t þ ci;2 sin ki x þ aki y 4ki þ 3ka i t arbitrary constants.
ð4:1:12Þ ði ¼ 1; 2; . . . ; NÞ, where both ci;1 and ci;2 are
Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
255
4.2. Wronskian condition for BKP II equation The second equation of the BKP hierarchy
ðut þ 15ux u3x þ 15u3x 15ux uy þ u5x Þx 5u3x;y 5uyy ¼ 0
ð4:2:1Þ
can be written as the bilinear form
ðD6x 5D2y 5D3x Dy þ 9Dx Dt Þf f ¼ 0;
ð4:2:2Þ
by a Cole–Hopf transformation
u ¼ 2ðlnf Þx :
ð4:2:3Þ
In this section, we present the first Wronskian condition for Eq. (4.2.2). Theorem 2. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ 2/3x ;
ð4:2:4Þ
/t ¼ 36/5x ;
ð4:2:5Þ
A/ ¼ /2x ;
ð4:2:6Þ
where A ¼ ðaij ÞNN is an arbitrary real matrix. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.2.2). Proof of Theorem 2. First, by Lemma 2 and conditions (4.2.4) and (4.2.5), we have
1j j Nd 1j ¼ D6ð1Þ j Nd 1j j Nd 1j; D6x j Nd
ð4:2:7Þ
D2y j Nd 1j j Nd 1j ¼ 4D2ð3Þ j Nd 1j j Nd 1j;
ð4:2:8Þ
1j j Nd 1j ¼ 2D3ð1Þ Dð3Þ j Nd 1j j Nd 1j; D3x Dy j Nd
ð4:2:9Þ
1j j Nd 1j ¼ 36Dð1Þ Dð5Þ j Nd 1j j Nd 1j Dx Dt j Nd
ð4:2:10Þ
and second from Lemma 3 and condition (4.2.6), we get
Dð2Þ Dð4Þ j Nd 1j j Nd 1j ¼ 0:
ð4:2:11Þ
Then substituting the results (4.2.7)–(4.2.10) into the left-side of Eq. (4.2.2), we obtain
1j j Nd 1j ¼ ½D6ð1Þ 20D2ð3Þ þ 10D3ð1Þ Dð3Þ 36Dð1Þ Dð5Þ j Nd 1j j Nd 1j ðD6x 5D2y 5D3x Dy þ Dx Dt Þj Nd 1 5 ¼ ½D6ð1Þ 20D3ð1Þ Dð3Þ 80D2ð3Þ þ 144Dð1Þ Dð5Þ 45D2ð1Þ D2ð2Þ þ ½D6ð1Þ þ 4D3ð1Þ Dð3Þ 32D2ð3Þ þ 36Dð2Þ Dð4Þ 4 4 1j j Nd 1j ¼ 0 9D2ð1Þ D2ð2Þ 45Dð2Þ Dð4Þ j Nd where we have made use of the identities (2.4) and (2.5) of Lemma 1 and the result (4.2.11). The proof is completed. h Soliton Let
2 k 1 A¼
2
k2
: .. . 2 kN
ð4:2:12Þ
By solving the Wronskian conditions (4.2.4)–(4.2.6), N-soliton soluton to Eq. (4.2.2) is given by
u ¼ 2ðWrð/1 ; /2 ; . . . ; /N ÞÞx ; ki x2k3i y36k5i t
where /i ¼ e Positon Let
ði ¼ 1; 2; . . . ; NÞ.
ð4:2:13Þ
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
2 k 1 A¼
2
k2
: .. . 2 kN
ð4:2:14Þ
By solving the Wronskian conditions (4.2.4)–(4.2.6), positon solution to Eq. (4.2.2) is given by
u ¼ 2ðWrð/1 ; /2 ; . . . ; /N ÞÞx ; with /i ¼ ci;1 cosðki x þ constants.
3 2ki y
5 36ki tÞ
ð4:2:15Þ þ ci;2 sinðki x þ
3 2ki y
5 36ki tÞ
ði ¼ 1; 2; . . . ; NÞ, where both ci;1 and ci;2 are arbitrary
4.3. Wronskian condition for generalized BKP equations Recently, the high-dimensional PDEs have attracted much interest from researchers. For example, some generalized KP 0 and BKP equations are often considered through different methods, such as improved GG -expansion method [20], Wronskian €cklund transformation, dressing method [22], multiple exp-function algorithm technique [21], pfaffian technique, bilinear Ba [23] and so on. The improved Wronskian technique described in this paper also provides us a easy way to search for the Wronskian condition for generalized bilinear KP and BKP equations. Some examples are given as follows. First of all, we consider the following (3 + 1)-dimensional generalized BKP equation [24,22]:
uyz uxxxy 3ðux uy Þx þ 3uxx þ 3uzz ¼ 0:
ð4:3:1Þ
When z ¼ x, Eq. (4.3.1) reduces to the BKP I equation. Under the dependent variable transformation:
u ¼ 2ðlnf Þx ;
ð4:3:2Þ
the above Eq. (4.3.1) can be mapped into the Hirota bilinear equation:
ðDt Dy D3x Dy þ 3D2x þ 3D2z Þf f ¼ 0:
ð4:3:3Þ
Theorem 3. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ r 2 /x ;
ð4:3:4Þ
/z ¼ r/xx ;
ð4:3:5Þ
/t ¼
3 / þ 4/xxx ; r2 x
ð4:3:6Þ
where r is an arbitrary nonzero real constant. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.3.3). Proof of Theorem 3. By Lemma 2 and conditions (4.3.4)–(4.3.6), we have
1j j Nd 1j ¼ ð3D2ð1Þ 4r 2 Dð1Þ Dð3Þ Þj Nd 1j j Nd 1j; Dt Dy j Nd
ð4:3:7Þ
D3x Dy j Nd 1j j Nd 1j ¼ r 2 D4ð1Þ j Nd 1j j Nd 1j;
ð4:3:8Þ
D2x j Nd 1j j Nd 1j ¼ D2ð1Þ j Nd 1j j Nd 1j;
ð4:3:9Þ
D2z j Nd 1j j Nd 1j ¼ r 2 D2ð2Þ j Nd 1j j Nd 1j:
ð4:3:10Þ
Then substituting the results (4.3.7)–(4.3.10) into the left-side of Eq. (4.3.3), we obtain
