Vertex operator for the non-autonomous ultradiscrete KP equation

V ertex op erator for t he non-aut o nomous ultradiscr e te KP equation Y oic hi Nak ata Graduate School of Mathematical Sciences, The U niversit y of T okyo, 3-8-1 Komaba, Meguro-k u, 153-8 914 T o kyo, Japan E-mail: ynak ata@ms .u-to kyo.ac.jp Abstract. W e prop ose an ultradis crete analogue of the v ertex o per ator in the case of the ultradiscrete KP equation–several other ultr adiscrete equations–which maps N - soliton solutions to N + 1-soliton ones. P A CS num b ers: 02.30 .Ik;05.45.Yv Keywor ds : In tegrable Systems; Solitons; D iscrete System s; Cellular automaton; KP equation 1. In tro duction The KP equation is widely considered as the paradigm of soliton equations. The main results of soliton theory , including first and foremost the celebrated Sato theory , were disco v ered in the study of this equation. The disco v ery o f the v ertex op erator, which maps N -soliton solutions to N + 1-soliton ones, figures prominen tly among these results. The discrete KP equation (or Hirot a -Miw a equation), whic h is a discretized v ersion of the KP equation, is also regarded as a fundamen tal discrete soliton equation. By restricting its solutions it reduces to many w ell- known discrete soliton equations as, f or example, the discrete KdV equation or the discrete T o da equation. Soliton Cellular Automata form a class of cellular automat a that exhibit soliton- lik e b ehav iour and p ossess a ric h structure including the existence of explicit N -soliton solutions and an infinite amount of conserv ed quantities [1], like most ordinary soliton equations. The “Box and Ball System” ( BBS) [2] is the main represen tativ e of this class. It is related t o the discrete soliton equations, through a limiting pro cedure called “ultradiscretization” [3]. The non-autonomous ultradiscrete KP equation is obtained b y ultradiscretizing the non-autonomous discre te KP equation [4]. T ok ohiro et al presen ted its N - soliton solution a nd described the dynamics of the BBS with sev eral kinds of balls V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 2 in [5]. Shinza w a and Hirota discuss ed the consistency conditions of the B¨ ac klund transformation fo r the autonomo us ultradiscrete KP equation in [6]. Recen tly , these systems dra w inc reasing in terest due to t he establishmen t of relationships to other mathematical topics, for example, to algebraic geometry a nd represen ta t io n theory . It is therefore fruitful to clarify the symmetries a nd the algebraic structure o f ultradiscrete s oliton eq uations, as w as done for the con tin uous ones. T ak ahashi and H irota presen ted an approac h based o n so-called “permanent type solutions” [7] (whic h are express ed as signature-free Casorati determinan ts) to discuss particular solutions of ultradiscrete systems. Nagai presen ted iden tities for p ermanen t t yp e solutions, whic h can b e considered as ultradiscre te analo g ues of Pl¨ uc k er relations for determinan ts in [8]. Another approac h fo r obtaining solutions is the v ertex op erator for the ultradiscrete KdV equation, whic h is prop osed b y the author in [9 ]. Th is approac h is b eliev ed to b e closely related to certain ty p es of symmetrie s for this system. In this pap er, w e pr o p ose a ve rtex op erator fo r the non-autonomous ultradiscrete KP equation a nd v arious ultr adiscrete soliton equations o bta ined by reduction. In section 2 w e first pro p ose a recursiv e represen tation of the soliton solutions of the non- autonomous ultradiscrete KP equation. In section 3, w e prop ose the v ertex op erator as an op erator represen ta t io n of the recursiv e one. In section 4 , w e presen t v a r io us reductions of this equation and discuss their v ertex op erators and solutions. Finally , in section 5, w e give some concluding remarks. 