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

Solutions of the (2+1)-dimensional KP , SK and KK equations generated b y gauge transformations from non-zero seeds Jingsong He , Xiao dong Li Departmen t of Mathematics Univ ersit y of Science and T ec hnology of China Hefei, 230026 Anh ui P .R. China Abstract By using gauge transformations, w e manage to obtain new solutions of (2 + 1)- dimensional Kadomtsev-P etviashvili(KP), Kaup-Kup ersc hmidt(KK) and Saw ada- Kotera(SK) equations from non-zero seeds. F or eac h of the preceding equations, a Galilean t yp e transformation b et w een these solutions u 2 and the previously known solutions u 0 2 generated from zero seed is given. W e presen t several explicit formulas of the single-soliton solutions for u 2 and u 0 2 , and further p oint out the tw o main differences of them under the same v alue of parameters, i.e., heigh t and lo cation of p eak line, which are demonstrated visibly in three figures. 1 In tro duction In the 1980s, Sato and his colleagues brough t us the famous Sato theory [1, 2]. Since then, the pseudo-differential op erator has b een pla ying an imp ortan t role in the researc h of the Kadom tsev-P etviash vili(KP) hierarc hy [3], whic h can yield man y imp ortan t nonlinear partial differential equations, such as the generalized nonlinear Sc hr¨ odinger equation, the KdV equation, the Sine-Gordon equation and the famous 1 KP equation. T o b e self-consisten t, we would lik e to giv e a brief review of the KP hierarc h y [1, 2, 3, 4]. Let L = ∂ + u 2 ∂ − 1 + u 3 ∂ − 2 + · · · , (1.1) b e a pseudo-differential op erator(ΨDO), here { u i } , u i = u i ( t 1 , t 2 , t 3 , . . . ) serv e as generators of a differen tial algebra A . The corresp onding generalized Lax equations are defined as ∂ L ∂ t n = [ B n , L ] , n = 1 , 2 , 3 , . . . , (1.2) whic h give rise to infinite num b er of partial differential equations of the KP hierar- c h y , B n is defined as B n = [ L n ] + . It can b e easily sho w ed that eq.(1.2) is equiv alent to the so-called Zakharo v-Shabat(ZS) equation [5] ∂ B m ∂ t n − ∂ B n ∂ t m + [ B m , B n ] = 0 , ( m, n = 2 , 3 , . . . ) . (1.3) The eigenfunction φ and conjugate eigenfunction ψ corresp onding to L are defined b y ∂ φ ∂ t n = B n φ, (1.4) ∂ ψ ∂ t n = − B ∗ n ψ . (1.5) The first non-trivial example is the KP equation giv en b y the t 2 -flo w and t 3 -flo w of the KP hierarc h y (4 u t − 12 uu x − u xxx ) x − 3 u y y = 0 , (1.6) in whic h x = t 1 , y = t 2 and t = t 3 . Supp ose L given by eq.(1.1) and L ∗ defined b y L ∗ = − ∂ + ∞ X i =1 ( − 1) i ∂ − i u i +1 . If L satisfies L ∗ + L = 0 , then L is called the Lax op erator of the CKP hierarch y [4, 6], and the corresp onding flo w equations of the CKP hierarch y are describ ed b y ∂ L ∂ t n = [ B n , L ] , n = 1 , 3 , 5 , · · · . (1.7) The first non-trivial example is the CKP equation [4, 7] u t = 5 9  ∂ − 1 x u y y + 3 u x ∂ − 1 x u y − 1 5 u xxxxx − 3 uu xxx − 15 2 u x u xx − 9 u 2 u x + u xxy + 3 uu y  (1.8) whic h is generated by t 3 -flo w and t 5 -flo w and also called the (2 + 1)-dimensional Kaup-Kup ersc hmidt(KK) equation [8]. Here x = t 1 , y = t 3 and t = t 5 . Moreov er, 2 L is called the Lax op erator of the BKP hierarch y [2, 9] if it satisfies L ∗ = − ∂ L∂ − 1 , and the flo w equations of the BKP hierarc h y associated with it are also describ ed b y eq.(1.7). The first non-trivial example is the BKP equation [10, 11] u t = 5 9  ∂ − 1 x u y y + 3 u x ∂ − 1 x u y − 1 5 u xxxxx − 3 uu xxx − 3 u x u xx − 9 u 2 u x + u xxy + 3 uu y  , (1.9) whic h is generated by t 3 -flo w and t 5 -flo w and also called the (2 + 1)-dimensional Sa w ada-Kotera(SK) equation [8]. Here x = t 1 , y = t 3 and t = t 5 . If we find a set of functions u 2 , u 3 , . . . which makes the corresp onding pseudo- differen tial operator L satisfies eq.