1j j Nd 1j ¼ r 2 ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j ¼ 0; ðDt Dy D3x Dy þ 3D2x þ 3D2z Þj Nd where we have made use of the identity (2.2) of Lemma 1. The proof is completed. h Next, we consider another (3 + 1)-dimensional generalized BKP equation [23,25]:
uty uxxxy 3ðux uy Þx þ 3uxz ¼ 0;
ð4:3:11Þ
Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
257
which reduces to the BKP I equation if z ¼ x. Under the dependent variable transformation u ¼ 2ðlnf Þx , the above equation is transformed into the Hirota bilinear form:
ðDt Dy D3x Dy þ 3Dx Dz Þf f ¼ 0:
ð4:3:12Þ
Theorem 4. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ r/xx ;
ð4:3:13Þ
/z ¼ r/xxxx ;
ð4:3:14Þ
/t ¼ 2/xxx ;
ð4:3:15Þ
where r is an arbitrary real constant. Then f ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.3.12). Proof of Theorem 4. By Lemma 2 and conditions (4.3.13)–(4.3.15), we have
1j j Nd 1j ¼ ð2rDð2Þ Dð3Þ Þj Nd 1j j Nd 1j; Dt Dy j Nd
ð4:3:16Þ
D3x Dy j Nd 1j
j Nd 1j ¼ rD3ð1Þ Dð2Þ j Nd 1j j Nd 1j;
ð4:3:17Þ
1j j Nd 1j ¼ rDð1Þ Dð4Þ j Nd 1j j Nd 1j; Dx Dz j Nd
ð4:3:18Þ
Then substituting the results (4.3.16)–(4.3.18) into the left-side of Eq. (4.3.12), we obtain
1j j Nd 1j ¼ r½ðD3ð1Þ þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j ¼ 0; ðDt Dy D3x Dy þ 3Dx Dz Þj Nd where we have made use of the identity (2.3) of Lemma 1. The proof is completed.
h
5. Concluding remarks We have discovered the Wronskian identities of the KP hierarchy and two lemmas on D-operator, which play key roles in our improved Wronskian technique. By utilizing this improved Wronskian technique, the Wronskian condition is presented for the BKP I equation and BKP II equation, respectively. Various types of Wronskian solutions can be considered and several sample solutions, especially the Soliton solutions in terms of Wronskian determinant, are listed. Besides, we also apply this technique to construct the Wronskian condition for generalized BKP equations. It is know that there are other interesting solutions such as pfaffian [22], Chain [25], Lump [26], multiple wave [23] and even complex traveling wave [27] type solutons to BKP equations. Because of the richness of the Wronskian solutions, we are looking forward to the possibility of deducing other type solutions from the Wronskian solutions. Moreover, we remark that the improved Wronskian technique can also be applied to more other bilinear KdV-type equations. Acknowledgments The authors would like to thanks the editors and reviewers for their careful check and meaningful comment. References [1] J. Satsuma, A Wronskian representation of n-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn. 40 (1979) 359–360. [2] W.X. Ma, Y.C. You, Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions, Trans. Am. Math. Soc. 357 (2005) 1753–1778. [3] N.C. Freeman, J.J.C. Nimmo, Soliton solutions of the Korteweg–de Vries and Kadomtsev–Petviashvili equations: the Wronskian technique, Phys. Lett. A 96 (1983) 443–446. [4] S. Sirianunpiboon, S.D. Howard, S.K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A 134 (1988) 31–33. [5] R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194. [6] W.X. Ma, Complexiton solutions to the Korteweg–de Vries equation, Phys. Lett. A 301 (2002) 35–44. [7] J.J.C. Nimmo, N.C. Freeman, A method of obtaining the soliton solution of the Boussinesq equation in terms of a Wronskian, Phys. Lett. A 95 (1983) 4–6. [8] W.X. Ma, C.X. Li, J.S. He, A second Wronskian formulation of the Boussinesq equation, Nonlinear Anal. 70 (2009) 4245–4258. [9] Y.Q. Yao, Y.Q. Liu, J. Ji, D.Y. Chen, Novel Wronskian solutions to Boussinesq equation, Commun. Theor. Phys. 48 (2007) 577–583. [10] C.X. Li, W.X. Ma, X.J. Liu, Y.B. Zeng, Wronskian solutions of the Boussinesq equation-solitons, negatons, positons and complexitons, Inverse Prob. 23 (2007) 5279–5296. [11] Y.N. Tang, W.X. Ma, L. Gao, Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo–Miwa equation, Appl. Math. Comput. 217 (2011) 8722– 8730. [12] Y.N. Tang, J.Y. Tu, W.X. Ma, Two new Wronskian conditions for the (3 + 1)-dimensional Jimbo–Miwa equation, Appl. Math. Comput. 218 (2012) 10050– 10055.
