Solutions of the (2+1)-dimensional KP, SK and KK equations generated by gauge transformations from non-zero seeds

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1 arxv: v1 [nln.si] 5 Nov 008 Solutons of the (+1)-dmensonal KP, SK and KK equatons generated by gauge transformatons from non-zero seeds Jngsong He, Xaodong L Department of Mathematcs Unversty of Scence and Technology of Chna Hefe, 3006 Anhu P.R. Chna Abstract By usng gauge transformatons, we manage to obtan new solutons of ( + 1)- dmensonal Kadomtsev-Petvashvl(KP), Kaup-Kuperschmdt(KK) and Sawada- Kotera(SK) equatons from non-zero seeds. For each of the precedng equatons, a Gallean type transformaton between these solutons u and the prevously known solutons u generated from zero seed s gven. We present several explct formulas of the sngle-solton solutons for u and u, and further pont out the two man dfferences of them under the same value of parameters,.e., heght and locaton of peak lne, whch are demonstrated vsbly n three fgures. 1 Introducton In the 1980s, Sato and hs colleagues brought us the famous Sato theory [1, ]. Snce then, the pseudo-dfferental operator has been playng an mportant role n the research of the Kadomtsev-Petvashvl(KP) herarchy [3], whch can yeld many mportant nonlnear partal dfferental equatons, such as the generalzed nonlnear Schrödnger equaton, the KdV equaton, the Sne-Gordon equaton and the famous 1

2 KP equaton. To be self-consstent, we would lke to gve a bref revew of the KP herarchy [1,, 3, 4]. Let L = + u 1 + u 3 +, (1.1) be a pseudo-dfferental operator(ψdo), here u }, u = u (t 1, t, t 3,...) serve as generators of a dfferental algebra A. The correspondng generalzed Lax equatons are defned as L t n = [B n, L], n = 1,, 3,..., (1.) whch gve rse to nfnte number of partal dfferental equatons of the KP herarchy, B n s defned as B n = [L n ] +. It can be easly showed that eq.(1.) s equvalent to the so-called Zakharov-Shabat(ZS) equaton [5] B m t n B n t m + [B m, B n ] = 0, (m, n =, 3,...). (1.3) The egenfuncton φ and conjugate egenfuncton ψ correspondng to L are defned by φ t n = B n φ, (1.4) ψ t n = Bnψ. (1.5) The frst non-trval example s the KP equaton gven by the t -flow and t 3 -flow of the KP herarchy (4u t 1uu x u xxx ) x 3u yy = 0, (1.6) n whch x = t 1, y = t and t = t 3. Suppose L gven by eq.(1.1) and L defned by L = + ( 1) u +1. =1 If L satsfes L + L = 0, then L s called the Lax operator of the CKP herarchy [4, 6], and the correspondng flow equatons of the CKP herarchy are descrbed by L t n = [B n, L], n = 1, 3, 5,. (1.7) The frst non-trval example s the CKP equaton [4, 7] u t = 5 ( x 1 u yy + 3u x x 1 u y u xxxxx 3uu xxx 15 ) u xu xx 9u u x + u xxy + 3uu y (1.8) whch s generated by t 3 -flow and t 5 -flow and also called the ( + 1)-dmensonal Kaup-Kuperschmdt(KK) equaton [8]. Here x = t 1, y = t 3 and t = t 5. Moreover,