(1.3), then we ha v e a solution of the KP hierarc h y . It’s a w ell-kno wn result that this set of solutions can be generated from one single function τ ( x ) as the follo wing w a y u 2 = ∂ 2 ∂ t 2 1 log τ , (1.10) u 3 = 1 2 [ ∂ 2 ∂ t 1 ∂ t 2 − ∂ 3 ∂ t 3 1 ] log τ , (1.11) . . . During the last tw o decades, in order to solve the KP hierarc h y , the gauge transformation was formally introduced in reference [12]. The basic idea b ehind gauge transformation is to find a transformation for the initial Lax op erator L (0) of the KP hierarc hy after whic h the new op erator L (1) and B (1) n still satisfies Lax equation eq.(1.2) and eq.(1.3) resp ectiv ely . Here L (1) = T ◦ L (0) ◦ T − 1 , B (1) n = ( L (1) ) n + , (1.12) T is a suitable pseudo-differential operator. There exist t w o kinds of gauge trans- formation op erators [12] T D ( φ (0) ) = φ (0) ∂ ( φ (0) ) − 1 , (1.13) T I ( ψ (0) ) = ( ψ (0) ) − 1 ∂ − 1 ψ (0) , (1.14) in whic h φ (0) , ψ (0) are eigenfun ction and conjugate eigenfunction of L (0) resp ectiv ely and they are also called the generating functions of the gauge transformation. T D is called differen tial type of gauge transformation, T I is called integral type of gauge transformation. After one gauge transformation T D , the new τ -function τ (1) = φ (0) τ (0) , (1.15) is transformed from an initial τ -function τ (0) asso ciated with the initial Lax op erator L (0) . A similar result can b e form ulated for the case of T I τ (1) = ψ (0) τ (0) . (1.16) 3 With the help of form ulas eq.(1.10), eq.(1.11), eq.(1.15) and eq.(1.16), w e can obtain new solutions { u (1) i } from the known seed solutions { u (0) i } in the L (0) . F or example, u (1) 2 = u (0) 2 + (log φ (0) ) xx b y the gauge transformation in eq.(1.15). By a successive application of gauge transformations, the determinant representation of τ ( n + k ) is giv en in [13] and further more u ( n + k ) 2 can b e deduced by using eq.(1.10). In the last decade, the metho d of gauge transformation has been developed b y sev eral researchers. The original form of this transformation proposed in reference [12] cannot b e applied directly to the sub-hierarc hies of the KP hierarc h y . So in [14, 15, 16], an impro v emen t was made which makes it applicable to the BKP and CKP hierarc hies, and in [17, 18, 19, 20, 21, 22] another improv ement w as made so that the gauge transformation can be used on the constrained KP hierarc h y . Besides gauge transformation, some other metho ds ha v e been used to solv e the KP , BKP , CKP equations. In [23], Hirota metho d was considered on the KP equation. Darboux transformation w as applied on this equation in Chapter 3 of [24]. N-soliton solutions of the BKP equation was obtained through Hirota metho d in [25, 26], lump solutions w as obtained through this metho d in [27], the same metho d was applied to the (2 + 1)-dimensional KK equations in [28] and 3-soliton solutions were obtained explicitly . Darb oux transformation w as applied to (2 + 1)-dimensional KK, SK equations in [29]. In [30], ¯ ∂ - dressing metho d w as used on the (2 + 1)-dimensional KK, SK equations and line solitons and line rational lumps were obtained. It is easy to recognize that all these known solutions are corresp onding to the solutions giv en by gauge transformation from zero seed. How ev er, solving the soliton equations starting from a ”non-zero seed” has not attracted enough atten tion. There are very few w orks on the KPI and KP