2. Recursiv e expression for the solution of the ult r adiscrete KP equation The non- autonomous ultradiscrete KP equation is written as T l,m +1 ,n + T l +1 ,m,n +1 = max ( T l +1 ,m,n + T l,m +1 ,n +1 − 2 R n , T l,m,n +1 + T l +1 ,m +1 ,n ) , (1) where R n ≥ 0 dep ends o nly on n . Theorem 1 The function T ( N ) l,m,n expr esse d as T ( N ) l,m,n = ( max ( T ( N − 1) l,m,n , 2 η N + T ( N − 1) l − 1 ,m +1 ,n ) ( N ≥ 1) 0 ( N = 0) (2) solves e q uation (1) for η N given by η N = C N + lP N − mQ N − n X 0 Ω N ,d . (3) Her e, P j i Ω N ,d stands for j X i Ω N ,d =                    j X d = i +1 Ω N ,d ( i < j ) 0 ( i = j ) − i X d = j +1 Ω N ,d ( i > j ) , (4) V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 3 and the p ar ameters P i , Q i and Ω i,n ( i = 1 , . . . , N ) s atisfy the r elations: P N ≥ P N − 1 ≥ . . . ≥ P 1 ≥ 0 (5) Q N ≥ Q N − 1 ≥ . . . ≥ Q 1 ≥ 0 (6) Ω i,n = min( Q i , R n − 1 ) . (7) Lemma 2 L et H ( N ) l,m,n = T ( N ) l,m + j +1 ,n + k + T ( N ) l + i +1 ,m,n − T ( N ) l +1 ,m + j,n + k − T ( N ) l + i,m +1 ,n (8) for i, j, k such that iP N + j Q N + n + k X n Ω N ,d ≥ . . . ≥ iP 1 + j Q 1 + n + k X n Ω 1 ,d ≥ 0 . (9) Then it holds that H ( N ) l,m,n ≤ 2 ( iP N + j Q N + n + k X n Ω N ,d ) . (10) Pro of By emplo ying the inequality max( a, b ) − max( c, d ) ≤ max( a − c, b − d ) , (11) w e obta in T ( N ) l,m + j +1 ,n + k − T ( N ) l + i,m +1 ,n ≤ ma x ( T ( N − 1) l,m + j +1 ,n + k − T ( N − 1) l + i,m +1 ,n , − 2( iP N − j Q N − n + k X n Ω N ,d ) + T ( N − 1) l − 1 ,m + j +2 ,n + k − T ( N − 1) l + i − 1 ,m +2 ,n ) (12) T ( N ) l + i +1 ,m,n − T ( N ) l +1 ,m + j,n + k ≤ ma x ( T ( N − 1) l + i +1 ,m,n − T ( N − 1) l +1 ,m + j,n + k , 2( iP N + j Q N + n + k X n Ω N ,d ) + T ( N − 1) l + i,m +1 ,n − T ( N − 1) l,m + j +1 ,n + k ) (13) Adding the inequalities yields H ( N ) l,m,n ≤ max ( H ( N − 1) l,m,n , 2( iP N + j Q N + n + k X n Ω N ,d ) , − 2( iP N + j Q N + n + k X n Ω N ,d ) + H ( N − 1) l,m,n + H ( N − 1) l − 1 ,m +1 ,n , H ( N − 1) l − 1 ,m +1 ,n ) (14) T aking in to accoun t the relations iP N + j Q N + P n + k n Ω N ,d ≥ iP N − 1 + j Q N − 1 + P n + k n Ω N − 1 ,d , it can b e sho wn inductiv ely that the four argumen ts in t his maxim um are all less than 2( iP N + j Q N + P n + k n Ω N ,d ).  Lemma 3 L et H ′ ( N ) l,m,n = T ( N ) l,m,n +1 + T ( N ) l,m +2 ,n − T ( N ) l,m +1 ,n − T ( N ) l,m +1 ,n +1 . (15) V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 4 One then has H ′ ( N ) l,m,n ≤ 2 ( Q N − Ω N ,n +1 ) (16) when one r e quir es that t he T ( i ) l,m,n ( i = 1 , . . . , N ) ar e solutions of (1). Esp e cial ly for Ω N ,n = Q N , the in e quality (16) b e c omes an e quality, i.e.: H ′ ( N ) l,m,n = 0 . Pro of When Ω N ,n = R n +1 , we obtain by virtue of the inequalit y (11): H ′ ( N ) l,m,n ≤ ma x ( H ′ ( N − 1) l,m,n , H ′ ( N − 1) l − 1 ,m +1 ,n , 2( Q N − Ω N ,n +1 ) + T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l,m +2 ,n − T ( N − 1) l − 1 ,m +2 ,n − T ( N − 1) l,m +1 ,n +1 , − 2( Q N − Ω N ,n +1 ) + H ( N − 1) l − 1 ,m,n +1 | ( i,j,k )=(0 , 1 , − 1) + H ′ ( N − 1) l − 1 ,m +1 ,n ) . (17) Ho w ev er, T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l,m +2 ,n − T ( N − 1) l − 1 ,m +2 ,n − T ( N − 1) l,m +1 ,n +1 ≤ 0 b ecause T ( N − 1) l,m,n satisfies (1). It