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[13] Y. Zhang, L.G. Jin, Y.L. Kang, Generalized Wronskian solutions for the (3 + 1)-dimensional Jimbo–Miwa equation, Appl. Math. Comput. 219 (2012) 2601–2610. [14] R. Hirota, M. Iwao, S. Tsujimoto, Soliton equations exhibiting pfaffian solutions, Glasg. Math. J. 43 (2001) 33–41. [15] R. Hirota, Soliton-solutions to the BKP equations. I: The pfaffian technique, J. Phys. Soc. Jpn 58 (1989) 2285–2296. [16] Q.P. Liu, M. Manas, The pfaffian solutions of the BKP hierarchy, Chin. Ann. Math. Ser. A 23 (2002) 693–698. [17] H.C. Hu, Soliton solutions for nonisospectral BKP equation, Commun. Theor. Phys. 49 (2008) 535–539. [18] R. Hirota, Soliton solutions to the BKP equations. II: The integral equation, J. Phys. Soc. Jpn 58 (1989) 2705–2712. [19] M. Jimbo, T. Miwa, Solitons and infinite dimensional lie algebras, Publ. RIMS, Kyoto Univ. 19 (1983) 943–1001. [20] H.Q. Zhang, A note on exact complex travelling wave solutions for (2 + 1)-dimensional B-type Kadomtsev–Petviashvili equation, Appl. Math. Comput. 216 (2010) 2771–2777. [21] W.X. Ma, A. Abdeljabbar, M.G. Asaad, Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation, Appl. Math. Comput. 217 (2011) 10016–10023. [22] M.G. Asaad, W.X. Ma, Pfaffian solutions to a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation and its modified counterpart, Appl. Math. Comput. 218 (2012) 5524–5542. [23] W.X. Ma, Z.N. Zhu, Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput. 218 (2012) 11871–11879. [24] W.X. Ma, Y. Zhang, Y.N. Tang, J.Y. Tu, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012) 7174–7183. [25] W.X. Ma, E.G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl 61 (2011) 950–959. [26] Y. Wu, X.X. Yang, Chain Solutions of the BKP Equation, vol. 23, International Academic Publishers, 1995. pp. 121–124. [27] H.C. Ma, Y. Wang, Z.Y. Qin, New exact complex traveling wave solutions for (2 + 1)-dimensional BKP equation, Appl. Math. Comput. 208 (2009) 564– 568.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Soliton solution to BKP equation in Wronskian form Yingli Kang, Yi Zhang, Ligang Jin ⇑ Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
a r t i c l e
i n f o
a b s t r a c t We find out a group of identities, named Wronskian identities of bilinear KP hierarchy here, as well as two useful properties of D-operator. This makes it possible to easily search for some new Wronskian solutions for PDE which owns bilinear form, and to simplify the process of the proof. As an application of this new method, we propose the first Wronskian condition for the BKP I equation and for the BKP II equation, respectively. In addition, we also discuss several generalized BKP equations and obtain their Wronskian conditions. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: D-operator KP hierarchy BKP equation Wronskian identities Wronskian solutions
1. Introduction For the nonlinear evolution equations that exhibit multisoliton solutions it has been usual to express the N-soliton solution in one of two ways. First, using inverse scattering or Hirota’s direct method, the N-soliton solution may be written in terms of Wronskian determinant. It has been revealed that a lot of soliton equations possess the N-soliton solution in terms of Wronskian determinant, such as KdV equation [1–6], KP equation [3], Boussinesq equation [7–10], Jimbo–Miwa equation [11–13] and so on. Second, for the BKP-type equation the N-soliton solution can be expressed as pfaffian [14–17]. Hirota [18] said that ‘‘in order to find soliton solutions for BKP-type equations by the inverse method, it is necessary to change the structure of the Gel’fand–Levitan integral equation because the solutions are expressed not as determinants but as pfaffians’’. However, in this letter we will construct Wronskian condition both for the first member and for the second member of the BKP hierarchy and express the N-soliton solution in terms of Wronskian determinant for the first time by an improved Wronskian technique. We express a Wronskian determinant by the compact Freeman and Nimmo’s notion [3]:
ð0Þ / 1 ð0Þ / 2 Wrð/1 ; /2 ; . . . ; /N Þ ¼ . . . ð0Þ / N
ð1Þ
/1
ð1Þ
/2 .. . ð1Þ /N
ðN1Þ /2 d .. .. ¼ j N 1j; . . ðN1Þ /N ðN1Þ
/1
ðN P 1Þ;
ð1:1Þ
where ð0Þ
/j
¼ /j ;
ðmÞ
/j
¼ @m x /j ;
ðm P 1; 1 6 j 6 NÞ:
ð1:2Þ
Next we sometimes denote jN 1j by s or j/ð0Þ ; /ð1Þ ; . . . ; /ðN1Þ j for convenience. 0 Let C ¼ jC 1 ; C 2 ; . . . ; C N j be a determinant with columns C 1 ; C 2 ; . . . ; C N where C i ¼ ðfi1 ; fi2 ; . . . ; fiN Þ , and every element in C, denoted fij ðxÞ, is a function with respect to variable x. Therefore, C is also a function with respect to x. Define that ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (L. Jin). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.085
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
@ ðmÞ fij ¼ @ m x fij ;
1 6 i;
j 6 N;
ð1:3aÞ 0
@ ðmÞ C i ¼ ð@ ðmÞ fi1 ; @ ðmÞ fi2 ; . . . ; @ ðmÞ fiN Þ ; @ ðmÞ C ¼
1 6 i 6 N;
ð1:3bÞ
N X jC 1 ; . . . ; C i1 ; @ ðmÞ C i ; C iþ1 ; . . . ; C N j;
ð1:3cÞ
i¼1
DpðmÞ DqðnÞ C C ¼ ð@ ðmÞ @ ðmÞ0 Þp ð@ ðnÞ @ ðnÞ0 Þq CðxÞCðx0 Þjx0 ¼x ;
ð1:3dÞ
where we similarly use notion @ ðmÞ0 when every element in C is a function with respect to variable x0 . In what follows, some examples, in which the determinant C is taken as a Wronskian determinant, are provided for a good understanding of the above definition (1.3).