I I equations with a non-decay initial bac kground[31, 32, 33] by dressing metho d and classical inv erse scattering metho d. On the other hand, gauge transformation from non-zero seeds w as not considered b efore to our knowledge. One p ossible reason is that in the case of the KdV equation, solutions obtained b y gauge transformation from zero seed can b e transformed to those solutions from non- zero seeds b y a Galilean transformation [34]. So far, we hav e not seen any similar discussions on solutions of (2 + 1)-dimensional KP , KK, SK equations. Therefore in this pap er, w e solv e these equations b y gauge transformation from non-zero seeds and manage to find out the relations b etw een new solutions and those from zero seed. The organization of this pap er is as follows. In section tw o w e consider the KP equation. In section three and section four, we discuss (2+1)-dimensional KK and SK equations resp ectively . Section five is dev oted to the conclusions and discussions. The notations w e use in this pap er is the same as in [18]. 4 2 Successiv e Gauge transformation for KP equa- tion It’s a natural thought to consider successive application of gauge transformation for KP hierarc h y . In [12, 13], a v ery useful theorem w as in tro duced ab out the result after successiv e gauge transformations. Lemma 1 ([12, 13]) . After n times T D and k times T I tr ansformations ( n ≥ k ) , we have : τ ( k + n ) = ψ ( k − 1+ n ) k · ψ ( k − 2+ 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) 2 , · · · , φ (0) n ) · τ (0) , (2.1) in which IW k,n ( ψ (0) k , ψ (0) k − 1 , · · · , ψ (0) 1 ; φ (0) 1 , φ (0) 2 , · · · , φ (0) n ) stands for IW k,n =                     R φ (0) 1 · ψ (0) k R φ (0) 2 · ψ (0) k · · · R φ (0) n · ψ (0) k R φ (0) 1 · ψ (0) k − 1 R φ (0) 2 · ψ (0) k − 1 · · · R φ (0) n · ψ (0) k − 1 . . . . . . · · · . . . R φ (0) 1 · ψ (0) 1 R φ (0) 2 · ψ (0) 1 · · · R φ (0) n · ψ (0) 1 φ (0) 1 φ (0) 2 · · · φ (0) n φ (0) 1 ,x φ (0) 2 ,x · · · φ (0) n,x . . . . . . · · · . . . ( φ (0) 1 ) ( n − k − 1) ( φ (0) 2 ) ( n − k − 1) · · · ( φ (0) n ) ( n − k − 1)                     φ (0) i and ψ (0) i ar e solutions of e q.(1.4) and e q.(1.5) asso ciate d with the initial value τ (0) , further we have u ( k + n ) 2 = (log IW k,n ) x,x + u (0) 2 . (2.2) By using the ab o ve theorem, we no w start to construct the new solutions of the KP equation in eq.(1.6) from non-zero seeds. T o the end, we choose the initial Lax op erator of the KP hierarch y to b e L (0) = ∂ + ∂ − 1 + ∂ − 2 + ∂ − 3 + · · · , suc h that all u (0) i = 1 and then the seed solution of the KP equation is u (0) = u (0) 2 = 1. W e know that the KP equation is generated b y t 2 -flo w and t 3 -flo w of the KP hierarc h y , so the generating functions φ (0) i and ψ (0) i for the gauge transformation satisfy ( φ (0) i,t 2 = B (0) 2 φ (0) i = ( ∂ 2 + 2) φ (0) i , B (0) 2 = ( L (0) ) 2 + φ (0) i,t 3 = B (0) 3 φ (0) i = ( ∂ 3 + 3 ∂ + 3) φ (0) i , B (0) 3 = ( L (0) ) 3 + (2.3) ( ψ (0) i,t 2 = − ( B (0) 2 ) ∗ ψ (0) i = − ( ∂ 2 + 2) ψ (0) i , ψ (0) i,t 3 = − ( B (0) 3 ) ∗ ψ (0) i = ( ∂ 3 + 3 ∂ − 3) ψ (0) i . (2.4) 5 Lemma 2. The solutions of e q.(2.3), e q.(2.4) ar e in form of φ (0) i = n X j =1 k j e β j − 3 α j +1 x + α j y + β j t , β j = β j ( α j ) , (2.5) ψ (0) i = m X j =1 e k j e f β j +3 − f α j +1 x + f α j y + f β j t , e β j = e β j ( f α j ) . (2.6) Her e α j , β j , f α j , e β j should satisfy the fol lowing r elations ( β j − 3) 2 = ( α j + 1) 2 ( α j − 2) , (2.7) ( e β j + 3) 2 = ( − f α j + 1) 2 ( − f α j − 2) . (2.8) Pr o of. W e assume the solutions of eq.(2.3) ha v e the form b φ = X ( x ) Y ( y ) T ( t ), then eq.