can then b e sho wn inductiv ely that all argumen ts in the maxim um are less than 2( Q N − Ω N ,n +1 ). On the other hand, when Ω N ,n = Q N , by virtue of (6 ), T ( N − 1) l,m +1 ,n is equal to T ( N − 1) l,m,n +1 for all l , m b ecause C i + lP i − ( m + 1) Q N − n X 0 Ω N ,d = C N + lP N − mQ N − n +1 X 0 Ω N ,d (18) for all i = 1 , . . . , N . W e thu s obtain that H ′ ( N ) l,m,n = ( T ( N ) l,m,n +1 − T ( N ) l,m +1 ,n ) + ( T ( N ) l,m +2 ,n − T ( N ) l,m +1 ,n +1 ) = 0 . (19)  Lemma 4 L et H ′′ ( N ) l,m,n = T ( N ) l,m,n +1 + T ( N ) l +2 ,m,n − T ( N ) l +1 ,m,n − T ( N ) l +1 ,m,n +1 . (20) One then has H ′′ ( N ) l,m,n ≤ 2 P N (21) when al l of T ( i ) l,m,n ( i = 1 , . . . , N ) a r e solutions of (1). Pro of The pro of is essen tially the same as t ha t of Lemma 3 when Ω N ,n = R n +1 .  W e no w hav e all the necess ary lemmas at o ur disp osal and pro ceed to the pro of of theorem 1. W e shall prov e the theorem inductiv ely . It is clear that T (0) l,m,n solv es equation (1) b ecause of the non- negativit y of R n . No w, let us a ssume that the theorem ho lds at 1 , . . . , N − 1. By substituting (2) in equation (1), each con tribution can b e written as T ( N ) l,m +1 ,n + T ( N ) l +1 ,m,n +1 = max ( T ( N − 1) l,m +1 ,n + T ( N − 1) l +1 ,m,n +1 , 2( P N − Ω N ,n +1 ) + 2 η N + T ( N − 1) l,m +1 ,n + T ( N − 1) l,m +1 ,n +1 , − 2 Q N + 2 η N + T ( N − 1) l − 1 ,m +2 ,n + T ( N − 1) l +1 ,m,n +1 , 4 η N + 2( P N − Q N − Ω N ,n +1 ) + T ( N − 1) l − 1 ,m +2 ,n + T ( N − 1) l,m +1 ,n +1 ) , (22) V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 5 for the left hand side of (1 ), and T ( N ) l +1 ,m,n + T ( N ) l,m +1 ,n +1 = max ( T ( N − 1) l,m,n + T ( N − 1) l,m +1 ,n +1 , 2 P N + T ( N − 1) l,m +1 ,n + T ( N − 1) l − 1 ,m +1 ,n +1 , − 2( Q N + Ω N ,n +1 ) + T ( N − 1) l +1 ,m,n + T ( N − 1) l,m +2 ,n +1 , 4 η N + 2( P N − Q N − Ω N ,n +1 ) + T ( N − 1) l,m +1 ,n + T ( N − 1) l − 1 ,m +2 ,n +1 ) (23) T ( N ) l,m,n +1 + T ( N ) l +1 ,m +1 ,n = max ( T ( N − 1) l,m,n +1 + T ( N − 1) l +1 ,m,n , 2( P N − Q N ) + T ( N − 1) l,m,n +1 + T ( N − 1) l,m +2 ,n , − 2Ω N ,n +1 + T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l +1 ,m +1 ,n , 4 η N + 2( P N − Q N − Ω N ,n +1 ) + T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l,m +2 ,n ) (24) for the righ t hand s ide. In thes e expressions it lo oks as if each of the maxim um op era t io ns in (2 2)–(24) has four argumen ts. How ev er, b y virtue of Lemma 2, the third argumen t in (22) and (23 ) cannot yield the maxim um b ecause it is alwa ys less tha n the second argumen t. Then, the relev an t a rgumen ts o f the maxim um in (2 2 ) are in fact T ( N − 1) l,m +1 ,n + T ( N − 1) l +1 ,m,n +1 (25) 2 η N + 2( P N − Ω N ,n +1 ) + T ( N − 1) l,m +1 ,n + T ( N − 1) l,m +1 ,n +1 (26) 4 η N + 2( P N − Q N − Ω N ,n +1 ) + T ( N − 1) l − 1 ,m +2 ,n + T ( N − 1) l,m +1 ,n +1 (27) and those in the maxim um o f the contributions in (23), (2 4), as they app ear in the righ t hand side of equation (1): max( T ( N − 1) l +1 ,m,n + T ( N − 1) l,m +1 ,n +1 − 2 R, T ( N − 1) l,m,n +1 + T ( N − 1) l +1 ,m,n ) (28) 2 η N + max(2 P N − 2 R n + T ( N − 1) l,m +1 ,n + T ( N − 1) l,m +1 ,n +1 , 2( P N − Q N ) + T ( N − 1) l,m,n +1 + T ( N − 1) l,m +2 ,n , − 2Ω N ,n +1 + T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l +1 ,m +1 ,n ) (29) 4 η N + 2( P N − Q N − Ω N ,n +1 ) + max( T ( N − 1) l,m +1 ,n + T ( N − 1) l − 1 ,m +2 ,n +1 − 2 R, T ( N − 1) l − 1 ,m +1 ,n +1 + T ( N − 1) l,m +2 ,n ) . (30) Here, (25) and (27) are iden tical to (2 8) and (30) b ecause b y assumption, T ( N − 1) l,m,n solv es the equation (1). By subtracting (26) from (29), we obtain max (2(Ω N ,n +1 − R n ) , 2(Ω N ,n +1 − Q N ) + H ′ ( N − 1) l,m,n , − 2 P N + H ′′ ( N − 1) l,m,n ) . (31) The third argumen t of this maximum is non- p ositiv e by virtue o f Lemma 4. In the case Ω N ,n +1 = Q N ≤ R n , Ω N − 1 has t o b e equal to Q N − 1 due to condition (6). Then, the first argumen t in (31) is also non- p ositiv e and the second argumen t is 0, due to L emma 3. In the case Ω N ,n +1 = R n , the first argumen t in the maximum in (31 ) is 0 and t he second argumen t is non-p ositiv e b y virtue of Lemma 3. Thus , (31) is equal to 0, in all p ossible cases (i.e., Ω N ,n +1 = Q N or Ω N ,n +1 = R n ). V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 6 W e hav e therefore shown that all arg umen ts of the maxim um in (2 2) which constitutes the left hand side of (1 ), hav e an equiv alent coun terpart a mong (28), (29) , (30), i.e. among the three arguments that contribute to the right hand side of (1). Hence, (1) is satisfied.  Please note that the pro of allo ws for the p ossibilit y that, at differen t v alues of n , Ω N ,n +1 satisfies differen t equalities (Ω N ,n +1 = R n or Ω N ,n +1 = Q N for different n ), b ecause the shift of the indep enden t v ariables induced b y (2) affects only l and m , not n . 3. V ertex op erator for the K P equation In this section w e prop o se an a lt ernat ive represen tation of the N -soliton solutions, generated by a v ertex op erator X . The 0- soliton solution T (; ; ) is written as: T (; ; ) := 0 (32) whereas the N + 1-soliton solution is g enerated from the N -soliton solution T ( P 1 , . . . , P N ; Q 1 , . . . , Q N ; C 1 , . . . , C N ) (written as T ( P ; Q ; C ) for brevit y) by X ( P N +1 , Q N +1 , C N +1 ) T ( P ; Q ; C ) := max ( T ( P ; Q ; C ) , 2 η N +1 + T ( P ; Q ; C − A N +1 )) (33) =: T ( P 1 , . . . , P N , P N +1 ; Q 1 , . . . , Q N , Q N +1 ; C 1 , . . . , C N , C N +1 ) , (34) where the parameters P N +1 , Q N +1 in the verte x op erator X m ust satisfy ( P i − P N +1 )( Q i − Q N +1 ) ≥ 0 . (35) The phase factor η N +1 is the same as in (3), and the interaction terms A N +1 = t ( A N +1 , 1 , . . . , A N +1 ,N ) are A i,j = min ( P i , P j ) + min ( Q i , Q j ) . (36) Prop osition 5 Th e action of the op er ator X is c ommutative. Pro of By calculating X (Ω b , η b ) X (Ω a , η a ) F ( Ω ; η ) directly , w e obta in X ( P b , Q b , C b ) X ( P a , Q a , C a ) T ( P ; Q ; C ) = max ( T ( P ; Q ; C ) , 2 η b + T ( P ; Q ; C − A b ) , 2 η a + T ( P ; Q ; C − A a ) , 2 η a + 2 η b − 2 A b,a + T ( P ; Q ; C − A a − A b )) . (37) F rom this relation it is clear that interc hanging the subscripts a and b do es not c hange the ov erall v alue of the maxim um.  Rewriting this prop osition yields the follo wing coro lla ry: V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 7 Corollary 6 The N -s oliton solution T ( P ; Q ; C ) is invariant under the p ermutation of its p a r ameters, i.e.: T ( P 1 , . . . , P N ; Q 1 , . . . , Q N ; C 1 , . . . , C N ) = T ( P σ (1) , . . . , P σ ( N ) ; Q σ (1) , . . . , Q σ ( N ) ; C σ (1) , . . . , C σ ( N ) ) ( σ ∈ S N ) (38) By virtue of corollary 6 , w e can fix the lab els of the parameters as in (5), (6) without loss of generality . By virtue of this ordering, the phase shifts in A i,j in the definition (33) simplify to min( P i , P N ) = P i , min( Q i , Q N ) = Q i ( i = 1 , . . . , N − 1) . (39) It should b e noted that the phase shifts η → η + P and η → η + Q are equiv alen t to shifts on the indep enden t v ariables l → l + 1 and m → m − 1, whic h sho ws that T ( P ; Q ; C ) is equiv alen t to T ( N ) l,m,n . 