On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation

On the remark able relations among PDEs in tegrable b y the in v erse sp ectral transform metho d, b y the metho d of c haracteristics and b y the Hopf-Cole transformation A. I. Zench uk 1 , § and P . M. Santini 2 , § 1 Center of Nonline ar Studies of the L andau Institute for T he or etic al Ph ysic s (International Ins titute of Nonline a r Scienc e) Kosygina 2, Mosc ow , Russia 119334 2 Dip artimen to di Fi sic a, Universit` a d i R om a ”L a Sapi e n za” and Istituto Nazional e di Fisic a Nucle ar e , Se z i one di R o ma 1 Piazz.le Aldo Mor o 2, I-00 185 R oma, Italy § e-mail: zenchuk@it p.ac.ru , paolo.santini @roma1.infn.it Octob er 24, 2018 Abstract W e establish deep and remark able connections among p artial differen tial equatio ns (PDEs) int egrable b y different m etho ds: the inv erse sp ectral transform metho d, t he metho d of characte ristics an d the Hopf-Cole tr an s formation. More concretely , 1) w e sho w that the in tegrabilit y prop er ties (Lax pair, infinitely-man y comm uting s ymmetries, large classes of analytic solutions) of (2+1)-dimensional P DEs int egrable by the Inv ers e Scattering T ransf orm metho d ( S -inte grable) can b e generated b y the in tegrabilit y prop- erties of the ( 1+1)-dimensional matrix B ¨ u rgers hierarc hy , int egrable by the matrix Hopf- Cole transf ormation ( C -in tegrable). 2) W e sho w that the in tegrabilit y prop er ties i) of S -in tegrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of the GL ( M , C ) self-dual Y ang Mills equations, and iii) of th e multidimensional Calogero equations can b e generated by the integrabilit y p rop erties of a recen tly introdu ced m ul- tidimensional matrix equation solv able b y the metho d of characte ristics. T o establish the ab o ve links, w e consider a blo c k F rob enius matrix reduction of the relev ant matrix fields, leading to in tegrable chai ns of m atrix equations for the blocks of s u c h a F rob enius matrix, follo wed by a systematic elimination p ro cedure of some of th ese blo c ks. The con- struction of large classes of solutions of the soliton equations fr om solutions of the matrix B ¨ ur gers hierarc hy turns out to b e intimately related to the co nstr u ction of solutions in Sato theory . 3) W e finally show that su itable generalizatio ns of the blo c k F rob enius ma- trix redu ction of the matrix B ¨ urgers hierarch y generates PDEs exhibiting in tegrabilit y prop erties in common with b oth S - and C - in tegrable equations. 1 In tro duction In tegrable nonlinear par t ia l differential equations (PDEs) can b e group ed into differen t classes, dep ending on their metho d of solution. W e distinguish the following three basic classes. 1. Equations solv able b y the metho d of characteristic s [1], hereafter called, for the sak e o f brevit y , C h -in tegrable, like the following matrix PDE in arbitrary dimensions [2 ]: w t + N X i =1 w x i ρ ( i ) ( w ) + [ B , w ] σ ( w ) = 0 , (1) 1 where w is a square matr ix and ρ ( i ) ( · ) , σ ( · ) are scalar functions represen table as p ositiv e p o w er series, or lik e the vec tor equations solv able by the generalized ho dogr aph metho d [3, 4, 5 , 6, 7, 8]. 2. Equations integrable b y a simple change of v ariables, often called C - in tegrable [9], lik e the matrix B ¨ urgers equation [10] w t − B w xx − 2 B w x w + [ w, B ]( w x + w 2 ) = 0 , (2) where B is any constan t s quare matrix, linearizable b y t he matrix v ersion of the Hopf-Cole transformation Ψ x = w Ψ [11]. 3. Equations in tegrable b y less elemen t a ry metho ds of sp ectral nature, the inv erse sp ectral transform (IST) [12, 13, 14, 15, 16] a nd the dressing metho d [17, 18, 19, 20, 16], often called S -inte grable [9] or soliton equations. Within this class of equations, w e distinguish four differen t sub classes, dep ending on the nature of the a sso ciated spectral theory . (a) Soliton e quations in (1+1)-dimensions lik e, for instance, the Kortew eg-de V ries (KdV) [21, 1 2] and the Nonlinear Sc hr¨ odinger (NLS) [22] equations, whose in v erse problems are lo cal R iemann- Hilb ert (R H) problems [13, 15]. (b) Their (2+1)- dimensional generalizations, like the Kadom tsev-P etviash vily (KP) [23 ] and Dav ey-Stew artson (D S) [24] equations, whose in ve rse problems a re nonlo cal RH [25, 26] or ¯ ∂ - pro blems [27]. (c) The self-dual Y ang- Mills (SD YM) equation [28, 29] and its generalizations in ar bi- trary dimensions. (d) Multidimensional PDEs asso ciated with one-para meter families of comm uting ve ctor fields, whose nov el IST, recen tly constructed in [3 0, 3 1 ], is c haracterized by nonlinear RH [3 0, 31] or ¯ ∂ [32] pro blems. D istinguished examples a r e the dispersion-less KP equation, the heav enly equation of Plebanski [33] and the fo llo wing inte grable syste m of PD Es in N + 4 dimensions [30]: ~ v t 1 z 2 − ~ v t 2 z 1 + N X i =1 ( ~ v z 1 · ∇ ~ x ) ~ v z 2 − N X i =1 ~ v z 2 · ∇ ~ x ~ v z 1 = ~ 0 , (3) where ~ v is an N -dimensional ve ctor and ∇ ~ x = ( ∂ x 1 , .., ∂ x N ). Eac h one of the ab o v e methods of solution allows one to solv e a particular class of PDEs and