(2.3) is equiv alen t to ( Y y Y = X xx X + 2 , T t T = X xxx X + 3 X x X + 3 . (2.9) Let Y y Y = α, T t T = β , (2.10) where α and β are constants, we hav e ( ( α − 2) X = X xx , ( β − 3) X = X xxx + 3 X x , (2.11) whic h can b e reduced to ( X x = ( α +1) ( α − 2) β − 3 X , X x = β − 3 α +1 X . (2.12) Under the consistency condition ( β − 3) 2 = ( α + 1) 2 ( α − 2) w e can obtain X ( x ) = c 1 e β − 3 α +1 x . (2.13) F rom eq.(2.10), w e ha v e Y ( y ) = c 2 e αy , T ( t ) = c 3 e β t , whic h infer the solutions of eq.(2.3) b φ = k e β − 3 α +1 x + αy + β t , (2.14) with the help of (2.13), where k = c 1 c 2 c 3 . By linear sup erp osition, the linear com bination of b φ in eq.(2.14) with resp ect to different α and β is still a solution of eq.(2.3), that is φ (0) i = n X j =1 k j b φ j = n X j =1 k j e β j − 3 α j +1 x + α j y + β j t (2.15) A similar pro cedure can b e applied to ψ (0) i whic h yields eq.(2.6). 6 Ha ving these results, it’s sufficien t to perform gauge transformation on L (0) . But according to lemma 1, the transformed τ -function ma y not b e satisfactory , since it ma y v anish on some p oin t. T o rule out this situation, w e need the follo wing theorem. Theorem 1. L et the gener ating functions of n-steps T D b e φ (0) m ( m = 1 , 2 , · · · , n ) in e q.(2.5) and r ewritten as φ (0) m = P p m i =1 k m,i exp a m,i x + α m,i y + β m,i t for simplicity, then the new τ -function τ ( n ) = IW 0 ,n · τ (0) = W n ( φ (0) 1 , φ (0) 2 , · · · , φ (0) n ) · τ (0) , (2.16) and W n ( φ (0) 1 , φ (0) 2 , · · · , φ (0) n ) > 0 if k m,i > 0 , a m,i < a m 0 ,j for al l m < m 0 and ∀ i, j . The tr ansforme d solution u ( n ) 2 of KP e quation is u ( n ) 2 = 1 +  log  W n ( φ (0) 1 , φ (0) 2 , · · · , φ (0) n )  xx (2.17) Pr o of. First, W n tak es the follo wing form W n =           φ (0) 1 φ (0) 2 · · · φ (0) n ∂ ∂ x φ (0) 1 ∂ ∂ x φ (0) 2 · · · ∂ ∂ x φ (0) n . . . . . . · · · . . . ∂ n − 1 ∂ x n − 1 φ (0) 1 ∂ n − 1 ∂ x n − 1 φ (0) 2 · · · ∂ n − 1 ∂ x n − 1 φ (0) n           n × n then w e expand the determinan t with resp ect to columns using the equation φ (0) m = p m X i =1 k m,i e a m,i x + α m,i y + β m,i t , m = 1 . . . n. then w e ha v e: W n = X 1 ≤ i q ≤ p q , q =1 ...n Π n j =1 k j,i j e a j,i j x + α j,i j y + β j,i j t          1 1 · · · 1 a 1 ,i 1 a 2 ,i 2 · · · a n,i n . . . . . . · · · . . . a n − 1 1 ,i 1 a n − 1 2 ,i 2 · · · a n − 1 n,i n          (2.18) Notice the V endermonde determinan ts in the ab o v e equation. Since k m,i > 0, the co efficien ts of these V endermonde determinan ts are p ositiv e. Using a m,i < a m 0 ,j for all m < m 0 and ∀ i, j , it’s easy to prov e that all V endermonde determinan ts in the ab o v e equation are p ositiv e, so W n > 0. Using eq.(2.16), eq.(2.2) and u (0) 2 = 1, we can obtain eq.(2.17). Next we give single-soliton solutions of the KP equation from a zero seed and a non-zero seed resp ectiv ely . Notations with prime are corresponding to the results of gauge transformation from a zero seed. The generating functions are  φ (0) 1  0 = k 0 e ξ 0 1 + k 0 e ξ 0 2 , (2.19) 7 φ (0) 1 = k e ξ 1 + k e ξ 2 , (2.20) where ξ 0 1 = β 0 1 α 0 1 x + α 0 1 y + β 0 1 t, (2.21) ξ 0 2 = β 0 2 α 0 2 x + α 0 2 y + β 0 2 t, (2.22) ξ 1 = ( β 1 − 3) α 1 + 1 x + α 1 y + β 1 t, (2.23) ξ 2 = ( β 2 − 3) α 2 + 1 x + α 2 y + β 2 t, (2.24) and ( α 0 i ) 3 = ( β 0 i ) 2 , ( β i − 3) 2 = ( α i + 1) 2 ( α i − 2), i = 1 , 2. The tw o single-solitons of the KP equation can b e written as ( u (1) 2 ) 0 = 1 4 ( β 0 1 α 0 1 − β 0 2 α 0 2 ) 2 sec h 2 ( ξ 0 2 − ξ 0 1 2 ) , (2.25) u (1) 2 = 1 + 1 4 ( β 1 − 3 α 1 + 1 − β 2 − 3 α 2 + 1 ) 2 sec h 2 ( ξ 1 − ξ 2 2 ) . (2.26) There are t w o differences b et w een u 2 and u 0 2 under the same parameters α : 1) the height of solitons, 2) the lo cation of the p eak line of the solitons, which are demonstrated visibly in figure 1. In figure 2, we demonstrate the solution obtained b y a t w o-step gauge transformation b y using eq.