3 L s called the Lax operator of the BKP herarchy [, 9] f t satsfes L = L 1, and the flow equatons of the BKP herarchy assocated wth t are also descrbed by eq.(1.7). The frst non-trval example s the BKP equaton [10, 11] u t = 5 ( x 1 u yy + 3u x x 1 u y 1 ) 9 5 u xxxxx 3uu xxx 3u x u xx 9u u x + u xxy + 3uu y, (1.9) whch s generated by t 3 -flow and t 5 -flow and also called the ( + 1)-dmensonal Sawada-Kotera(SK) equaton [8]. Here x = t 1, y = t 3 and t = t 5. If we fnd a set of functons u, u 3,... whch makes the correspondng pseudodfferental operator L satsfes eq.(1.3), then we have a soluton of the KP herarchy. It s a well-known result that ths set of solutons can be generated from one sngle functon τ(x) as the followng way u = t log τ, (1.10) 1 u 3 = 1 [ 3 t 1 t t 3 ] log τ, (1.11) 1. Durng the last two decades, n order to solve the KP herarchy, the gauge transformaton was formally ntroduced n reference [1]. The basc dea behnd gauge transformaton s to fnd a transformaton for the ntal Lax operator L (0) of the KP herarchy after whch the new operator L (1) and B n (1) stll satsfes Lax equaton eq.(1.) and eq.(1.3) respectvely. Here L (1) = T L (0) T 1, B (1) n = (L (1) ) n +, (1.1) T s a sutable pseudo-dfferental operator. There exst two knds of gauge transformaton operators [1] T D ( ) = ( ) 1, (1.13) T I (ψ (0) ) = (ψ (0) ) 1 1 ψ (0), (1.14) n whch, ψ (0) are egenfuncton and conjugate egenfuncton of L (0) respectvely and they are also called the generatng functons of the gauge transformaton. T D s called dfferental type of gauge transformaton, T I s called ntegral type of gauge transformaton. After one gauge transformaton T D, the new τ-functon τ (1) = τ (0), (1.15) s transformed from an ntal τ-functon τ (0) assocated wth the ntal Lax operator L (0). A smlar result can be formulated for the case of T I τ (1) = ψ (0) τ (0). (1.16) 3

4 Wth the help of formulas eq.(1.10), eq.(1.11), eq.(1.15) and eq.(1.16), we can obtan new solutons u (1) } from the known seed solutons u (0) } n the L (0). For example, u (1) = u (0) + (log ) xx by the gauge transformaton n eq.(1.15). By a successve applcaton of gauge transformatons, the determnant representaton of τ (n+k) s gven n [13] and further more u (n+k) can be deduced by usng eq.(1.10). In the last decade, the method of gauge transformaton has been developed by several researchers. The orgnal form of ths transformaton proposed n reference [1] cannot be appled drectly to the sub-herarches of the KP herarchy. So n [14, 15, 16], an mprovement was made whch makes t applcable to the BKP and CKP herarches, and n [17, 18, 19, 0, 1, ] another mprovement was made so that the gauge transformaton can be used on the constraned KP herarchy. Besdes gauge transformaton, some other methods have been used to solve the KP, BKP, CKP equatons. In [3], Hrota method was consdered on the KP equaton. Darboux transformaton was appled on ths equaton n Chapter 3 of [4]. N-solton solutons of the BKP equaton was obtaned through Hrota method n [5, 6], lump solutons was obtaned through ths method n [7], the same method was appled to the ( + 1)-dmensonal KK equatons n [8] and 3-solton solutons were obtaned explctly. Darboux transformaton was appled to ( + 1)-dmensonal KK, SK equatons n [9]. In [30], - dressng method was used on the ( + 1)-dmensonal KK, SK equatons and lne soltons and lne ratonal lumps were obtaned. It s easy to recognze that all these known solutons are correspondng to the solutons gven by gauge transformaton from zero seed. However, solvng the solton equatons startng from a non-zero seed has not attracted enough attenton. There are very few works on the KPI and KP II equatons wth a non-decay ntal background[31, 3, 33] by dressng method and classcal nverse scatterng method. On the other hand, gauge transformaton from non-zero seeds was not consdered before to our knowledge. One possble reason s that n the case of the KdV equaton, solutons obtaned by gauge transformaton from zero seed can be transformed to those solutons from nonzero seeds by a Gallean transformaton [34]. So far, we have not seen any smlar dscussons on solutons of ( + 1)-dmensonal KP, KK, SK equatons. Therefore n ths paper, we solve these equatons by gauge transformaton from non-zero seeds and manage to fnd out the relatons between new solutons and those from zero seed. The organzaton of ths paper s as follows. In secton two we consder the KP equaton. In secton three and secton four, we dscuss (+1)-dmensonal KK and SK equatons respectvely. Secton fve s devoted to the conclusons and dscussons. The notatons we use n ths paper s the same as n [18]. 4