is no t a pplicable to other classes. Recen tly , sev eral v aria n ts of the classic al dressing metho d ha v e b een suggested, a llo wing to unify the integration algorithms for C - and S - integrable PDEs [34], f o r C - a nd C h - in tegrable PDEs [35], and for S - and C h - integrable PDEs [36]. In particular, the relation b et w een the matrix PDE (1), in tegra ble by the metho d of c haracteristics, and the GL ( M , C ) SD YM equation has b een recen tly established in [3 7 ]. As a conse quence of this result, it w as sho wn that the SDYM equation admits an infinite class of low er-dimensional reductions whic h are in tegrable b y the metho d of c haracteristics. In this pap er we extend the results of [37], show ing the existence of remark ably deep relations among S -, C - and C h - in tegrable systems. More precisely , we do the follo wing. 2 1. W e sho w (in § 2 ) that t he in tegrability prop erties (Lax pair, infinitely-man y commu ting symmetries , large classes of a nalytic solutions) of the C - in tegrable (1+1)- dimensional ma- trix B ¨ urgers hierarc h y can b e used to generate the in tegrability prop erties of S - in tegrable PDEs in (2+1) dimensions, like the N- wa ve, KP , and DS equations; this result is ac hieve d using a blo ck F r o b enius matrix reduction of the relev ant matrix field of t he matrix B ¨ urgers hierarc h y , leading to integrable c hains o f matrix equations for the blo c ks of suc h a F rob e- nius matrix, follow ed b y a systematic elimination pro cedure of some of these blo cks . The construction of large classes of solutions o f the soliton equations from solutions of t he ma- trix B ¨ urgers hierarch y turns out to b e in timately related to the construction of solutions in Sato theory [38, 39 , 40, 41]. On the w ay bac k, starting with the Lax pair eigenfunctions of the deriv ed S -integrable sys tems, w e sho w that the co efficien ts of their asymptotic ex- pansions, for large v alues of the sp ectral parameter, coincide with the elemen ts of the ab ov e in tegrable c hains, obtaining an in teresting sp ectral meaning of suc h c hains. It follo ws that, compiling these co efficien ts into the F r o b enius matr ix, one constructs the C - integrable matrix Burgers hierarc h y and its solutions from the eigenfunctions of the S -in tegrable systems. 2. W e sho w (in § 3 ) that the in tegrability prop erties o f the m ultidimensional matrix equation (1), solv able by the metho d of c ha r a cteristics, can b e used to generate the in tegrability prop erties of (a) S -inte grable PDEs in (1+1) dimensions, like the N -wa ve , KdV, mo dified KdV (mKdV), and NLS equations (in § 3.2); (b) S -in tegrable m ultidimensional generalizations of the GL ( M , C ) SDYM equations (in § 3.3); this deriv a t io n from the simpler and basic matrix equation ( 1 ), allows one to unco v er for free t w o imp orta n t prop erties of such equations: a con venie nt parametrization, giv en in terms of the blo ck s of the F rob enius matrix, allow ing one to reduce by ha lf the n um b er o f equations, and the existence of a large class of solutions describing the gradien t catastrophe of multidimens ional w av es. (c) S -in tegrable m ultidimensional Calog ero equations [42 , 43 , 44, 45, 46] (in § 3.4). As b efore, these results are obtained considering a blo c k F rob enius matrix reduction, leading to in tegrable c hains, follo w ed by a systematic elimination pro cedure of some of their elemen ts. Vice-v ersa, suc h c hains are satisfied b y the co efficien ts of the asymptotic expansion, for la rge v alues of the sp ectral parameter, of the eigenfunctions of the soliton equations. 3. W e show (in § 4) that a pro p er generalization of the blo c k F rob enius matrix reduction of the matrix B¨ urgers hierarch y can b e used to construct the in tegrabilit y prop erties of non- linear PDEs exhibiting pro p erties in common with b oth S - and C - integrable equations. Figure 1 b elo w sho ws the diagram summarizing t he connections discussed in § 2 and § 3. W e end t his in tro duction men tioning previous w ork related to our main findings. i) The matrix Burgers equation (2) with B = I , together with the blo ck F ro b enius matrix reduction (13), ha v e b een used in [47 ] to construct some explicit solutions of the linear Sc hr¨ o ding er and diffusion equations. ii) As already men tioned, once the connections illustrated in § 2 are exploited to construct large classes of solutions of soliton equations from simpler solutions of the matrix B ¨ urgers hierarc h y , the corresp onding f ormalism turns out to b e intimately related to the construction o f solutions of soliton equations in Sato theory . 