(2.17) and φ (0) 1 = e 2 y +3 t + e x +3 y +7 t , (2.27) φ (0) 2 = e √ 2 x +4 y +(3+5 √ 2) t + e √ 6 x +8 y +(3+9 √ 6) t . (2.28) Corollary 1. Ther e exists a Galile an typ e tr ansformation u 0 2 7− → u 2 ( x, y , t ) = 1 + u 0 2 ( x + 3 t, y , t ) . (2.29) b etwe en u 0 2 in e q.(2.25) and u 2 in e q.(2.26). Ob viously , this result is consistent with the Galilean transformation [34] of the KdV equation b y a dimensional reduction. 3 Gauge transformation for (2+1)-dimensional KK equation Gauge transformation of the CKP hierarch y is somewhat different from that of the KP hierarc h y , b ecause a transformed Lax op erator L (1) b y one-step gauge transfor- mation has to satisfy ( L (1) ) ∗ + L (1) = 0. T o meet this requiremen t, w e introduce the follo wing lemma. 8 Lemma 3 ([16]) . 1. The appr opriate gauge tr ansformation T n + k is given by n = k and gener ating functions ψ (0) i = φ (0) i for i = 1 , 2 , · · · , n . 2. The τ -function τ ( n + n ) CKP of the CKP hier ar chy has the form τ ( n + n ) CKP = IW n,n ( φ (0) n , φ (0) n − 1 , · · · , φ (0) 1 ; φ (0) 1 , φ (0) 2 , · · · , φ (0) n ) · τ (0) CKP =        R φ (0) n · φ (0) 1 · · · R φ (0) n · φ (0) n . . . · · · . . . R φ (0) 1 · φ (0) 1 · · · R φ (0) 1 · φ (0) n        · τ (0) CKP . (3.1) and further we have u ( n + n ) 2 = u (0) 2 + (log IW n,n ) xx . (3.2) T o solve the (2+1)-dimensional KK equation from non-zero seed solution, we c ho ose a initial Lax op erator L (0) of the CKP hierarc h y to b e L (0) = ∂ + ∂ − 1 + ∂ − 3 + ∂ − 5 + · · · . Since the (2+1)-dimensional KK equation is ge nerated by t 3 -flo w and t 5 -flo w of the CKP hierarc h y , w e solv e ( φ (0) i,t 3 = B (0) 3 φ (0) i = ( ∂ 3 + 3 ∂ ) φ (0) i , B (0) 3 = ( L (0) ) 3 + , φ (0) i,t 5 = B (0) 5 φ (0) i = ( ∂ 5 + 5 ∂ 3 + 15 ∂ ) φ (0) i , B (0) 5 = ( L (0) ) 5 + , (3.3) in order to obtain the eigenfunctions. Lemma 4. The solutions of e q.(3.3) ar e φ (0) i = n X j =1 k j e α 3 j − 18 α j +9 β j α 2 j + α j β j +81 x + α j y + β j t , β j = β j ( α j ) , (3.4) her e α j , β j should satisfy the r elation α 5 j − 25 α 3 j + 30 β j α 2 j + 1215 α j − β 3 j − 243 β j = 0 . (3.5) Pr o of. First, we assume the solution of eq.(3.3) has the form b φ = X ( x ) Y ( y ) T ( t ) then w e ha v e ( Y y Y = X xxx X + 3 X x X , T t T = X xxxxx X + 5 X xxx X + 15 X x X . (3.6) Let Y y Y = α, T t T = β , (3.7) 9 where α and β are constants, eq.(3.6) b ecome ( X xxx = α X − 3 X x , X xxxxx = β X − 15 X x − 5 X xxx , (3.8) whic h can b e further reduced to ( 9 X xx − ( α + β ) X x + α 2 X = 0 , α X xx + 9 X x + (2 α − β ) X = 0 . (3.9) Com bining the t w o equations in eq.(3.9) together, w e ha v e ( α 2 + α β + 81) X x = ( α 3 − 18 α + 9 β ) X . (3.10) The solution of eq.(3.10) X ( x ) = c 1 e α 3 − 18 α +9 β α 2 + αβ +81 x . (3.11) By substituting eq.(3.11) bac k in to eq.(3.8), w e ha v e α 5 − 25 α 3 + 30 β α 2 + 1215 α − β 3 − 243 β = 0 , (3.12) that means if α and β satisfy eq.(3.12), then eq.(3.11) is the solution of eq.(3.8). F rom eq.(3.7), w e ha v e Y ( y ) = c 2 e αy , T ( t ) = c 3 e β t , together with eq.