5 Successve Gauge transformaton for KP equaton It s a natural thought to consder successve applcaton of gauge transformaton for KP herarchy. In [1, 13], a very useful theorem was ntroduced about the result after successve gauge transformatons. Lemma 1 ([1, 13]). After n tmes T D and k tmes T I transformatons (n k), we have : τ (k+n) = ψ (k 1+n) k ψ (k +n) k 1 ψ (n) 1 τ (n) = IW k,n (ψ (0) k, ψ(0) k 1,, ψ(0) 1 ; φ(0) 1, φ(0),, φ(0) n ) τ (0), (.1) n whch IW k,n (ψ (0) k, ψ(0) k 1,, ψ(0) 1 ; φ(0) 1, φ(0),, φ(0) n ) stands for (0) φ 1 ψ (0) (0) k φ ψ (0) (0) k φ n ψ (0) k (0) φ 1 ψ (0) (0) k 1 φ ψ (0) (0) k 1 φ n ψ (0) k 1... (0) IW k,n = φ 1 ψ (0) (0) 1 φ ψ (0) (0) 1 φ n ψ (0) 1 1 n 1,x,x n,x... ( 1 )(n k 1) ( )(n k 1) ( n ) (n k 1) and ψ (0) are solutons of eq.(1.4) and eq.(1.5) assocated wth the ntal value τ (0), further we have u (k+n) = (log IW k,n ) x,x + u (0). (.) By usng the above theorem, we now start to construct the new solutons of the KP equaton n eq.(1.6) from non-zero seeds. To the end, we choose the ntal Lax operator of the KP herarchy to be L (0) = , such that all u (0) = 1 and then the seed soluton of the KP equaton s u (0) = u (0) = 1. We know that the KP equaton s generated by t -flow and t 3 -flow of the KP herarchy, so the generatng functons and ψ (0) for the gauge transformaton satsfy,t = B (0) φ(0) = ( + ), B (0) = (L (0) ) +,t 3 = B (0) 3 φ(0) = ( ), B (0) (.3) 3 = (L (0) ) 3 + ψ (0),t = (B (0) ) ψ (0) = ( + )ψ (0), ψ (0),t 3 = (B (0) 3 ) ψ (0) = ( )ψ (0) (.4). 5

6 Lemma. The solutons of eq.(.3), eq.(.4) are n form of β j 3 α = k j e j +1 x+α jy+β j t, βj = β j (α j ), (.5) ψ (0) = j=1 m fβ j +3 fα k j e j +1 x+fα jy+ β f j t, βj = β j ( α j ). (.6) j=1 Here α j, β j, α j, β j should satsfy the followng relatons (β j 3) = (α j + 1) (α j ), (.7) ( β j + 3) = ( α j + 1) ( α j ). (.8) Proof. We assume the solutons of eq.(.3) have the form φ = X(x) Y (y) T (t), then eq.(.3) s equvalent to Yy Y T t T = Xxx = Xxxx X X +, Let Y y Y = α, T t T where α and β are constants, we have (α ) X = X xx, whch can be reduced to (β 3) X = X xxx + 3 X x, Xx = X x + 3 Xx X + 3. (.9) (α+1) (α ) β 3 X, = β, (.10) (.11) = β 3 α+1 X. (.1) Under the consstency condton (β 3) = (α + 1) (α ) we can obtan From eq.(.10), we have whch nfer the solutons of eq.(.3) X(x) = c 1 e β 3 α+1 x. (.13) Y (y) = c e αy, T (t) = c 3 e βt, φ = k e β 3 α+1 x+αy+βt, (.14) wth the help of (.13), where k = c 1 c c 3. By lnear superposton, the lnear combnaton of φ n eq.(.14) wth respect to dfferent α and β s stll a soluton of eq.(.3), that s β j 3 α = k j φj = k j e j +1 x+α jy+β j t (.15) j=1 j=1 A smlar procedure can be appled to ψ (0) 6 whch yelds eq.(.6).

7 Havng these results, t s suffcent to perform gauge transformaton on L (0). But accordng to lemma 1, the transformed τ-functon may not be satsfactory, snce t may vansh on some pont. To rule out ths stuaton, we need the followng theorem. Theorem 1. Let the generatng functons of n-steps T D be m (m = 1,,, n) n eq.(.5) and rewrtten as m = p m =1 k m, exp a m,x+α m, y+β m, t for smplcty, then the new τ-functon τ (n) = IW 0,n τ (0) = W n ( 1, φ(0),, φ(0) n ) τ (0), (.16) and W n ( 1, φ(0),, φ(0) n ) > 0 f k m, > 0, a m, < a m,j for all m < m and, j. The transformed soluton u (n) of KP equaton s )) u (n) = 1 + ( log ( W n ( 1, φ(0),, φ(0) n ) Proof. Frst, W n takes the followng form 1 n W n = x φ(0) 1 x φ(0) x φ(0) n... n 1 n 1 x n 1 1 x n 1 n 1 n xx x n 1 n n then we expand the determnant wth respect to columns usng the equaton then we have: W n = m = 1 q p q, q=1...n p m =1 k m, e a m, x+α m, y+β m, t, m = 1... n Π n j=1k j,j e a j, j x+α j,j y+β j,j t a 1,1 a, a n,n... a n 1 1, 1 a n 1, a n 1 n, n (.17) (.18) Notce the Vendermonde determnants n the above equaton. Snce k m, > 0, the coeffcents of these Vendermonde determnants are postve. Usng a m, < a m,j for all m < m and, j, t s easy to prove that all Vendermonde determnants n the above equaton are postve, so W n > 0. Usng eq.(.16), eq.(.) and u (0) = 1, we can obtan eq.(.17). Next we gve sngle-solton solutons of the KP equaton from a zero seed and a non-zero seed respectvely. Notatons wth prme are correspondng to the results of gauge transformaton from a zero seed. The generatng functons are ( 1 ) = k e ξ 1 + k e ξ, (.19) 7