1j ¼ @ x j Nd 1j ¼ j Nd 2; Nj; @ ð1Þ j Nd
ð1:4Þ
@ ð2Þ j Nd 1j ¼ j Nd 3; N 1; Nj þ j Nd 2; N þ 1j;
ð1:5Þ
1j ¼ j Nd 3; N 1; Nj þ j Nd 2; N þ 1j; @ 2x j Nd
ð1:6Þ
D2ð2Þ s s ¼ 2ð@ 2ð2Þ sÞs 2ð@ ð2Þ sÞ2 4; N 2; N 1; N þ 1j j Nd 3; N 1; N 5; N 3; N 2; N 1; Nj j Nd ¼ 2ðj Nd 2
þ 2j þ 2j Nd 3; N; N þ 1j þ j Nd 2; N þ 3jÞðj Nd 1jÞ 2ðj Nd 3; N 1; Nj þ j Nd 2; N þ 1jÞ :
ð1:7Þ
Notice that @ ðiÞ j Nd 1j does not equal to @ ix j Nd 1j for all integer i > 1. 2. Wronskian identities of bilinear KP hierarchy The KP hierarchy [19] is listed as follows
ðD41 4D1 D3 þ 3D22 Þf f ¼ 0; ½ðD31 þ 2D3 ÞD2 3D1 D4 f f ¼ 0; ðD61 20D31 D3 80D23 þ 144D1 D5 45D21 D22 Þf f ¼ 0;
ð2:1Þ
ðD61 þ 4D31 D3 32D23 þ 36D2 D4 9D21 D22 Þf f ¼ 0: .. . If Di is substituted for DðiÞ and let f ¼ j Nd 1j, then every equation in (2.1) is transformed into an identity. Especially, for the equations of degree four, degree five and degree six in (2.1), we have the following Lemma. Lemma 1. Using the above notions, it holds that
ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j 0; ½ðD3ð1Þ ðD6ð1Þ ðD6ð1Þ
ð2:2Þ
þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j 0;
20D3ð1Þ Dð3Þ
þ
4D3ð1Þ Dð3Þ
80D2ð3Þ
32D2ð3Þ
þ 144Dð1Þ Dð5Þ
þ 36Dð2Þ Dð4Þ
ð2:3Þ
1j 45D2ð1Þ D2ð2Þ Þj Nd
1j 9D2ð1Þ D2ð2Þ Þj Nd
j Nd 1j 0;
j Nd 1j 0:
ð2:4Þ ð2:5Þ
Proof of Lemma 1. After calculation , the identity (2.2) can be transformed into a Plücker relation, and the identity (2.3) can be transformed into a linear combination of two Plücker relations, and either of identities (2.3) and (2.4) can be transformed into a linear combination of five Plücker relations , that is,
ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j ¼ 24ðj Nd 1jj Nd 3; N; N þ 1j j Nd 2; Njj Nd 3; N 1; N þ 1j 2; N þ 1jj Nd 3; N 1; NjÞ 0; þ j Nd ½ðD3ð1Þ þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j ¼ 12ðj Nd 3; N; N þ 2jj Nd 1j j Nd 3; N 1; N þ 2jj Nd 2; Nj 3; N 1; Njj Nd 2; N þ 2jÞ 12ðj Nd 4; N 2; N; N þ 1jj Nd 1j j Nd 4; N 2; N 1; N þ 1jj Nd 2; Nj þ j Nd þ j Nd 4; N 2; N 1; Njj Nd 2; N þ 1jÞ 0;
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Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
½D6ð1Þ 20D3ð1Þ Dð3Þ 80D2ð3Þ þ 144Dð1Þ Dð5Þ 45D2ð1Þ D2ð2Þ j Nd 1j j Nd 1j ¼ 720½j Nd 1jj Nd 4; N 2; N; N þ 2j 2; Njj Nd 4; N 2; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 4; N 2; N 1; Nj 360½j Nd 1jj Nd 5; N 3; N j Nd 5; N 3; N 2; N 1; N þ 1j þ j Nd 5; N 3; N 2; N 1; Nj 2; N; N þ 1j j Nd 2; Njj Nd 2; N þ 1jj Nd 360½j Nd 1jj Nd 4; N 1; N; N þ 1j j Nd 3; N 1; Njj Nd 4; N 2; N 1; N þ 1j þ j Nd 3; N 1; N þ 1jj Nd 4; N 2; N 1; Nj 360½j Nd 1jj Nd 3; N; N þ 3j j Nd 2; Njj Nd 3; N 1; N þ 3j þ j Nd 2; N þ 3jj Nd 3; N 1; Nj 1jj Nd 3; N þ 1; N þ 2j j Nd 2; N þ 1jj Nd 3; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 3; N 1; N þ 1j 0; 360½j Nd and