4. Reduction to v arious ultradiscrete soliton equations In this section w e presen t some examples of reductions of the ultradiscrete KP equation to 1 + 1 dimensional ultradiscrete equations a nd we giv e the v ertex op erators f o r these equations. 4.1. The Box and Bal l System and its varieties By restricting T l,m,n to T l,m,n = F l − M m n (40) and denoting s = l − M m and n = j , t he non-autonomous ultradiscrete KP equation (1) is reduced to the so-called non-autonomous ultra discrete h ungry KdV equation: F s + M +1 j +1 + F s j = max( F s + M +1 j + F s j +1 − 2 R j , F s +1 j + F s + M j +1 ) . (41) By means of the dep enden t v ariable transformat ion B t i,j = 1 2 ( F s +1 j + F s j +1 − F s +1 j +1 − F s j ) , (42) and denoting s = M t + i , (41) is transformed into B t +1 i,j = min  R j − i − 1 X k =1 B t +1 k ,j − M X k = i B t k ,j , j − 1 X n = −∞ ( B t i,n − B t +1 i,n )  , (43) whic h describes the dynamics of a Box and Ball System with M kinds o f balls, as presen ted in [5]. This system is required to satisfy the following b oundary conditions: B t i,j = 0 for j ≪ 0 (44) In particular, in the case of M = 1 it reduces to an extension of the standard BBS [2], with v ar iable size of b oxes at eac h site. V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 8 In our represen t ation (33), the reduction (4 0) is equiv alen t to the pa r ameter restriction: M P N = Q N . (45) It should b e noted that our represen tation satisfies the b oundary condition (44) b ecause the first argumen t of max in (33) is nev er chos en for sufficien tly small j . Then, the v ertex op erator for (41) can b e written as X ( P N +1 , C N +1 ) T ( P ; C ) := max ( T ( P ; C ) , 2 η N +1 + T ( P ; C − A N +1 )) , (46) where the phase factor η N +1 is η N = C N + sP N − j X 0 Ω N ,d , (47) and Ω N ,j and the in teraction terms A i,j are expressed as Ω N ,j = min( R j − 1 , M P N ) , A i,j = ( M + 1) min( P i , P j ) . (48) 4.2. The ultr adiscr ete T o da e quation By restricting T l,m,n to T l,m,n = F l + n m + n , (49) R n = co nst. and denoting t = l + n a nd s = m + n , (1) is reduced to the ultradiscrete T o da equation: F t s +1 + F t +2 s +1 = max( F t +1 s +2 + F t +1 s − 2 R, 2 F t +1 s +1 ) (50) By means of the dep enden t v ariable transformat ion U t s = 1 2 ( F t s +2 − 2 F t s +1 + F t s ) , (51) (50) is transformed into U t +2 s +1 − 2 U t +1 s +1 + U t s +1 = max ( U t +1 s +2 − R , 0) − 2 max( U t +1 s +1 − R , 0) + max( U t +1 s − R , 0) , (52 ) whic h describ es the dynamics of the T o da t yp e cellular automaton presen ted in [10]. In our represen t ation (33), the reduction (4 9) is equiv alen t to the pa r ameter restriction: Ω N = Q N − P N i.e. P N = Q N − Ω N = max ( Q N − R, 0) (53) The verte x op erator of (50) can b e expresse d as X ( P N +1 , C N +1 ) T ( P ; C ) := max ( T ( P ; C ) , 2 η N +1 + T ( P ; C − A N +1 )) , (54) where the phase factor η N +1 is η N = C N + t max( Q N − R, 0) − sQ N , (55) and the in teraction term A i,j is written as A i,j = min ( Q i , Q j ) + ma x(min ( Q i , Q j ) − R , 0) (56) V ertex op er ator for the non-autonom ous ultr adiscr e te KP e quation 9 5. Concluding Remarks In this pap er, we prop osed a recursiv e represen tation of the N - soliton solutions and v ertex op erato rs f or the ultra discrete KP equation. W e also prop osed expressions for v arious ultradiscrete equations, obtained b y reduction from the KP equation. 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