8 1 = k e ξ 1 + k e ξ, (.0) where ξ 1 = β 1 x + α α 1 y + β 1 t, (.1) 1 ξ = β x + α α y + β t, (.) ξ 1 = (β 1 3) α x + α 1 y + β 1 t, (.3) ξ = (β 3) α + 1 x + α y + β t, (.4) and (α )3 = (β ), (β 3) = (α + 1) (α ), = 1,. The two sngle-soltons of the KP equaton can be wrtten as (u (1) ) = 1 4 ( β 1 α 1 β α ) sech ( ξ ξ 1 ), (.5) u (1) = ( β 1 3 α β 3 α + 1 ) sech ( ξ 1 ξ ). (.6) There are two dfferences between u and u under the same parameters α: 1) the heght of soltons, ) the locaton of the peak lne of the soltons, whch are demonstrated vsbly n fgure 1. In fgure, we demonstrate the soluton obtaned by a two-step gauge transformaton by usng eq.(.17) and 1 = e y+3 t + e x+3 y+7 t, (.7) = e x+4 y+(3+5 ) t + e 6 x+8 y+(3+9 6) t. (.8) Corollary 1. There exsts a Gallean type transformaton u u (x, y, t) = 1 + u (x + 3 t, y, t). (.9) between u n eq.(.5) and u n eq.(.6). Obvously, ths result s consstent wth the Gallean transformaton [34] of the KdV equaton by a dmensonal reducton. 3 Gauge transformaton for (+1)-dmensonal KK equaton Gauge transformaton of the CKP herarchy s somewhat dfferent from that of the KP herarchy, because a transformed Lax operator L (1) by one-step gauge transformaton has to satsfy (L (1) ) + L (1) = 0. To meet ths requrement, we ntroduce the followng lemma. 8

9 Lemma 3 ([16]). 1. The approprate gauge transformaton T n+k s gven by n = k and generatng functons ψ (0) = for = 1,,, n.. The τ-functon τ (n+n) CKP τ (n+n) of the CKP herarchy has the form CKP = IW n,n ( n, n 1,, φ(0) 1 ; φ(0) (0) φ n (0) 1 φ n n =.. (0) φ 1 (0) 1 φ 1 n and further we have 1, φ(0),, φ(0) n ) τ (0) CKP τ (0) CKP. (3.1) u (n+n) = u (0) + (log IW n,n ) xx. (3.) To solve the (+1)-dmensonal KK equaton from non-zero seed soluton, we choose a ntal Lax operator L (0) of the CKP herarchy to be L (0) = Snce the (+1)-dmensonal KK equaton s generated by t 3 -flow and t 5 -flow of the CKP herarchy, we solve,t 3 = B (0) 3 φ(0) = ( ), B (0) 3 = (L (0) ) 3 +,,t 5 = B (0) 5 φ(0) = ( ), B (0) (3.3) 5 = (L (0) ) 5 +, n order to obtan the egenfunctons. Lemma 4. The solutons of eq.(3.3) are = α 3 j 18α j +9β j α k j e x+α jy+β j t j +α j β j +81, βj = β j (α j ), (3.4) j=1 here α j, β j should satsfy the relaton α 5 j 5α 3 j + 30β j α j + 115α j β 3 j 43β j = 0. (3.5) Proof. Frst, we assume the soluton of eq.(3.3) has the form φ = X(x) Y (y) T (t) then we have Yy Y = Xxxx X + 3 Xx X, T t T = Xxxxxx X + 5 Xxxx Xx X + 15 X. (3.6) Let Y y Y = α, T t T = β, (3.7) 9