5; N 3; N 2; N; N þ 1j ½D6ð1Þ þ 4D3ð1Þ Dð3Þ 32D2ð3Þ þ 36Dð2Þ Dð4Þ 9D2ð1Þ D2ð2Þ j Nd 1j j Nd 1j ¼ 72½j Nd 1jj Nd j Nd 2; Njj Nd 2; N þ 1jj Nd 1jj Nd 4; N 5; N 3; N 2; N 1; N þ 1j þ j Nd 5; N 3; N 2; N 1; Nj þ 144½j Nd 2; N; N þ 2j j Nd 2; Njj Nd 4; N 2; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 4; N 2; N 1; Nj 216½j Nd 1jj Nd 4; N 3; N 1; Njj Nd 4; N 2; N 1; N þ 1j þ j Nd 3; N 1; N þ 1jj Nd 4; N 2; N 1; Nj 1; N; N þ 1j j Nd 216½j Nd 1jj Nd 3; N þ 1; N þ 2j j Nd 2; N þ 1jj Nd 3; N 1; N þ 2j þ j Nd 2; N þ 2jj Nd 3; N 1; N þ 1j þ 72½j Nd 1jj Nd 3; N; N þ 3j j Nd 2; Njj Nd 3; N 1; N þ 3j þ j Nd 2; N þ 3jj Nd 3; N 1; Nj 0: Therefore, all of the identities (2.2)–(2.5) hold. h
3. Two useful properties of D-operator Now we are ready to give two useful properties of D-operator which play an important role in our solving method: Lemma 2. If the functions /0i s with respect to variables {x, y, z} in Wronskian determinant (1.1) satisfy
@ y / ¼ a1 @ x þ a2 @ 2x þ þ am @ m x /; @ z / ¼ b1 @ x þ b2 @ 2x þ þ bn @ nx /; 0
where / ¼ ð/1 ; /2 ; . . . ; /N Þ and
1j Dpy Dqz j Nd
a0i s;
0 bi s
ð3:1Þ ð3:2Þ
are arbitrary functions independent on x, then
j Nd 1j ¼ ða1 Dð1Þ þ a2 Dð2Þ þ þ am DðmÞ Þp ðb1 Dð1Þ þ b2 Dð2Þ þ þ bn DðnÞ Þ j Nd 1j j Nd 1j: q
ð3:3Þ
Proof of Lemma 2. Without loss of generality, we can assume that m P n. By the definition (1.3) and the conditions (3.1) and (3.2), we have
Dpy Dqz s s ¼ ð@ y @ y0 Þp ð@ z @ z0 Þq sðx; y; zÞsðx; y0 ; z0 Þjy0 ¼y;z0 ¼z p ¼ a1 ð@ ð1Þ @ ð1Þ0 Þ þ a2 ð@ ð2Þ @ ð2Þ0 Þ þ am ð@ ðmÞ @ ðmÞ0 Þ q b1 ð@ ð1Þ @ ð1Þ0 Þ þ b2 ð@ ð2Þ @ ð2Þ0 Þ þ bn ð@ ðnÞ @ ðnÞ0 Þ sðx; y; zÞsðx0 ; y; zÞjx0 ¼x ( ! ) m n X Y Y x ¼ ½ai ð@ i @ i0 Þki ½bj ð@ j @ j0 Þ j sðx; y; zÞsðx0 ; y; zÞjx0 ¼x k1 þk2 þþkm ¼p x1 þx2 þþxn ¼q ki ;xi ¼0;1;2;...
¼
X k1 þk2 þþkm ¼p x1 þx2 þþxn ¼q ki ;xi ¼0;1;2;...
i¼1
(" m Y
ðai Di Þki
i¼1
j¼1
n Y
# ðbj Dj Þ
xj
)
ss
j¼1
¼ ða1 Dð1Þ þ a2 Dð2Þ þ þ am DðmÞ Þp ðb1 Dð1Þ þ b2 Dð2Þ þ þ bn DðnÞ Þ
q
s s:
It means that the conclusion (3.3) holds. h
Lemma 3. If the functions /0i s in Wronskian determinant (1.1) satisfy
A/ ¼ ðc1 @ x þ c2 @ 2x þ þ cm @ m x Þ/;
ð3:4Þ
253
Y. Kang et al. / Applied Mathematics and Computation 224 (2013) 250–258
where / ¼ ð/1 ; /2 ; . . . ; /N Þ0 ; c0i s are arbitrary functions independent on x and A ¼ ðaij ÞNN is an arbitrary real matrix, then for any non-negative integer n,
1j j Nd 1j ¼ 0: DðnÞ ðc1 Dð1Þ þ c2 Dð2Þ þ þ cm DðmÞ Þj Nd
ð3:5Þ
Proof of Lemma 3. Let us compute the value of TrðAÞs as follows:
ð0Þ / 1 . . . N X a /ð0Þ TrðAÞs ¼ ii i i¼1 .. . ð0Þ /N
ð1Þ /1
.. . ð1Þ
aii /i .. .
ð1Þ
/N
/ð0Þ 1 . . .. N . X . N X ð0Þ ðN1Þ aij /j aii /i ¼ i¼1 j¼1 .. . .. . ðN1Þ /N /ð0Þ ðN1Þ /1
N
ð1Þ
/1 .. . N X ð1Þ aij /j j¼1
.. . ð1Þ
/N
.. . N X ðN1Þ aij /j ; j¼1 .. . ðN1Þ /N
ðN1Þ
/1
ð3:6Þ
where we have used the elementary row operations of a determinant. Then under condition (3.3), the above expression (3.5) equals to
ð0Þ /1 . m .. N X X ð0Þ ck @ kx /i i¼1 k¼1 .. . ð0Þ /N
ð1Þ
m X
ð1Þ
ck @ kx /i
k¼1
.. .