3 for the w (j) ’s integrable chains for the w (j) ’s integrable chains w w w (2) (3) I 0 0 0 I 0 (1) for the w (j) ’s integrable chains + t w x j ρ w j (w) N−1 j=1 +[B,w] =0 (w) matrix PDEs integrable by Lax pairs, symmetries N − dimensional and solutions ρ j =0 =0 σ for the w (j) ’s integrable chains of some w ’s (j) of some w ’s (j) of some w ’s (j) of some w ’s (j) (1+1) − dimensional C−integrable matrix PDES (N−wave, Burgers, 3rd order Burgers, ...) for matrix w Lax pairs, symmetries and solutions Frobenius Lax pairs, symmetries S−integrable matrix PDES (2+1) − dimensional and solutions (1+1) − dimensional and solutions Lax pairs, symmetries w= the method of characteristics: Σ σ Frobenius block matrix w S−integrable matrix PDES Calogero systems in multidimensions Lax pairs, symmetries and solutions Lax pairs, symmetries and solutions GL(M,C) − SDYM and its (2N) − dimensional generalizations block matrix reduction matrix reduction Frobenius block (N−wave, DS, KP, ..) (N−wave, NLS, KdV, ..) elimination elimination elimination elimination Fig. 1 The remark able relations amo ng PDEs integrable b y the inv erse sp ectral tr a nsform metho d, b y the metho d of c haracteristics and b y the Hopf-Cole transformation. 2 Relation b et ween C - and S -in tegrability Usually C - and S - integrable systems are considered a s completely in tegrable systems with differen t in tegrability f eat ur es. In this section w e show t he remark able relations b etw een them. 2.1 C - int egrable PDEs It is well kno wn that the hierarc h y of C -integrable system s associated with the matrix Hopf - Cole transformation Ψ x = w Ψ (4) can b e generated b y the compatibilit y condition b etw een equation (4) and the following hi- erarc hies of linear comm uting flo ws (the hierarc h y generated by higher x -deriv ative s a nd its 4 replicas): Ψ t nm = B ( nm ) ∂ n x Ψ , n, m ∈ N + , (5) where Ψ and w a re square matrix functions and B ( nm ) , n, m ∈ N + are constant comm uting square matrices. The inte grability conditions yield the fo llowing hierarc h y of C -inte grable equations and its replicas: w t nm + [ w , B ( nm ) W ( n ) ] − B ( nm ) W ( n ) x = 0 , (6) where W ( n ) = W ( n − 1) x + W ( n − 1) w , n ∈ N + , W (0) = I , W (1) = w , W (2) = w x + w 2 , W (3) = w xx + 2 w x w + w w x + w 3 , . . . . (7) and I is the iden tit y mat r ix. The first three examples, together with their comm uting replicas, read: 1. n = 1: a C - in tegrable N - w a v e equation in (1+1)-dimensions: w t 1 m − B (1 m ) w x + [ w , B (1 m ) ] w = 0 , (8) 2. n = 2: the matrix B ¨ urgers equation: w t 2 m − B (2 m ) w xx − 2 B (2 m ) w x w + [ w , B (2 m ) ]( w x + w 2 ) = 0 ; (9) 3. n = 3: the 3-rd order matrix B ¨ urgers equation: w t 3 m − B (3 m ) w xxx − 3 B (3 m ) w xx w + [ w , B (3 m ) ]( w xx + w w x + 2 w x w + w 3 ) − 3 B (3 m ) w x ( w x + w 2 ) = 0 . (10) The w ay of generating solutions of the C - integrable PDEs (6) is elemen ta ry: tak e the general solution of equations (5 ) : Ψ( ~ x ) = Z Γ e k x + P j,m ≥ 1 B ( j m ) t j m k j ˆ Ψ( k ) d Ω( k ) , (11) where Γ is an arbitra r y contour in the complex k -pla ne, Ω( k ) is an a rbitrary measure and ˆ Ψ( k ) is an arbitrary matrix function of the sp ectral pa r a meter k , and ~ x is the vec tor of all indep enden t v ariables: ~ x = { x, t nm ; n, m ∈ N + } . Then w = Ψ x Ψ − 1 (12) solv es (6). 5 2.2 Blo c k F rob enius matrix structure, in tegrable c hains and S -in tegrable PDEs It turns out that the C -integrable hierarc h y of (1+ 1)-dimensional PDEs (6), including the N - w a v e, B ¨ urgers and third order B ¨ urgers equations (8)- (10) as distinguished examples, generates a corresp onding hierarch y of S -in tegrable (2 +1)-dimensional PDEs, including the celebrated N - w a v e, DS a nd KP equations resp ectiv ely . This is p ossible, due t o the remark able f a ct that eqs. (4) and (5) are compatible with the following blo ck F rob enius matrix structure of the matrix function w : w =      w (1) w (2) w (3) · · · I M 0 M 0 M · · · 0 M I M 0 M · · · . . . . . . . . . . . .      , (13) where I M and 0 M are the M × M iden tit y a nd zero matrices, M ∈ N + , a nd w ( j ) , j ∈ N + are M × M matrix functions. This blo c k structure of w is consisten t with eqs.(4) and (5) (and therefore with the whole C -in tegrable hierarc h y (6)) iff matrix Ψ is a blo c k W ronskian matrix : Ψ =      Ψ (11) Ψ (12) Ψ (13) · · · ∂ − 1 x Ψ (11) ∂ − 1 x Ψ (12) ∂ − 1 x Ψ (13) · · · ∂ − 2 x Ψ (11) ∂ − 2 x Ψ (12) ∂ − 2 x Ψ (13) · · · . . . . . . . . . . . .      , (14) and B ( im ) = diag( ˜ B ( im ) , ˜ B ( im ) , · · · ) , (15) where the blo c ks Ψ ( ij ) , i, j ∈ N + are M × M matrices, and ˜ B ( im ) , i ∈ N + are constan t comm uting M × M matrices. In equations (13) -(15), the matr ices w , Ψ and B ( im ) are c hosen to b e ∞ × ∞ square matrices containing an infinite n um b er o f finite blo c ks; only in dealing with the construction of explicit solutions, it is conv enien t to consider a finite num b er o f blo c ks. Substituting the expressions (13) a nd (15) into the nonlinear PDEs (8-10), one obtains the follo wing (b y construction) in tegrable infinite c hains of PDEs , for n, m ∈ N + : w ( n ) t 1 m − ˜ B (1 m ) w ( n ) x + [ w ( n +1) , ˜ B (1 m ) ] + [ w (1) , ˜ B (1 m ) ] w ( n ) = 0 , (16) w ( n ) t 2 m − ˜ B (2 m ) w ( n ) xx − 2 ˜ B (2 m ) w ( n +1) x − 2 ˜ B (2 m ) w (1) x w ( n ) + (17) [ w (1) , ˜ B (2 m ) ]( w (1) w ( n ) + w ( n ) x + w ( n +1) ) + [ w (2) , ˜ B (2 m ) ] w ( n ) + [ w ( n +2) , ˜ B (2 m ) ] = 0 , w ( n ) t 3 m − ˜ B (3 m )  w ( n ) xxx + 3( w (1) xx w ( n ) + w ( n +1) xx ) + 3 w (1) x ( w (1) w ( n ) + w ( n +1) + w ( n ) x ) + (18) 3 w (2) x w ( n ) + 3 w ( n +2) x  + [ w (1) , ˜ B (3 m ) ]  w ( n ) xx + 2( w (1) x w ( n ) + w ( n +1) x ) + w (1) ( w ( n ) x + w (1) w ( n ) + w ( n +1) ) + w (2) w ( n ) + w ( n +2)  + [ w (2) , ˜ B (3 m ) ]  w ( n ) x + w (1) w ( n ) + w ( n +1)  + [ w (3) , ˜ B (3 m ) ] w ( n ) + [ w ( n +3) , ˜ B (3 m ) ] = 0 . F rom these chains , whose sp ectral nature will b e un v eiled in § 2.4, o ne constructs, through a systematic elimination of some of the blo c ks w ( j ) , the target S -in tegrable PDEs. Here w e consider the follo wing basic examples. 