10 where α and β are constants, eq.(3.6) become X xxx = α X 3 X x, X xxxxx = β X 15 X x 5 X xxx, whch can be further reduced to 9 X xx (α + β) X x + α X = 0, α X xx + 9 X x + ( α β) X = 0. (3.8) (3.9) Combnng the two equatons n eq.(3.9) together, we have The soluton of eq.(3.10) (α + α β + 81) X x = (α 3 18 α + 9 β) X. (3.10) By substtutng eq.(3.11) back nto eq.(3.8), we have X(x) = c 1 e α 3 18α+9β α x +αβ+81. (3.11) α 5 5α βα + 115α β 3 43β = 0, (3.1) that means f α and β satsfy eq.(3.1), then eq.(3.11) s the soluton of eq.(3.8). From eq.(3.7), we have together wth eq.(3.11) we have Y (y) = c e αy, T (t) = c 3 e βt, φ = k e α3 18α+9β α +αβ+81 x+αy+βt, (3.13) where k = c 1 c c 3. Usng the lnear superposton as we dd n lemma, we can obtan α 3 j 18α j +9β j α = k j φj = k j e x+α jy+β j t j +α j β j +81. (3.14) j=1 j=1 Smlar to the prevous secton about KP equaton, we need the followng theorem to assure that the solutons we get are wthout sngulartes. Theorem. Let egenfunctons m take the form as n lemma 4 m = k m, e a m,x+α m, y+β m, t, (3.15) =1 where m = 1,, f k m, > 0, a 1, < a,j, then IW, (, φ(0) soluton of the (+1)-dmensonal KK equaton can be wrtten as 1 ; φ(0) 1, φ(0) ) < 0. The u (+) = 1 + (log IW, ) xx (3.16) 10

11 Proof. We rewrte 1 and n eq.(3.15) as 1 = n =1 R e ax, = n =1 S e bx. Here the values of R and S are greater than zero. Then we have ( 1 ) = n,j=1 R e R (a +a j )x j a +a j, (3.17) ( ) = n,j=1 S S j e (b +b j )x b +b j, (3.18) 1 φ(0) = n,j=1 R S j e (a +b j )x a +b j. (3.19) Snce a < b j for, j = 1... n, t s easy prove the followng nequalty R R j e (a +a j )x a + a j S k S l e (bk+bl)x b k + b l > R S k R j S l e (a+bk)x a + b k e (aj+bl)x a j + b l, (3.0) where 1, j, k, l n, then (0) φ 1 φ(0) (0) (φ 1 ) (0) (φ ) (0) φ 1 φ(0) = ( 1 φ(0) ) ( 1 ) ( ) < 0. (3.1) can be drectly verfed by usng eq.(3.17), eq.(3.18), eq.(3.19). Eq.(3.16) can be obtaned by eq.(3.) and u (0) = 1. Remark 1. For T (1+1) = T I T D, wth the generatng functon 1 as n eq.(3.15), t s easy to show that τ (1+1) = ( ( 1 ) ) τ (0) (3.) s postve. The correspondng new soluton of the (+1)-dmensonal KK equaton can be represented as u (1+1) = 1 + (log ( 1 ) ) xx (3.3) Here we gve the sngle-solton soluton of the (+1)-dmensonal KK equaton from the generatng functon 1 = e ξ 1 + e ξ, (3.4) where ξ = α3 18α +9β α +α β +81 x + α y + β t, the soluton s u (1+1) = 1 + (a 1 a ) ( e a 1 + a ξ 1 ξ a 1 + e ξ ξ1 a ) (e ξ1 ξ + e ξ ξ 1 ) ( eξ 1 ξ a 1 + eξ ξ 1 a + a 1 +a ), (3.5) 11