(3.11) w e ha v e b φ = k e α 3 − 18 α +9 β α 2 + αβ +81 x + αy + β t , (3.13) where k = c 1 c 2 c 3 . Using the linear sup erposition as w e did in lemma 2, we can obtain φ (0) i = n X j =1 k j b φ j = n X j =1 k j e α 3 j − 18 α j +9 β j α 2 j + α j β j +81 x + α j y + β j t . (3.14) Similar to the previous section ab out KP equation, we need the follo wing theo- rem to assure that the solutions w e get are without singularities. Theorem 2. L et eigenfunctions φ (0) m take the form as in lemma 4 φ (0) m = n X i =1 k m,i e a m,i x + α m,i y + β m,i t , (3.15) wher e m = 1 , 2 , if k m,i > 0 , a 1 ,i < a 2 ,j , then IW 2 , 2 ( φ (0) 2 , φ (0) 1 ; φ (0) 1 , φ (0) 2 ) < 0 . The solution of the (2+1)-dimensional KK e quation c an b e written as u (2+2) 2 = 1 + (log IW 2 , 2 ) xx (3.16) 10 Pr o of. W e rewrite φ (0) 1 and φ (0) 2 in eq.(3.15) as ( φ (0) 1 = P n i =1 R i e a i x , φ (0) 2 = P n i =1 S i e b i x . Here the v alues of R i and S i are greater than zero. Then we hav e Z ( φ (0) 1 ) 2 = P n i,j =1 R i R j e ( a i + a j ) x a i + a j , (3.17) Z ( φ (0) 2 ) 2 = P n i,j =1 S i S j e ( b i + b j ) x b i + b j , (3.18) Z φ (0) 1 φ (0) 2 = P n i,j =1 R i S j e ( a i + b j ) x a i + b j . (3.19) Since a i < b j for i, j = 1 . . . n , it’s easy pro v e the follo wing inequality R i R j e ( a i + a j ) x a i + a j S k S l e ( b k + b l ) x b k + b l > R i S k R j S l e ( a i + b k ) x a i + b k e ( a j + b l ) x a j + b l , (3.20) where 1 ≤ i, j, k , l ≤ n , then      R φ (0) 1 φ (0) 2 R ( φ (0) 1 ) 2 R ( φ (0) 2 ) 2 R φ (0) 1 φ (0) 2      = ( Z φ (0) 1 φ (0) 2 ) 2 − Z ( φ (0) 1 ) 2 Z ( φ (0) 2 ) 2 < 0 . (3.21) can b e directly verified b y using eq.(3.17), eq.(3.18), eq.(3.19). Eq.(3.16) can b e obtained b y eq.(3.2) and u (0) 2 = 1. Remark 1. F or T (1+1) = T I T D , with the gener ating function φ (0) 1 as in e q.(3.15), it’s e asy to show that τ (1+1) = ( Z ( φ (0) 1 ) 2 ) τ (0) (3.22) is p ositive. The c orr esp onding new solution of the (2+1)-dimensional KK e quation c an b e r epr esente d as u (1+1) 2 = 1 + (log Z ( φ (0) 1 ) 2 ) xx (3.23) Here w e give the single-soliton solution of the (2+1)-dimensional KK equation from the generating function φ (0) 1 = e ξ 1 + e ξ 2 , (3.24) where ξ i = α 3 i − 18 α i +9 β i α 2 i + α i β i +81 x + α i y + β i t , the solution is u (1+1) 2 = 1 + ( a 1 − a 2 ) 2 a 1 + a 2 ( e ξ 1 − ξ 2 2 a 1 + e ξ 2 − ξ 1 2 a 2 ) ( e ξ 1 − ξ 2 2 + e ξ 2 − ξ 1 2 ) ( e ξ 1 − ξ 2 2 a 1 + e ξ 2 − ξ 1 2 a 2 + 2 a 1 + a 2 ) 2 , (3.25) 11 where a i = α 3 i − 18 α i +9 β i α 2 i + α i β i +81 . The solution ( u (1+1) 2 ) 0 generated from zero seed hav e the form ( u (1+1) 2 ) 0 = ( a 0 1 − a 0 2 ) 2 a 0 1 + a 0 2 ( e ξ 0 1 − ξ 0 2 2 a 0 1 + e ξ 0 2 − ξ 0 1 2 a 0 2 ) ( e ξ 0 1 − ξ 0 2 2 + e ξ 0 2 − ξ 0 1 2 ) ( e ξ 0 1 − ξ 0 2 2 a 0 1 + e ξ 0 2 − ξ 0 1 2 a 0 2 + 2 a 0 1 + a 0 2 ) 2 , (3.26) where ξ 0 i = ( α 0 i ) 2 β 0 i x + α 0 i y + β 0 i t , a 0 i = ( α 0 i ) 2 β 0 i and ( α 0 i ) 5 = ( β 0 i ) 3 . The differences b et w een u (1+1) 2 and ( u (1+1) 2 ) 0 under the same v alue of parameters are show ed in figure 3. By taking φ (0) 1 = e 0 . 0001999999974 x +0 . 0006 y +0 . 003 t + e 0 . 0006666665679 x +0 . 002 y +0 . 01 t + e 0 . 003333320988 x +0 . 01 y +0 . 05 t + e 0 . 006666567904 x +0 . 02 y +0 . 1 t , (3.27) φ (0) 2 = e 1 . 218304787 x +5 . 463203409 y +30 t + e 0 . 4917724251 x +1 . 594247576 y +8 t + e 0 . 6835764081 x +2 . 370148557 y +12 t + e 0 . 970831384 x +3 . 827515914 y +20 t . (3.28) in eq.(3.16), w e can obtain solution of the (2 + 1)-dimensional KK equation which is plotted in figure 4. 4 Gauge transformation for (2+1)-dimensional SK equation The pro cedure of this section is mostly the same as the previous section except that the transformed Lax op erator L (1) b y one-step gauge transformation should satisfy ( L (1) ) ∗ = − ∂ L (1) ∂ − 1 , so we need lemma 5 ab out gauge transformation for BKP hierarc h y . Lemma 5 ([16]) . 