6 (2+1)-dimensional N -w a v e equation. Fixing n = 1 in eqs.(16), and choosing m = 1 , 2, one obtains the fo llo wing complete system o f equations fo r w ( i ) , i = 1 , 2 : w (1) t 1 m − ˜ B (1 m ) w (1) x + [ w (1) , ˜ B (1 m ) ] w (1) + [ w (2) , ˜ B (1 m ) ] = 0 , m = 1 , 2 . (19) Eliminating w (2) from equations (19) o ne obtains the classical (2+1)-dimensional S -in tegrable N - w a v e equation: [ w (1) t 11 , ˜ B (12) ] − [ w (1) t 12 , ˜ B (11) ] − ˜ B (11) w (1) x ˜ B (12) + ˜ B (12) w (1) x ˜ B (11) + (20) [[ w (1) , ˜ B (11) ] , [ w (1) , ˜ B (12) ]] = 0 . DS-t yp e equation. Cho osing n = 1 , 2 in equations (16), n = 1 in eq.(17 ), m = 1 in b oth equations, and simplifying the notation a s follo ws: t j = t j 1 , ˜ B ( j ) = ˜ B ( j 1) , j ∈ N + , ( 2 1) one obtains the fo llo wing complete system o f equations fo r w ( i ) , i = 1 , 2 , 3 : w (1) t 1 − ˜ B (1) w (1) x + [ w (2) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (1) = 0 , (22) w (2) t 1 − ˜ B (1) w (2) x + [ w (3) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (2) = 0 , (23) w (1) t 2 − ˜ B (2) w (1) xx − 2 ˜ B (2) w (2) x − 2 ˜ B (2) w (1) x w (1) + (24) [ w (1) , ˜ B (2) ]( w (1) w (1) + w (1) x + w (2) ) + [ w (2) , ˜ B (2) ] w (1) + [ w (3) , ˜ B (2) ] = 0 . Using eqs.(22) and (23), one can eliminate w (3) and w (2) from eq.(24). In the case ˜ B (2) = α ˜ B (1) ( α is a scalar), t his results in the follo wing equation for w (1) : [ w (1) t 2 , ˜ B (1) ] + α  w (1) t 1 t 1 − ˜ B (1) w (1) xx ˜ B (1) + [[ w (1) , ˜ B (1) ] , w (1) t 1 ] + (25) B (1) w x [ B (1) , w ] − [ B (1) , w ] w x B (1)  = 0 . In the simplest case of square matrices ( M = 2), with ˜ B (1) = β diag(1 , − 1) ( β is a scalar constan t), this equation reduces to the D S system: ˜ β q t 2 − 1 2 ( q xx + 1 β 2 q t 1 t 1 ) − 2( ϕq + 2 r q 2 ) = 0 , (26) − ˜ β r t 2 − 1 2 ( r xx + 1 β 2 r t 1 t 1 ) − 2( ϕr + 2 q r 2 ) = 0 , ϕ xx − 1 β 2 ϕ t 1 t 1 + 4( r q ) xx = 0 , where q = w (1) 12 , r = w (1) 21 , ϕ = ( w (1) 11 + w (1) 22 ) x , ˜ β = 1 αβ . (27) If ˜ β = i , this system admits the reduction r = ¯ q : iq t 2 − 1 2 ( q xx + 1 β 2 q t 1 tu 1 ) − 2( ϕq + 2 ¯ q q 2 ) = 0 , (28) ϕ xx − 1 β 2 ϕ t 1 t 1 + 4( ¯ q q ) xx = 0 , b ecoming DS-I and DS-I I if β 2 = − 1 and β 2 = 1 resp ectiv ely . 7 KP . T o deriv e the celebrated KP equation, c ho ose M = 1, tak e eqs.(17) with n = 1 , 2, and eq. (18) with n = 1 , ˜ B (2) = β , ˜ B (3) = − 1, where β is a scalar parameter, obtaining: w (1) t 2 − β  w (1) xx + 2 w (1) w (1) x + 2 w (2) x  = 0 , (29) w (2) t 2 − β  w (2) xx + 2 w (2) w (1) x + 2 w (3) x  = 0 , w (1) t 3 + w (1) xxx + 3  ( w (1) ) 2 w (1) x + ( w (1) x ) 2 + w (1) w (1) xx  + 3  w (2) xx + w (2) w (1) x + w (1) w (2) x  + 3 w (3) x = 0 , where w e ha v e set m = 1 and used again the nota tions (21). After eliminating w (2) and w (3) , one obta ins t he scalar p o ten tial KP for u = w (1) , y = t 2 , t = t 3 :  u t + 1 4 u xxx + 3 2 u 2 x  x + 3 4 β 2 u y y = 0 . (30) KP-I and KP-I I corresp ond to β 2 = − 1 and β 2 = 1 resp ectiv ely . 2.3 Lax pairs for the S -in tegrable systems Also the Lax pairs for the S -in tegrable syste ms deriv ed in § 2.2 can b e constructed in a similar w a y , from the system (4), (5). W e first observ e tha t, due to equation (4), equations (5) can b e rewritten as Ψ t nm = B ( nm ) W ( n ) Ψ . (31) Due to the blo c k F rob enius structure of w , it is con v enien t to w ork with the duals of equations (4) and ( 3 1): ˜ Ψ x = − ˜ Ψ w , (32) ˜ Ψ t nm = − ˜ Ψ B ( nm ) W ( n ) . (33) Substituting (13) and (15) in to equations (32-33), one o btains a system of linear chains for the blo c ks of matrix ˜ Ψ. The first few equations in v olving the blo c ks of the first row read: ˜ Ψ (1 n ) x + ˜ Ψ (11) w ( n ) + ˜ Ψ (1( n +1)) = 0 , (34) ˜ Ψ (1 n ) t 1 m + ˜ Ψ (11) ˜ B (1 m ) w ( n ) + ˜ Ψ (1( n +1)) ˜ B (1 m ) = 0 , (35) ˜ Ψ (1 n ) t 2 m + ˜ Ψ (11) ˜ B (2 m ) ( w ( n ) x + w (1) w ( n ) + w ( n +1) ) + ˜ Ψ (12) ˜ B (2 m ) w ( n ) + ˜ Ψ (1( n +2)) ˜ B (2 m ) = 0 , (36) ˜ Ψ (1 n ) t 3 m +  ˜ Ψ (11) ˜ B (3 m )  w ( n ) xx + 2( w (1) x w ( n ) + w ( n +1) x ) + w (1) ( w ( n ) x + w (1) w ( n ) + w ( n +1) ) + (37) w (2) w ( n ) + w ( n +2)  + ˜ Ψ (12) ˜ B (3 m )  w ( n ) x + w (1) w ( n ) + w ( n +1)  + ˜ Ψ (13) ˜ B (3 m ) w ( n ) + ˜ Ψ (1( n +3)) ˜ B (3 m ) = 0 , where n ∈ N + and ˜ Ψ ( ij ) is the ( i, j )-blo c k of matrix ˜ Ψ. 8 Lax pair for the N -w a v e equation. Setting n = 1 into eqs.