ð1Þ /N
ð0Þ ð1Þ ðN1Þ / /1 /1 1 .. .. .. .. . m X m N . . . X X ðN1Þ k ð1Þ k ðN1Þ ck @ kx /i ck @ kx /ð0Þ ¼ @ / @ / x i x i i k¼1 .. .. k¼1 i¼1 .. .. . . . ð0Þ ð1Þ ðN1Þ . / / / ðN1Þ N N N /N /ð0Þ /ði1Þ @ k /ðiÞ /ðiþ1Þ 1 x 1 1 1 ði1Þ ðiþ1Þ k ðiÞ m N1 ð0Þ X X / / @ /2 x /2 2 2 ¼ ck . .. .. .. . i¼0 . k¼1 . . . ð0Þ ði1Þ ðiÞ ðiþ1Þ k / /N @ x /N /N N
/1 .. .
ðN1Þ
/1
ðN1Þ /1 ðN1Þ /2 .. ; . ðN1Þ /
ð3:7Þ
N
which gives that
TrðAÞs ¼ ðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þs:
ð3:8Þ
Through a similar calculation, we have
TrðAÞð@ ðnÞ sÞ ¼ ðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þð@ ðnÞ sÞ:
ð3:9Þ
Substituting (3.7) and (3.8) into an apparently established identity
ð@ ðnÞ sÞ½TrðAÞs ¼ s½TrðAÞð@ ðnÞ sÞ;
ð3:10Þ
we have
ð@ ðnÞ sÞðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þs ¼ sðc1 @ ð1Þ þ c2 @ ð2Þ þ þ cm @ ðmÞ Þð@ ðnÞ sÞ:
ð3:11Þ
So we can get
c1 ½ð@ ðnÞ @ ð1Þ sÞs ð@ ðnÞ sÞð@ ð1Þ sÞ þ c2 ½ð@ ðnÞ @ ð2Þ sÞs ð@ ðnÞ sÞð@ ð2Þ sÞ þ þ cm ½ð@ ðnÞ @ ðmÞ sÞs ð@ ðnÞ sÞð@ ðmÞ sÞ ¼ 0
ð3:12Þ
and then
Dn ðc1 Dð1Þ þ c2 Dð2Þ þ þ cm DðmÞ Þs s ¼ 0:
ð3:13Þ
The proof is completed. h
4. Application of the improved Wronskian technique In this section, the Wronskian condition will be constructed for the first member and the second member of BKP hierarchy and for the high-dimensional equations, respectively. Based on these Wronskian conditions, various types of Wronskian solutions, such as soliton, positon, negaton, complexiton, rational solutions and so on (see Refs. [6,13,11,10,2] for detail), can be considered. Some sample solutions will be provided for illumination.
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4.1. Wronskian condition for BKP I equation The bilinear form of the first equation of BKP hierarchy reads
h i ðDt D3x ÞDy þ 3D2x f f ¼ 0:
ð4:1:1Þ
Next the first Wronskian condition for Eq. (4.1.1) will be constructed. Theorem 1. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ a/x ;
ð4:1:2Þ
3 /t ¼ 4/3x /x ; a
ð4:1:3Þ
A/ ¼ /2x ;
ð4:1:4Þ
where A ¼ ðaij ÞNN is an arbitrary real matrix and a is arbitrary real nonzero constant. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.1.1). Proof of Theorem 1. By Lemma 2 and conditions (4.1.2) and (4.1.3), we have
Dt Dy j Nd 1j j Nd 1j ¼ ð4aDð1Þ Dð3Þ 3D2ð1Þ Þj Nd 1j j Nd 1j;
ð4:1:5Þ
1j D3x Dy j Nd
ð4:1:6Þ
1j 3D2x j Nd
j Nd 1j ¼
j Nd 1j ¼
aD4ð1Þ j Nd 1j
3D2ð1Þ j Nd 1j
j Nd 1j;
j Nd 1j
ð4:1:7Þ
and using Lemma 3 and condition (4.1.4), we get
D2ð2Þ j Nd 1j j Nd 1j ¼ 0:
ð4:1:8Þ
Then substituting the results (4.1.5)–(4.1.7) into the left-side of Eq. (4.1.1), we obtain
½ðDt D3x ÞDy þ 3D2x j Nd 1j j Nd 1j ¼ ½aðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þ þ 3aDð2Þ2 j Nd 1j j Nd 1j ¼ 0 where we have made use of the identities (2.2) of Lemma 1 and result (4.1.8). The proof is completed. h Soliton Let
2 k 1 A¼
2
k2
: .. . 2 kN
ð4:1:9Þ
By solving the Wronskian conditions (4.1.2)–(4.1.4), N-soliton soluton to Eq. (4.1.1) is given by
f ¼ Wrð/1 ; /2 ; . . . ; /N Þ; ki xþaki yþð4k3i 3aki Þt
where /i ¼ e Positon Let
2 k 1 A¼
2
k2
ð4:1:10Þ ði ¼ 1; 2; . . . ; NÞ.
: .. . 2 kN
ð4:1:11Þ
By solving the Wronskian conditions (4.1.2)–(4.1.4), positon to Eq. (4.1.1) is given by
f ¼ Wrð/1 ; /2 ; . . . ; /N Þ;
3 3 with /i ¼ ci;1 cos ki x þ aki y 4ki þ 3ka i t þ ci;2 sin ki x þ aki y 4ki þ 3ka i t arbitrary constants.