1. The appr opriate gauge tr ansformation T n + k is given by n = k and gener ating functions ψ (0) i = φ (0) i,x for i = 1 , 2 , . . . , n . 2. The τ -function τ ( n + n ) BKP of the BKP hier ar chy has the form τ ( n + n ) BKP = IW n,n ( φ (0) n,x , φ (0) n − 1 ,x , . . . , φ (0) 1 ,x ; φ (0) 1 , φ (0) 2 , . . . , φ (0) n ) · τ (0) BKP =         R φ (0) n,x · φ (0) 1 · · · R φ (0) n,x · φ (0) n . . . · · · . . . R φ (0) 1 ,x · φ (0) 1 · · · R φ (0) 1 ,x · φ (0) n         · τ (0) BKP . (4.1) and we have u ( n + n ) 2 = u (0) 2 + (log IW n,n ) xx . (4.2) 12 With this theorem, we can write do wn the solutions of the (2+1)-dimensional SK equation explicitly after successive application of gauge transformations. W e tak e the initial Lax op erator L (0) of the BKP hierarc h y as L (0) = ∂ + ∂ − 1 + ∂ − 3 + ∂ − 5 + · · · . The corresp onding eigenfunction φ (0) i and conjugate eigenfunction ψ (0) i = φ (0) i,x are giv en b y lemma 4 and lemma 5, i.e. φ (0) i = n X j =1 k j e α 3 j − 18 α j +9 β j α 2 j + α j β j +81 x + α j y + β j t , (4.3) ψ (0) i = n X j =1 k j α 3 j − 18 α j + 9 β j α 2 j + α j β j + 81 e α 3 j − 18 α j +9 β j α 2 j + α j β j +81 x + α j y + β j t , β j = β j ( α j ) . (4.4) Similar as section t w o and section three, w e need the follo wing theorem to assure that the new τ -function we get after gauge transformations will not v anish at an y p oin t. Theorem 3. L et eigenfunction φ (0) m take the form as in e q.(4.3) P n i =1 k m,i e a m,i x + α m,i y + β m,i t , m = 1 , 2 , if 0 < 3 · a 1 ,i < a 2 ,j , then we have IW 2 , 2 ( φ (0) 2 ,x , φ (0) 1 ,x ; φ (0) 1 , φ (0) 2 ) < 0 . The solution c an b e written as u (2+2) 2 = 1 + (log IW 2 , 2 ) xx . (4.5) Pr o of. φ (0) 1 and φ (0) 2 can b e rewritten as ( φ (0) 1 = P n i =1 R i e a i x , φ (0) 2 = P n i =1 S i e b i x , where the v alue of R i and S i are greater than zero, then w e ha v e ( φ (0) 1 ) 2 2 = 1 2 n X i,j =1 R i R j e ( a i + a j ) x , (4.6) ( φ (0) 2 ) 2 2 = 1 2 n X i,j =1 S i S j e ( b i + b j ) x , (4.7) Z φ (0) 1 ,x φ (0) 2 = n X i,j =1 R i S j a i a i + b j e ( a i + b j ) x , (4.8) Z φ (0) 2 ,x φ (0) 1 = n X i,j =1 R j S i b i a j + b i e ( a j + b i ) x . (4.9) The follo wing inequalit y ( a i + b k ) ( a j + b l ) > 4 a i b l , 13 is trivial if w e use 0 < 3 · a 1 ,i < a 2 ,j whic h means 0 < 3 · a i < b k , together with eq.(4.6), eq.(4.7), eq.(4.8) and eq.(4.9), w e can pro v e       R φ (0) 1 φ (0) 2 ,x ( φ (0) 2 ) 2 2 ( φ (0) 1 ) 2 2 R φ (0) 1 ,x φ (0) 2       = ( Z φ (0) 1 ,x φ (0) 2 )( Z φ (0) 2 ,x φ (0) 1 ) − ( φ (0) 1 ) 2 ( φ (0) 2 ) 2 4 < 0 , (4.10) b y a direct calculation. Eq.(4.5) can b e obtained b y eq.(4.2) and u (0) 2 = 1. Remark 2. F or T 1+1 = T I T D , with the gener ating function φ (0) 1 as in e q.(4.3), it’s e asy to show that τ (1+1) = ( φ (0) 1 ) 2 2 τ (0) (4.11) is p ositive. The c orr esp onding new solution of the (2+1)-dimensional SK e quation c an b e r epr esente d as u (1+1) 2 = 1 + (log( ( φ (0) 1 ) 2 2 )) xx (4.12) T o obtain a single-soliton solution of the (2+1)-dimensional SK equation, w e start from a generating function φ (0) 1 = e ξ + e − ξ , (4.13) and the solution is u (1+1) 2 = 1 + 2 a 2 sec h 2 ( ξ ) , (4.14) here ξ = α 3 − 18 α +9 β α 2 + αβ +81 x + α y + β t and a = α 3 − 18 α +9 β α 2 + αβ +81 . A solution generated from zero seed is ( u (1+1) 2 ) 0 = 2 ( a 0 ) 2 sec h 2 ( ξ 0 ) , (4.15) in which ξ 0 = ( α 0 ) 2 β 0 x + α 0 y + β 0 t , ( α 0 ) 5 = ( β 0 ) 3 and a 0 = ( α 0 ) 2 β 0 . The differences b et w een u (1+1) 2 and ( u (1+1) 2 ) 0 are show ed in figure 5. In figure 6, we plot the solution of the (2 + 1)-dimensional SK equation by taking φ (0) 1 = e 0 . 009999666694 x +0 . 02999999998 y +0 . 