(34,35) and eliminating ˜ Ψ (12) , one obtains (the dual of ) the Lax pair for the N - w a v e equation (20): ˜ ψ t 1 m − ˜ ψ x ˜ B (1 m ) + ˜ ψ [ ˜ B (1 m ) , w (1) ] = 0 , m = 1 , 2 , (38) where ˜ ψ = ˜ Ψ (11) . The dual of it, is the w ell-kno wn Lax pair of the N -w a v e equation (20) : ψ t 1 m − ˜ B (1 m ) ψ x − [ ˜ B (1 m ) , w (1) ] ψ = 0 , m = 1 , 2 . (39) Of course, the compatibility condition of eqs.(38) and/o r eqs. (39 ) yields t he nonlinear system (20). Lax pair for DS. In this paragra ph we set m = 1 in the integrable c hains, and use the notation (21). The first equation of the dual of the L ax pair is eq. (38) with m = 1 . T o deriv e the second equation, w e set m = n = 1 in to eq. (36), and eliminate the fields ˜ Ψ (12) , ˜ Ψ (13) , using eq. (34) with n = 1 , 2 . In this w ay one obtains the dual of the Lax pair for DS-type equations: ˜ ψ t 1 − ˜ ψ x ˜ B (1) + ˜ ψ [ ˜ B (1) , w (1) ] = 0 , (40) ˜ ψ t 2 + ˜ ψ xx ˜ B (2) + ˜ ψ x [ w (1) , ˜ B (2) ] + ˜ ψ ˜ s = 0 , ˜ s = [ ˜ B (2) , w (2) ] + w (1) x ˜ B (2) + ˜ B (2) w (1) x + [ ˜ B (2) , w (1) ] w (1) . Therefore the Lax pa ir reads: ψ t 1 − ˜ B (1) ψ x − [ ˜ B (1) , w (1) ] ψ = 0 , ψ t 2 − ˜ B (2) ψ xx + [ w (1) , ˜ B (2) ] ψ x − s ( y ) ψ = 0 , s = [ ˜ B (2) , w (2) ] + 2 ˜ B (2) w (1) x + [ ˜ B (2) , w (1) ] w (1) . (41) The compatibility conditions of equations (40) o r (41) yie ld a nonlinear system equiv alent to the system (22-24). Lax pair for KP . In this paragra ph we use the notatio ns (21) as w ell. The first equations of the dual of the Lax pair for KP are the scalar v ersions of eq.(40b) and eq.(41b) resp ectiv ely , with ˜ B (2) = β and ˜ B (3) = − 1. T o write the second equation of the Lax pair for KP , w e mus t tak e the scalar v ersion of eq.(37) with m = n = 1 , and eliminate ˜ Ψ (1 i ) , i = 2 , 3 , 4 using eq.s(36 ) for n = 1 , 2. As a result, the dual of the Lax pair reads 1 β ˜ ψ t 2 + ˜ ψ xx + 2 ˜ ψ u x = 0 , (42) ˜ ψ t 3 + ˜ ψ xxx + 3 ˜ ψ x u x − 3 2 ˜ ψ  u t 2 β − u xx  = 0 , and the Lax pa ir is 1 β ψ t 2 − ψ xx − 2 u x ψ = 0 , (43) ψ t 3 + ψ xxx + 3 u x ψ x + 3 2  u t 2 β + u xx  ψ = 0 . 9 2.4 F rom the Lax pairs of S -in tegrable PDEs to C -in tegrable P DEs As usual in the IST for (2+1)- dimensional soliton equations, one in tro duces the sp ectral pa- rameter λ in to the Lax pairs (39),(41) and (43) as follo ws ψ ( λ ; ~ x ) = χ ( λ ; ~ x ) e λxI + P i,m ≥ 1 ˜ B ( im ) t im λ i , (44) obtaining, resp ectiv ely , the follo wing sp ectral systems for the new eigenfunction χ : χ t 1 m − ˜ B (1 m ) χ x − λ [ ˜ B (1 m ) , χ ] − [ ˜ B (1 m ) w (1) ] χ = 0 , (45) χ t 1 − ˜ B (1) χ x − λ [ ˜ B (1) , χ ] − [ ˜ B (1) w (1) ] χ, (46) χ t 2 − ˜ B (2) χ xx + λ 2 [ χ, ˜ B (2) ] − 2 λ ˜ B (2) χ x + [ w (1) , ˜ B (2) ]( χ x + λχ ) − sχ = 0 , (47) s = [ ˜ B (2) , w (2) ] + 2 ˜ B (2) w (1) x + [ ˜ B (2) , w (1) ] w (1) . χ t 2 − β  2 λχ x + χ xx + 2 χw x  = 0 , (48) χ t 3 + 3 λχ xx + 3 λ 2 χ x + χ xxx + 3 λχw x + 3 2 χ  w t 2 β + w xx  = 0 . It is no w easy to v erify that the co efficien ts of the λ - large expansion of the eigenfunction χ satisfy the infinite chains (16- 18): χ ( λ ; ~ x ) = I − X n ≥ 1 w ( n ) ( ~ x ) λ i . (49) Therefore we ha ve obtained the sp ect ral in terpretation of such c hains. In addition, since the infinite c hains (16-18) for the w ( n ) ’s are equiv a len t, via the F rob enius structure (13), to C - integrable systems, w e hav e a lso shown how to go bac kw ar d, from S - to C - in tegrability . 2.5 Construction of solutions and Sato theory In order to construct solutions of the S- in tegrable PDEs generated in § 2.2 from the eleme ntary solution sche me (11),(12) of the matrix Burgers hierarc h y , w e consider the matrices w and Ψ to b e finite matrices consisting of n 0 × n 0 blo c ks (this can b e done assuming that Ψ (1( n 0 +1)) = 0), where n 0 is an a rbitrary p ositive integer greater tha n the num b er of blo cks w ( j ) ′ s in v olve d in the S-in tegrable PDE under consideration. T aking into a ccoun t the structures o f w and Ψ giv en b y eqs.(13) and (14) respective ly , w e hav e that w =         w (1) w (2) · · · · · · w ( n 0 ) I M 0 M · · · · · · 0 M 0 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 M I M 0 M         , Ψ =      Ψ (11) Ψ (12) · · · Ψ (1 n 0 ) ∂ − 1 x Ψ (11) ∂ − 1 x Ψ (12) · · · ∂ − 1 x Ψ (1 n 0 ) . . . . . . . . . . . . ∂ − n 0 +1 x Ψ (11) ∂ − n 0 +1 x Ψ (12) · · · ∂ − n 0 +1 x Ψ (1 n 0 )      . (50) 10 W e remark that the n 0 blo c ks Ψ (1 j ) , j = 1 , . . . , n 0 of Ψ are defined, via (11), b y equations Ψ (1 j ) ( ~ x ) = Z e k x + P i,m ≥ 1 ˜ B ( im ) t im k i ˆ Ψ (1 j ) ( k ) d Ω( k ) , j = 1 , . . . , n 0 (51) in terms of t he arbitrar y sp ectral functions ˆ Ψ (1 j ) , while the remaining blo c ks are constructed through the equations Ψ ( ij ) = ∂ i − 1 x Ψ (1 j ) . Then, via (12), the compo nen ts of the M × M blo c ks w ( i ) are express ed in terms of Ψ through the compact formula w ( i ) αβ = (Ψ x Ψ − 1 ) α ( iM − M + β ) , α , β = 1 . . . , M , i = 1 , . . . , n 0 . (52) This form ula is intimately connected to those obtained via Sato theory . T o see it, we consider the simplest case of scalar blo ck s ( M = 1), containing the example of the KP equation. Then equation (52) b ecomes w ( i ) = △ ( i ) △ , (53) where △ =          ∂ n 0 − 1 x f (1) ∂ n 0 − 1 x f (2) · · · ∂ n 0 − 1 x f ( n 0 ) ∂ n 0 − 2 x f (1) ∂ n 0 − 2 x f (2) · · · ∂ n 0 − 2 x f ( n 0 ) . . . . . . . . . . . . f (1) f (2) · · · f ( n 0 )          , (54) △ ( i ) =                ∂ n 0 − 1 x f (1) ∂ n 0 − 1 x f (2) · · · · · · · · · · · · ∂ n 0 − 1 x f ( n 0 ) . . . . . . . . . . . . . . . . . . . . . ∂ n 0 − i x f (1) ∂ n 0 − i x f (2) · · · · · · · · · · · · ∂ n 0 − i x f ( n 0 ) ∂ n 0 x f (1) ∂ n 0 x f (2) · · · · · · · · · · · · ∂ n 0 x f ( n 0 ) ∂ n 0 − i − 2 x f (1) ∂ n 0 − i − 2 x f (2) · · · · · · · · · · · · ∂ n 0 − i − 2 x f ( n 0 ) . . . . . . . . . . . . . . . . . . . . . f (1) f (2) · · · · · · · · · · · · f ( n 0 )                , (55) where f ( j ) = ∂ − ( n 0 − 1) x Ψ (1 j ) , j = 1 , . . . , n 0 , equiv alen t to the formula obtained using Sato theory [41]. 