ð4:1:12Þ ði ¼ 1; 2; . . . ; NÞ, where both ci;1 and ci;2 are
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4.2. Wronskian condition for BKP II equation The second equation of the BKP hierarchy
ðut þ 15ux u3x þ 15u3x 15ux uy þ u5x Þx 5u3x;y 5uyy ¼ 0
ð4:2:1Þ
can be written as the bilinear form
ðD6x 5D2y 5D3x Dy þ 9Dx Dt Þf f ¼ 0;
ð4:2:2Þ
by a Cole–Hopf transformation
u ¼ 2ðlnf Þx :
ð4:2:3Þ
In this section, we present the first Wronskian condition for Eq. (4.2.2). Theorem 2. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ 2/3x ;
ð4:2:4Þ
/t ¼ 36/5x ;
ð4:2:5Þ
A/ ¼ /2x ;
ð4:2:6Þ
where A ¼ ðaij ÞNN is an arbitrary real matrix. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.2.2). Proof of Theorem 2. First, by Lemma 2 and conditions (4.2.4) and (4.2.5), we have
1j j Nd 1j ¼ D6ð1Þ j Nd 1j j Nd 1j; D6x j Nd
ð4:2:7Þ
D2y j Nd 1j j Nd 1j ¼ 4D2ð3Þ j Nd 1j j Nd 1j;
ð4:2:8Þ
1j j Nd 1j ¼ 2D3ð1Þ Dð3Þ j Nd 1j j Nd 1j; D3x Dy j Nd
ð4:2:9Þ
1j j Nd 1j ¼ 36Dð1Þ Dð5Þ j Nd 1j j Nd 1j Dx Dt j Nd
ð4:2:10Þ
and second from Lemma 3 and condition (4.2.6), we get
Dð2Þ Dð4Þ j Nd 1j j Nd 1j ¼ 0:
ð4:2:11Þ
Then substituting the results (4.2.7)–(4.2.10) into the left-side of Eq. (4.2.2), we obtain
1j j Nd 1j ¼ ½D6ð1Þ 20D2ð3Þ þ 10D3ð1Þ Dð3Þ 36Dð1Þ Dð5Þ j Nd 1j j Nd 1j ðD6x 5D2y 5D3x Dy þ Dx Dt Þj Nd 1 5 ¼ ½D6ð1Þ 20D3ð1Þ Dð3Þ 80D2ð3Þ þ 144Dð1Þ Dð5Þ 45D2ð1Þ D2ð2Þ þ ½D6ð1Þ þ 4D3ð1Þ Dð3Þ 32D2ð3Þ þ 36Dð2Þ Dð4Þ 4 4 1j j Nd 1j ¼ 0 9D2ð1Þ D2ð2Þ 45Dð2Þ Dð4Þ j Nd where we have made use of the identities (2.4) and (2.5) of Lemma 1 and the result (4.2.11). The proof is completed. h Soliton Let
2 k 1 A¼
2
k2
: .. . 2 kN
ð4:2:12Þ
By solving the Wronskian conditions (4.2.4)–(4.2.6), N-soliton soluton to Eq. (4.2.2) is given by
u ¼ 2ðWrð/1 ; /2 ; . . . ; /N ÞÞx ; ki x2k3i y36k5i t
where /i ¼ e Positon Let
ði ¼ 1; 2; . . . ; NÞ.
ð4:2:13Þ
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2 k 1 A¼
2
k2
: .. . 2 kN
ð4:2:14Þ
By solving the Wronskian conditions (4.2.4)–(4.2.6), positon solution to Eq. (4.2.2) is given by
u ¼ 2ðWrð/1 ; /2 ; . . . ; /N ÞÞx ; with /i ¼ ci;1 cosðki x þ constants.
3 2ki y
5 36ki tÞ
ð4:2:15Þ þ ci;2 sinðki x þ
3 2ki y
5 36ki tÞ
ði ¼ 1; 2; . . . ; NÞ, where both ci;1 and ci;2 are arbitrary
4.3. Wronskian condition for generalized BKP equations Recently, the high-dimensional PDEs have attracted much interest from researchers. For example, some generalized KP 0 and BKP equations are often considered through different methods, such as improved GG -expansion method [20], Wronskian €cklund transformation, dressing method [22], multiple exp-function algorithm technique [21], pfaffian technique, bilinear Ba [23] and so on. The improved Wronskian technique described in this paper also provides us a easy way to search for the Wronskian condition for generalized bilinear KP and BKP equations. Some examples are given as follows. First of all, we consider the following (3 + 1)-dimensional generalized BKP equation [24,22]:
uyz uxxxy 3ðux uy Þx þ 3uxx þ 3uzz ¼ 0:
ð4:3:1Þ
When z ¼ x, Eq. (4.3.1) reduces to the BKP I equation. Under the dependent variable transformation:
u ¼ 2ðlnf Þx ;
ð4:3:2Þ
the above Eq. (4.3.1) can be mapped into the Hirota bilinear equation:
ðDt Dy D3x Dy þ 3D2x þ 3D2z Þf f ¼ 0:
ð4:3:3Þ
Theorem 3. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ r 2 /x ;
ð4:3:4Þ
/z ¼ r/xx ;
ð4:3:5Þ
/t ¼
3 / þ 4/xxx ; r2 x
ð4:3:6Þ
where r is an arbitrary nonzero real constant. Then fN ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.3.3). Proof of Theorem 3. By Lemma 2 and conditions (4.3.4)–(4.3.6), we have
1j j Nd 1j ¼ ð3D2ð1Þ 4r 2 Dð1Þ Dð3Þ Þj Nd 1j j Nd 1j; Dt Dy j Nd
ð4:3:7Þ
D3x Dy j Nd 1j j Nd 1j ¼ r 2 D4ð1Þ j Nd 1j j Nd 1j;
ð4:3:8Þ
D2x j Nd 1j j Nd 1j ¼ D2ð1Þ j Nd 1j j Nd 1j;
ð4:3:9Þ
D2z j Nd 1j j Nd 1j ¼ r 2 D2ð2Þ j Nd 1j j Nd 1j:
ð4:3:10Þ
Then substituting the results (4.3.7)–(4.3.10) into the left-side of Eq. (4.3.3), we obtain
1j j Nd 1j ¼ r 2 ðD4ð1Þ 4Dð1Þ Dð3Þ þ 3D2ð2Þ Þj Nd 1j j Nd 1j ¼ 0; ðDt Dy D3x Dy þ 3D2x þ 3D2z Þj Nd where we have made use of the identity (2.2) of Lemma 1. The proof is completed. h Next, we consider another (3 + 1)-dimensional generalized BKP equation [23,25]:
uty uxxxy 3ðux uy Þx þ 3uxz ¼ 0;
ð4:3:11Þ
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257
which reduces to the BKP I equation if z ¼ x. Under the dependent variable transformation u ¼ 2ðlnf Þx , the above equation is transformed into the Hirota bilinear form:
ðDt Dy D3x Dy þ 3Dx Dz Þf f ¼ 0:
ð4:3:12Þ
Theorem 4. Assuming that /i ¼ /i ðx; y; tÞ ði ¼ 1; 2; . . . ; NÞ has continuous derivative up to any order and satisfies the following linear differential equations
/y ¼ r/xx ;
ð4:3:13Þ
/z ¼ r/xxxx ;
ð4:3:14Þ
/t ¼ 2/xxx ;
ð4:3:15Þ
where r is an arbitrary real constant. Then f ¼ Wrð/1 ; /2 ; . . . ; /N Þ defined by (1.1) solves Eq. (4.3.12). Proof of Theorem 4. By Lemma 2 and conditions (4.3.13)–(4.3.15), we have
1j j Nd 1j ¼ ð2rDð2Þ Dð3Þ Þj Nd 1j j Nd 1j; Dt Dy j Nd
ð4:3:16Þ
D3x Dy j Nd 1j
j Nd 1j ¼ rD3ð1Þ Dð2Þ j Nd 1j j Nd 1j;
ð4:3:17Þ
1j j Nd 1j ¼ rDð1Þ Dð4Þ j Nd 1j j Nd 1j; Dx Dz j Nd
ð4:3:18Þ
Then substituting the results (4.3.16)–(4.3.18) into the left-side of Eq. (4.3.12), we obtain
1j j Nd 1j ¼ r½ðD3ð1Þ þ 2Dð3Þ ÞDð2Þ 3Dð1Þ Dð4Þ j Nd 1j j Nd 1j ¼ 0; ðDt Dy D3x Dy þ 3Dx Dz Þj Nd where we have made use of the identity (2.3) of Lemma 1. The proof is completed.
h
5. Concluding remarks We have discovered the Wronskian identities of the KP hierarchy and two lemmas on D-operator, which play key roles in our improved Wronskian technique. By utilizing this improved Wronskian technique, the Wronskian condition is presented for the BKP I equation and BKP II equation, respectively. Various types of Wronskian solutions can be considered and several sample solutions, especially the Soliton solutions in terms of Wronskian determinant, are listed. Besides, we also apply this technique to construct the Wronskian condition for generalized BKP equations. It is know that there are other interesting solutions such as pfaffian [22], Chain [25], Lump [26], multiple wave [23] and even complex traveling wave [27] type solutons to BKP equations. Because of the richness of the Wronskian solutions, we are looking forward to the possibility of deducing other type solutions from the Wronskian solutions. Moreover, we remark that the improved Wronskian technique can also be applied to more other bilinear KdV-type equations. Acknowledgments The authors would like to thanks the editors and reviewers for their careful check and meaningful comment. References [1] J. Satsuma, A Wronskian representation of n-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn. 40 (1979) 359–360. [2] W.X. Ma, Y.C. You, Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions, Trans. Am. Math. Soc. 357 (2005) 1753–1778. [3] N.C. Freeman, J.J.C. Nimmo, Soliton solutions of the Korteweg–de Vries and Kadomtsev–Petviashvili equations: the Wronskian technique, Phys. Lett. A 96 (1983) 443–446. [4] S. Sirianunpiboon, S.D. Howard, S.K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A 134 (1988) 31–33. [5] R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194. [6] W.X. Ma, Complexiton solutions to the Korteweg–de Vries equation, Phys. Lett. A 301 (2002) 35–44. [7] J.J.C. Nimmo, N.C. Freeman, A method of obtaining the soliton solution of the Boussinesq equation in terms of a Wronskian, Phys. Lett. A 95 (1983) 4–6. [8] W.X. Ma, C.X. Li, J.S. He, A second Wronskian formulation of the Boussinesq equation, Nonlinear Anal. 70 (2009) 4245–4258. [9] Y.Q. Yao, Y.Q. Liu, J. Ji, D.Y. Chen, Novel Wronskian solutions to Boussinesq equation, Commun. Theor. Phys. 48 (2007) 577–583. [10] C.X. Li, W.X. Ma, X.J. Liu, Y.B. Zeng, Wronskian solutions of the Boussinesq equation-solitons, negatons, positons and complexitons, Inverse Prob. 23 (2007) 5279–5296. [11] Y.N. Tang, W.X. Ma, L. Gao, Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo–Miwa equation, Appl. Math. Comput. 217 (2011) 8722– 8730. [12] Y.N. Tang, J.Y. Tu, W.X. Ma, Two new Wronskian conditions for the (3 + 1)-dimensional Jimbo–Miwa equation, Appl. Math. Comput. 218 (2012) 10050– 10055.
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