15 t + e 0 . 01333254332 x +0 . 03999999992 y +0 . 2 t + e 0 . 006666567904 x +0 . 02 y +0 . 1 t , (4.16) φ (0) 2 = e 0 . 5924749002 x +1 . 985399095 y +10 t + e 0 . 06656825084 x +0 . 1999997386 y + t + e 1 . 218304787 x +5 . 463203409 y +30 t , (4.17) in eq.(4.5). Corollary 2. F or the (2+1)-dimensional KK e quation and (2+1)-dimensional SK e quation, ther e exist a c ommon Galile an typ e tr ansformation b etwe en ( u (1+1) 2 ) 0 (gen- er ate d fr om zer o se e d) and u (1+1) 2 (gener ate d fr om non-zer o se e d), i.e. u 0 2 ( x, y , t ) 7− → u 2 ( x, y , t ) = 1 + u 0 2 ( x + 3 y + 15 t, y + 5 t, t ) . (4.18) 14 5 Conclusions and Discussions By now w e hav e obtained new solutions u ( n ) 2 in theorem 1 for KP equation, u (2+2) 2 in theorem 2 for (2+1)-dimensional KK equation and u (2+2) 2 in theorem 3 for (2+1)- dimensional SK equation b y using the the gauge transformations of the KP hierar- c h y , CKP hierarc h y and BKP hierarc h y resp ectively . The corresp onding generating functions of the gauge transformations previously men tioned are explicitly expressed in lemma 2 and lemma 4. F or these three equations, the single-soliton u (1) 2 (or u (1+1) 2 ) generated from non-zero seeds and ( u (1) 2 ) 0 (or ( u (1+1) 2 ) 0 ) generated from zero seed are constructed. The main differences b et w een the u 2 and ( u 2 ) 0 are heigh t and locations of the peak line under the same v alue of parameters, whic h are demonstrated visi- bly in figures 1, 2 and 3. W e also found a Galilean t yp e transformation in eq.(2.29) b et w een ( u (1) 2 ) 0 and u (1) 2 for the KP equation, and another one in eq.(4.18) b et ween ( u (1+1) 2 ) 0 and u (1+1) 2 for the (2+1)-dimensional KK and SK equations. T o guaran tee the new solutions u 2 generated by gauge transformations is smo oth, in other w ords, the transformed τ -function do esn’t v anish at an y p oint, we only consider the W n in theorem 1 and IW 2 , 2 in theorem 2 and theorem 3. The corollary 1 and corollary 2 sho w that we can establish a one-parameter transformation group (sp ecifically , Galilean t yp e transformation) of the solutions of these three equations b y setting the seeds u (0) 2 =  (arbitrary constan t) instead of u (0) 2 = 1. The adv an tage of this new metho d to find one-parameter group is to a v oid solving the c haracteristic line equation, whic h is not easy to solv e, as usual approac h of Lie p oint transformation. W e will try to do this in the future. 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Ablowitz, On the initial v alue problem for the KPI I equa- tion with data that do es not deca y along a line, Nonlinearit y 17 (2004), 1843- 1866. 17 [34] C.Tian, ”Symmetry” in Soliton Theory and Its Applications Edited b y C.H.Gu (Springer 1995) p192-229. 18 Figure 1: Single-soliton solutions at t = 1 of the KP equation. The lo w er one is ( u (1) 2 ) 0 with k 0 = 1, α 0 1 = 2 . 7225 and α 0 2 = 3 . 24; the higher one is ( u (1) 2 − 1) with parameters k = 1, α 1 = 2 . 7225 and α 2 = 3 . 24. Figure 2: Two-soliton solution at t = 0 of the KP equation. 19 Figure 3: Single-soliton solutions at t = 1 of the (2+1)-dimensional KK equation. The higher one is ( u (1+1) 2 ) 0 with α 0 1 = 0 . 970299 and α 0 2 = 0 . 075; the lo w er one is ( u (1+1) 2 − 1) with parameters α 1 = 0 . 970299 and α 2 = 0 . 075. Figure 4: Solution at t = 0 of the (2+1)-dimensional KK equation. 20 Figure 5: Single-soliton solutions at t = 1 of (2+1)-dimensional SK equation. The higher one is ( u (1+1) 2 ) 0 with α 0 = 4 . 096 ,the low er one is ( u (1+1) 2 − 1) with parameters α = 4 . 096 . Figure 6: Solution at t = 0 of (2+1)-dimensional SK equation. 21