3 Relation b et ween C h - and S -in tegrability F ollowing the same strategy illustrated in § 2, in this sec tion we establish the deep relatio ns b et w een the matrix PDE (1), recen tly in tro duced in [2] and in tegrated there b y the method of c ha racteristics, a nd i) (1+1)-dimensional S -in tegrable soliton equations lik e the KdV and NLS equations; ii) the GL ( M , C ) - SDYM equation a nd its multidimen sional generalizations; iii) the multidime nsional Calogero systems [42, 43 , 44, 45, 46]. In this section, matrix w mus t b e diagonalizable. 3.1 Matrix equations in tegrable b y the metho d of c haracteristics Consider the f ollo wing matrix eigen v alue problem w ( ~ x )Ψ(Λ ; ~ x ) = Ψ(Λ; ~ x )Λ( ~ x ) , (56) 11 for the matrix w ( ~ x ), where Λ( ~ x ) is the diagonal matrix of eigen v alues, Ψ(Λ; ~ x ) is a suitably normalized matrix of eigen vec tors, and asso ciate with it the follo wing flo ws for Ψ(Λ; ~ x ): Ψ t mk + P N j =1 Ψ x j k ρ ( mj k ) (Λ) − B ( mk ) Ψ σ ( mk ) (Λ) = 0 , m ∈ N + , k = 1 , 2 , (57) where B ( mk ) are constan t commuting matrices as in § 2 and ~ x is the v ector o f all indep enden t v ariables: ~ x = ( x j k , t nm , j, n, m ∈ N + , k = 1 , 2 ). The compatibilit y b etw een the flows (5 7) implies the f ollo wing comm uting quasilinear PDEs for the eigen v alues: Λ t mk + P N j =1 Λ x j k ρ ( mj k ) (Λ) = 0 , m ∈ N + , k = 1 , 2; (58) the additional compatibilit y with the eigenv a lue problem (56) implies the following nonlinear PDEs: w t mk + N P i =1 w x ik ρ ( mik ) ( w ) + [ w , B ( mk ) ] σ ( mk ) ( w ) = 0 , m ∈ N + , k = 1 , 2 , (59) comm uting replicas of equation (1). The wa y of solving equations (59 ) is a s follows [2]. Consider the g eneral solution of equations (58) with k = 1 , 2, c haracterized b y the follow ing non-differen tial equations: Λ = E x 11 I − X m ≥ 1 ρ ( m 11) (Λ) t m 1 , . . . , x N 1 I − X m ≥ 1 ρ ( mN 1) (Λ) t m 1 ; (60) x 12 I − X m ≥ 1 ρ ( m 12) (Λ) t m 2 , . . . , x N 2 I − X m ≥ 1 ρ ( mN 2) (Λ) t m 2 ! , where E is an arbitr a ry diagonal matrix function of 2 N argumen ts; then t he general solution of the linear mat r ix PDEs (57) for Ψ, with the conv enien t parametrization Ψ ii = 1, is g iv en by Ψ αβ = F αβ x 11 − X m ≥ 1 ρ ( m 11) (Λ β ) t m 1 , . . . , x N 1 − X m ≥ 1 ρ ( mN 1) (Λ β ) t m 1 ; (61) x 12 I − X m ≥ 1 ρ ( m 12) (Λ β ) t m 2 , . . . , x N 2 − X m ≥ 1 ρ ( mN 2) (Λ β ) t m 2 ! × e 2 P k =1 P m ≥ 1 B ( mk ) α σ ( mk ) (Λ β ) t mk  , α , β = 1 , 2 , . . . where F αβ are arbitrary scalar functions of 2 N argumen ts, with F αα = 1. Then w = ΨΛΨ − 1 (62) solv es the nonlinear PDEs (59). No w w e pro ceed as in § 2, assuming for w the same blo c k F rob enius matrix structure (1 3), consisten t with equations (5 6) and (57) (and then with the hierarchies (59)) iff the matrices B ( mk ) are giv en as in (15), the diagonal matrix of eigen v alues Λ( ~ x ) has the blo c k-structure Λ( ~ x ) = dia g [Λ (1) ( ~ x ) , Λ (2) ( ~ x ) , · · · ] , (63) 12 and Ψ =        Ψ (11) Ψ (22) Λ (2) Ψ (33) Λ (3) 2 · · · Ψ (11) (Λ (1) ) − 1 Ψ (22) Ψ (33) Λ (3) · · · Ψ (11) (Λ (1) ) − 2 Ψ (22) (Λ (2) ) − 1 Ψ (33) · · · Ψ (11) (Λ (1) ) − 3 Ψ (22) (Λ (2) ) − 2 Ψ (33) (Λ (3) ) − 1 · · · . . . . . . . . . . . .        (64) Using the same strategy as in § 2 , we show that, 1. if ρ ( mj k ) = 0, k = 1 , 2, implying Λ = const , equations (59) generate classical (1+1)- dimensional S -in tegrable PDEs lik e the N -w av e, NLS, KdV and mKdV equations; 2. if B ( mk ) = 0 (or σ ( mk ) = 0), k = 1 , 2 , equations (59) generate the GL ( M , C )- SD YM equation and its (2 N + 2) - dimensional generalization; 3. if B ( m 1) = 0 (or σ ( m 1) = 0) a nd ρ ( mi 2) = 0, equations (59) generate Calogero systems. 3.2 De riv ation of (1+1)-dimensional S -int egrable PDEs Let ρ ( mj k ) = 0 in (57-59), implying Λ = const (isosp ectral flo ws), and let σ ( mk ) (Λ) = Λ m , i.e.: w Ψ = ΨΛ , (65) Ψ t mk − B ( mk ) ΨΛ m = 0 , m, k ∈ N + . (66) The compatibility conditio ns for the system (6 5-66) yields, fo r m ∈ N + : w t mk + [ w , B ( mk ) ] w m = 0 . (67) W e remark that these equations a re equiv alent to eqs.(6) with ∂ j x w = 0, ∀ j . Consequen tly , the discrete c hains generated b y the eq.(67), with m = 1, k = 1 , 2 and m = 2 , 3, k = 1, a re given b y the eqs.