12 where a = α3 18α +9β α +α. The soluton (u(1+1) β +81 ) generated from zero seed have the form (u (1+1) ) = (a 1 a ) a 1 + a ξ ( e 1 ξ a 1 ( eξ 1 ξ a 1 ξ + e ξ 1 a + eξ ξ 1 a ) (e ξ 1 ξ + e ξ ξ 1 ), (3.6) + ) a 1 +a where ξ = (α ) x+α β y +β t, a = (α ) and (α β )5 = (β )3. The dfferences between u (1+1) and (u (1+1) ) under the same value of parameters are showed n fgure 3. By takng x+0.00 y+0.01 t 1 = e x y t + e +e x+0.01 y+0.05 t + e x+0.0 y+0.1 t, (3.7) x y+8 t = e x y+30 t + e +e x y+1 t + e x y+0 t.(3.8) n eq.(3.16), we can obtan soluton of the ( + 1)-dmensonal KK equaton whch s plotted n fgure 4. 4 Gauge transformaton for (+1)-dmensonal SK equaton The procedure of ths secton s mostly the same as the prevous secton except that the transformed Lax operator L (1) by one-step gauge transformaton should satsfy (L (1) ) = L (1) 1, so we need lemma 5 about gauge transformaton for BKP herarchy. Lemma 5 ([16]). 1. The approprate gauge transformaton T n+k s gven by n = k and generatng functons ψ (0) =,x for = 1,,..., n.. The τ-functon τ (n+n) BKP τ (n+n) of the BKP herarchy has the form n,x, BKP = IW n,n ( n 1,x,..., φ(0) 1,x ; φ(0) (0) φ n,x (0) 1 φ n,x n =.. (0) φ 1,x φ(0) (0) 1 φ 1,x φ(0) n 1, φ(0),..., φ(0) n ) τ (0) BKP τ (0) BKP. (4.1) and we have u (n+n) = u (0) + (log IW n,n ) xx. (4.) 1

13 Wth ths theorem, we can wrte down the solutons of the (+1)-dmensonal SK equaton explctly after successve applcaton of gauge transformatons. We take the ntal Lax operator L (0) of the BKP herarchy as L (0) = The correspondng egenfuncton and conjugate egenfuncton ψ (0) = gven by lemma 4 and lemma 5,.e. = ψ (0) = j=1,x are α 3 j 18α j +9β j α k j e x+α jy+β j t j +α j β j +81, (4.3) j=1 αj 3 k 18α α j + 9β 3 j 18α j +9β j j j αj + α jβ j + 81 e α x+α jy+β j t j +α j β j +81, βj = β j (α j ). (4.4) Smlar as secton two and secton three, we need the followng theorem to assure that the new τ-functon we get after gauge transformatons wll not vansh at any pont. Theorem 3. Let egenfuncton m take the form as n eq.(4.3) n =1 k m, e a m,x+α m, y+β m, t, m = 1,, f 0 < 3 a 1, < a,j, then we have IW, (,x, φ(0) 1,x ; φ(0) 1, φ(0) Proof. 1 and can be rewrtten as 1 = n =1 R e ax, = n =1 S e bx, ) < 0. The soluton can be wrtten as u (+) = 1 + (log IW, ) xx. (4.5) where the value of R and S are greater than zero, then we have The followng nequalty ( 1 ) ( ) 1,x φ(0) =,x φ(0) 1 = = 1 = 1 R R j e (a +a j )x, (4.6),j=1 S S j e (b +b j )x, (4.7),j=1 a R S j e (a +b j )x, (4.8) a + b j,j=1 b R j S e (a j+b )x. (4.9) a j + b,j=1 (a + b k ) (a j + b l ) > 4 a b l, 13

14 s trval f we use 0 < 3 a 1, < a,j whch means 0 < 3 a < b k, together wth eq.(4.6), eq.(4.7), eq.(4.8) and eq.(4.9), we can prove (0) φ 1 φ(0) ( ),x (0) φ = ( 1,x φ(0) )(,x φ(0) 1 ) (φ(0) 1 ) ( ) < 0, (4.10) 4 ( 1 ) 1,x φ(0) by a drect calculaton. Eq.(4.5) can be obtaned by eq.(4.) and u (0) = 1. Remark. For T 1+1 = T I T D, wth the generatng functon 1 as n eq.(4.3), t s easy to show that τ (1+1) = (φ(0) 1 ) τ (0) (4.11) s postve. The correspondng new soluton of the (+1)-dmensonal SK equaton can be represented as u (1+1) 1 ) = 1 + (log( (φ(0) )) xx (4.1) To obtan a sngle-solton soluton of the (+1)-dmensonal SK equaton, we start from a generatng functon and the soluton s 1 = e ξ + e ξ, (4.13) u (1+1) = 1 + a sech (ξ), (4.14) here ξ = α3 18α+9β α +αβ+81 x+α y +β t and a = α3 18α+9β. A soluton generated from zero α +αβ+81 seed s (u (1+1) ) = (a ) sech (ξ ), (4.15) n whch ξ = (α ) x + α y + β t, (α ) 5 = (β ) 3 and a = (α ). The dfferences β β between u (1+1) and (u (1+1) ) are showed n fgure 5. In fgure 6, we plot the soluton of the ( + 1)-dmensonal SK equaton by takng x y+0. t 1 = e x y+0.15 t + e +e x+0.0 y+0.1 t, (4.16) x y+t = e x y+10 t + e +e x y+30 t, (4.17) n eq.(4.5). Corollary. For the (+1)-dmensonal KK equaton and (+1)-dmensonal SK equaton, there exst a common Gallean type transformaton between (u (1+1) ) (generated from zero seed) and u (1+1) (generated from non-zero seed),.e. u (x, y, t) u (x, y, t) = 1 + u (x + 3y + 15t, y + 5t, t). (4.18) 14