(16-18) with ∂ j x w = 0, ∀ j : w ( n ) t 1 k + [ w ( n +1) , ˜ B (1 k ) ] + [ w (1) , ˜ B (1 k ) ] w ( n ) = 0 , k = 1 , 2 , (68) w ( n ) t 2 + [ w (1) , ˜ B (2) ]( w (1) w ( n ) + w ( n +1) ) + [ w (2) , ˜ B (2) ] w ( n ) + [ w ( n +2) , ˜ B (2) ] = 0 , (69) w ( n ) t 3 + [ w (1) , ˜ B (3) ]  w (1) ( w (1) w ( n ) + w ( n +1) ) + w (2) w ( n ) + w ( n +2)  + (70) [ w (2) , ˜ B (3) ]  w (1) w ( n ) + w ( n +1)  + [ w (3) , ˜ B (3) ] w ( n ) + [ w ( n +3) , ˜ B (3) ] = 0 , for n ∈ N + , where, in equations ( 69),(70), we hav e used the simplifying notation (21). (1+1)-dimensional N -wa ve equation. Setting n = 1 in eqs . ( 6 8), and eliminating w (2) , one obtains the well-kno wn S - in tegrable N - w a v e system in (1 +1)-dimensions: [ w (1) t 11 , ˜ B (12) ] − [ w (1) t 12 , ˜ B (11) ] + [[ w (1) , ˜ B (11) ] , [ w (1) , ˜ B (12) ]] = 0 . (71) 13 NLS. Eq.(68) with k = 1, n = 1 , 2 and eq.(69) with n = 1 are a complete sys tem of PDEs for w ( j ) , j = 1 , 2 , 3: w (1) t 1 + [ w (2) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (1) = 0 , (72) w (2) t 1 + [ w (3) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (2) = 0 , w (1) t 2 + [ w (1) , ˜ B (2) ]( w (1) w (1) + w (2) ) + [ w (2) , ˜ B (2) ] w (1) + [ w (3) , ˜ B (2) ] = 0 In the case ˜ B (2) = α ˜ B (1) ( α is a scalar constan t) this system results in the fo llo wing equation for w (1) : [ w (1) t 2 , ˜ B (1) ] + w (1) t 1 t 1 − α [ w (1) t 1 w (1) , ˜ B (1) ] + α  [ w (1) , ˜ B (1) ] w (1)  t 1 = 0 (73) If, in addition, M = 2 , ˜ B (1) = diag(1 , − 1) , (74) this equation yields the celebrated NLS system: 1 α q t 2 − 1 2 q τ 1 τ 1 − 4 r q 2 = 0 , (75) 1 α r t 2 + 1 2 r τ 1 τ 1 + 4 q r 2 = 0 for the off-diago nal elemen ts of w (1) : q = w (1) 12 , r = w (1) 21 . The NLS equation iq t 2 + 1 2 q τ 1 τ 1 + 4 q 2 ¯ q = 0 corresp onds to the r eduction r = ¯ q , α = i . KdV and mKdV. Eq.(68) with k = 1, n = 1 , 2 , 3, and eq.(70) with n = 1 yield a complete system of PDEs for w ( j ) , j = 1 , 2 , 3 , 4: w (1) t 1 + [ w (2) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (1) = 0 , (76) w (2) t 1 + [ w (3) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (2) = 0 , w (3) t 1 + [ w (4) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (3) = 0 , w (1) t 3 + [ w (1) , ˜ B (3) ]  w (1) ( w (1) w (1) + w (2) ) + w (2) w (1) + w (3)  + [ w (2) , ˜ B (3) ]  w (1) w (1) + w (2)  + [ w (3) , ˜ B (3) ] w (1) + [ w (4) , ˜ B (3) ] = 0 . In the case ˜ B (3) = − ˜ B (1) , this system r educes to the t w o coupled matrix equations w (1) t 1 + [ w (2) , ˜ B (1) ] + [ w (1) , ˜ B (1) ] w (1) = 0 , − [ ˜ B (1) , w (1) t 1 ] = w (2) t 1 t 1 + [ ˜ B (1) , w (1) t 1 ( w (1) w (1) + w (2) ) + w (2) t 1 w (1) ] −  [ ˜ B (1) , w (1) ] w (2)  t 1 . (77) If, in addition, the choice (7 4) is made, the system (77) b ecomes the mKdV system: q t 3 + 1 4 q t 1 t 1 t 1 + 6 q t 1 q r = 0 , (78) r t 3 + 1 4 r t 1 t 1 t 1 + 6 r t 1 r q = 0 , where q = w (1) 12 , r = w (1) 21 , reducing to the K dV equation q t 3 + 1 4 q t 1 t 1 t 1 + 6 q q t 1 = 0 a nd to the mKdV equation q t 3 + 1 4 q t 1 t 1 t 1 + 6 q 2 q t 1 = 0 if r = 1 and r = q resp ectiv ely . 14 3.2.1 Lax pairs for S -in t egrable P DEs in (1+1)-dimensions As in § 2, in order to deriv e the Lax pairs for t he ab o v e S - in tegrable PDEs in (1+1 )-dimensions, it is conv enien t to write the system (65 ,66) in t he equiv alent fo r m w Ψ = ΨΛ , (79) Ψ t mk − B ( mk ) w m Ψ = 0 , m ∈ N + , (80) and consider the dua l system ˜ Ψ w = Λ ˜ Ψ , (81) ˜ Ψ t mk + ˜ Ψ B ( mk ) w ( m ) = 0 , m ∈ N + . (82) T aking in to account the blo ck structure of the matr ix w giv en b y eq.(13) and considering the first ro ws of eqs.(81,82), w e obtain the f o llo wing sp ectral c hains, for n ∈ N + : ˜ Ψ (1 n ) E = ˜ Ψ (11) w ( n ) + ˜ Ψ (1( n +1)) , (83) ˜ Ψ (1 n ) t 1 k + ˜ Ψ (11) ˜ B (1 k ) w ( n ) + ˜ Ψ (1( n +1)) ˜ B (1 k ) = 0 , (84) ˜ Ψ (1 n ) t 2 k + ˜ Ψ (11) ˜ B (2 k ) ( w (1) w ( n ) + w ( n +1) ) + ˜ Ψ (12) ˜ B (2 k ) w ( n ) + ˜ Ψ (1( n +2)) ˜ B (2 k ) = 0 , (85) ˜ Ψ (1 n ) t 3 k + ˜ Ψ (11) ˜ B (3 k )  w (1) ( w (1) w ( n ) + w ( n +1) ) + w (2) w ( n ) + w ( n +2)  (86) + ˜ Ψ (12) ˜ B (3 k )  w (1) w ( n ) + w ( n +1)  + ˜ Ψ (13) ˜ B (3 k ) w ( n ) + ˜ Ψ (1( n +3)) ˜ B (3 k ) = 0 . where E = Λ (1) . Setting n = 1 into eqs.(83,84) a nd eliminating ˜ Ψ (12) , one gets the dual o f the Lax pair for the (1+ 1)-dimensional N - w a v e equation (71) ( ˜ ψ = ˜ Ψ (11) ): ˜ ψ t 1 k + E ˜ ψ ˜ B (1 k ) + ˜ ψ [ ˜ B (1 k ) , w (1) ] = 0 , k = 1 , 2 . (87) Eq.(87) with k = 1, written in terms of (21), is the first equation of the dual Lax pair also fo r eqs.(72) and (76). Setting k = n = 1 in to eq.(85) and eliminating ˜ Ψ (12) , ˜ Ψ (13) using eq.(83 ) , one gets the second equation of the dual L a x pair for (72): ˜ ψ t 2 + E 2 ˜ ψ ˜ B (2) + E ˜ ψ [ ˜ B (2) , w (1) ] + ˜ ψ s = 0 , (88) s = [ ˜ B (2) , w (2) ] + [ ˜ B (2) , w (1) ] w (1) . The second equation of the dual Lax pair for the eq.(76) results from the eq.(86), k = n = 1 after eliminating ˜ Ψ (1 j ) , j = 2 , 3 , 4 using eq.