15 5 Conclusons and Dscussons By now we have obtaned new solutons u (n) n theorem 1 for KP equaton, u (+) n theorem for (+1)-dmensonal KK equaton and u (+) n theorem 3 for (+1)- dmensonal SK equaton by usng the the gauge transformatons of the KP herarchy, CKP herarchy and BKP herarchy respectvely. The correspondng generatng functons of the gauge transformatons prevously mentoned are explctly expressed n lemma and lemma 4. For these three equatons, the sngle-solton u (1) (or u(1+1) ) generated from non-zero seeds and (u (1) ) (or (u (1+1) ) ) generated from zero seed are constructed. The man dfferences between the u and (u ) are heght and locatons of the peak lne under the same value of parameters, whch are demonstrated vsbly n fgures 1, and 3. We also found a Gallean type transformaton n eq.(.9) between (u (1) ) and u (1) for the KP equaton, and another one n eq.(4.18) between (u (1+1) ) and u (1+1) for the (+1)-dmensonal KK and SK equatons. To guarantee the new solutons u generated by gauge transformatons s smooth, n other words, the transformed τ-functon doesn t vansh at any pont, we only consder the W n n theorem 1 and IW, n theorem and theorem 3. The corollary 1 and corollary show that we can establsh a one-parameter transformaton group (specfcally, Gallean type transformaton) of the solutons of these three equatons by settng the seeds u (0) = ɛ(arbtrary constant) nstead of u (0) = 1. The advantage of ths new method to fnd one-parameter group s to avod solvng the characterstc lne equaton, whch s not easy to solve, as usual approach of Le pont transformaton. We wll try to do ths n the future. On the other hand, f we can choose some more complcated ntal Lax operator L (0) n whch u (0) } are not constants and we are able to solve the correspondng generatng functons, then we can get some other new solutons. Of course, the calculaton s much tedous although the dea s straghtforward. The present work s the frst step to ths dffcult purpose. Acknowledgement Ths work s supported partly by the NSFC grant of Chna under No We thank Professor L Yshen(USTC, Chna) for many valuable suggestons on ths paper. References [1] Y. Ohta, J. Satsuma, D. Takahash and T. Tokhro, An elementry ntroducton to Sato theory, Prog. Theor. Phys. Suppl. 94 (1988) [] E.Date, M. Kashwara, M. Jmbo and T. Mwa, Transformaton groups for solton equatons n Nonlnear ntegrable systems - classcal and quantum 15

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19 Fgure 1: Sngle-solton solutons at t = 1 of the KP equaton. The lower one s (u (1) ) wth k = 1, α 1 =.75 and α = 3.4; the hgher one s (u (1) 1) wth parameters k = 1, α 1 =.75 and α = 3.4. Fgure : Two-solton soluton at t = 0 of the KP equaton. 19

20 Fgure 3: Sngle-solton solutons at t = 1 of the (+1)-dmensonal KK equaton. The hgher one s (u (1+1) ) wth α 1 = and α = 0.075; the lower one s (u (1+1) 1) wth parameters α 1 = and α = Fgure 4: Soluton at t = 0 of the (+1)-dmensonal KK equaton. 0

21 Fgure 5: Sngle-solton solutons at t = 1 of (+1)-dmensonal SK equaton. The hgher one s (u (1+1) ) wth α = 4.096,the lower one s (u (1+1) 1) wth parameters α = Fgure 6: Soluton at t = 0 of (+1)-dmensonal SK equaton. 1