(83). In view of conditions (74 ) the complete dual sp ectral system reads: ˜ ψ t 1 + E ˜ ψ  1 0 0 − 1  + ˜ ψ  0 2 q − 2 r 0  = 0 , (89) ˜ ψ t 3 − E 3 ˜ ψ  1 0 0 − 1  − E 2 ˜ ψ  0 2 q − 2 r 0  + E ˜ ψ  − 2 q r − q τ 1 − r τ 1 2 q r  − ˜ ψ  r q τ 1 − q r τ 1 1 2 q τ 1 τ 1 + 4 q 2 r − 1 2 r τ 1 τ 1 − 4 r 2 q q r τ 1 − r q τ 1  = 0 15 The duals of the equations (87-89) read: ψ t 1 k − ˜ B (1 k ) ψ E − [ ˜ B (1 k ) , w (1) ] ψ = 0 , k = 1 , 2 , (90) ψ t 2 − ˜ B (2) ψ E 2 − [ ˜ B (2) , w (1) ] ψ E − sψ = 0 , (91) ψ t 1 +  1 0 0 − 1  ψ E −  0 2 q − 2 r 0  ψ = 0 , (92) ψ t 3 −  1 0 0 − 1  ψ E 3 +  0 2 q − 2 r 0  ψ E 2 +  − 2 q r − q τ 1 − r τ 1 2 q r  ψ E +  r q τ 1 − q r τ 1 1 2 q τ 1 τ 1 + 4 q 2 r − 1 2 r τ 1 τ 1 − 4 r 2 q q r τ 1 − r q τ 1  ψ = 0 Eqs. (90) a re the Lax pair of the N -wa ve eq.(71), eq. (90) with k = 1 and eq.(91) are the Lax pair of eq.(73), reducing to the NLS system if (7 4) ho lds, a nd eqs.(92) are Lax pair of the system (78) reducing to either KdV or mKdV. 3.2.2 F rom the Lax pairs of ( 1+1)-dimensional S -in t egr able PDEs to the Ch- in tegrable eqs.(67) W e sho w that the eqs.(67 ) ma y b e deriv ed from the sp ectral problems obtained in § 3.2.1. Let ψ (Λ; x ) = χ (Λ ; x ) e P i,k ≥ 1 ( − 1) i ˜ B ( ik ) t ik Λ i . (93) Then equation (90-92) yield: χ t 1 k + Λ[ ˜ B (1 k ) , χ ] − [ ˜ B (1 k ) , w (1) ] χ = 0 , k = 1 , 2 , (94) χ t 2 − Λ 2 [ ˜ B (2) , χ ] − [ w (1) , ˜ B (2) ]Λ χ − sχ = 0 , (95) s = [ ˜ B (2) , w (2) ] + [ ˜ B (2) , w (1) ] w (1) . χ t 1 +  1 0 0 − 1  , χ  Λ −  0 2 q − 2 r 0  χ = 0 , (96) χ t 3 −  1 0 0 − 1  , χ  Λ 3 +  0 2 q − 2 r 0  χ Λ 2 + (97)  − 2 q r − q τ 1 − r τ 1 2 q r  χ Λ +  r q τ 1 − q r τ 1 1 2 q τ 1 τ 1 + 4 q 2 r − 1 2 r τ 1 τ 1 − 4 r 2 q q r τ 1 − r q τ 1  χ It is now easy to v erify , as in § 2, that the co efficien ts of the Λ - large expansion of the eigen- function χ satisfy the infinite c hains (6 8 ), (6 8 ), (6 9), (7 0): χ (Λ; y ) → I − X j ≥ 1 ( − 1) j w ( j ) Λ − j , (9 8) clarifying their sp ectral meaning. In addition, r ecompiling suc h co efficien ts in to the blo c k F rob e- nius matrix, w e reconstruct the matrix equations (67) from the Lax pairs of (1+1)-dimensional S -in tegrable PDEs. 16 3.3 Deriv ation of the SD YM equation and of its m ulti-dimensional generalizatio ns No w we tak e, in equations (56),(57), B ( nk ) = 0 ( o r σ ( kj ) = 0), m = 1, ρ (1 j k ) (Λ) = Λ j , t 11 = t , t 12 = τ , x j 1 = x j , x j 2 = y j , obtaining t he system w Ψ = ΨΛ , (99) Ψ t + N X j =1 Ψ x j Λ j = 0 , Ψ τ + N X j =1 Ψ y j Λ j = 0 , whose compatibilit y condition yields w t + N X j =1 w x j w j = 0 , (100) w τ + N X j =1 w y j w j = 0 . W e pro ceed as in the previous sections but, b efore considering the deriv ation in the general case, quite complicated, w e illustrate the simplest tw o examples. 3.3.1 N = 1 : t he GL ( M , C ) − S D Y M equation The compatibilit y condition of the system (99) yields w t + w x 1 w = 0 , w τ + w y 1 w = 0 . (101) Let w and Ψ b e giv en by eqs.(13) and (64) resp ective ly . The first row s of the matrix equations (101) generate the chains, fo r n ∈ N + [37]: w ( n ) t + w (1) x 1 w ( n ) + w ( n +1) x 1 = 0 , (102) w ( n ) τ + w (1) y 1 w ( n ) + w ( n +1) y 1 = 0 . Setting n = 1 and eliminating w (2) , w e derive the w ell-know n GL ( M , C ) − S D Y M equation: w (1) ty 1 − w (1) τ x 1 + [ w (1) x 1 , w (1) y 1 ] = 0 . (103) T o derive the Lax pair of (103), w e write first the dual of system (99) in the conv enien t form: ˜ Ψ w = Λ ˜ Ψ , (104) ˜ Ψ t + ˜ Ψ x 1 w = 0 , ˜ Ψ τ + ˜ Ψ y 1 w = 0 . Using again (13) and (64), the first ro ws of equations (104) app ear in the form: ˜ Ψ (11) w ( n ) + ˜ Ψ (1( n +1)) = E ˜ Ψ (1 n ) , (105) ˜ Ψ (1 n ) t + ˜ Ψ (11) x 1 w ( n ) + ˜ Ψ (1( n +1)) x 1 = 0 , ˜ Ψ (1 n ) τ + ˜ Ψ (11) y 1 w ( n ) + ˜ Ψ (1( n +1)) y 1 = 0 , 17 where E = Λ (1) . Setting n = 1 in eqs.(105) and eliminating ˜ Ψ (12) , one obtains the dual of the Lax pair of ( 1 03) for the sp ectral function ˜ ψ = ˜ Ψ (11) : ˜ ψ t + ( E ˜ ψ ) x 1 = ˜ ψ w (1) x 1 , (106 ) ˜ ψ τ + ( E ˜ ψ ) y 1 = ˜ ψ w (1) y 1 . Then the Lax pair of (103) reads: ψ t + ψ x 1 E + w (1) x 1 ψ = 0 , (10 7) ψ τ + ψ y 1 E + w (1) y 1 ψ = 0 . Vice-v ersa, it is easy to v erify t ha t the co efficien ts of the E larg e expansion of the eigen- function ψ in (107) are the elemen ts of the chain (102 ) : ψ ( E ; ~ x ) → I − X j ≥ 1 w ( j ) E − j , (108) obtaining the sp ectral meaning of such chains. As a conseq uence, one reconstructs eqs.(101) from the Lax pair (10 7) of the SD YM equation. 3.3.2 N = 2 : a generalization of the S D Y M equation in 6 dimensions The compatibilit y condition of the system (99) yields now w t + w x 1 w + w x 2 w 2 = 0 , w τ + w y 1 w + w y 2 w 2 = 0 . (109) The corresp onding c hains read: w ( n ) t + w (1) x 1 w ( n ) + w ( n +1) x 1 + w (1) x 2 ( w (1) w ( n ) + w ( n +1) ) + w (2) x 2 w ( n ) + w ( n +2) x 2 = 0 , (110) w ( n ) τ + w (1) y 1 w ( n ) + w ( n +1) y 1 + w (1) y 2 ( w (1) w ( n ) + w ( n +1) ) + w (2) y 2 w ( n ) + w ( n +2) y 2 = 0 . Setting n = 1 and n = 2 in (110) a nd eliminating the fields w (3) , w (4) , one obtains the following in tegrable system of t w o nonlinear PDEs in 6 dimensions for the fields w (1) , w (2) : w (1) x 2 τ − w (1) y 2 t + w (2) x 2 y 1 − w (2) x 1 y 2 + w (1) x 2 y 1 w (1) − w (1) x 1 y 2 w (1) + w (1) y 1 w (1) x 2 − w (1) x 1 w (1) y 2 + [ w (2) y 2 , w (1) x 2 ] +[ w (1) y 2 , w (2) x 2 ] + w (1) y 2  w (1) 2  x 2 − w (1) x 2  w (1) 2  y 2 = 0 , w (1) x 1 τ − w (1) y 1 t + [ w (1) y 1 , w (1) x 1 ] + w (2) x 2 τ − w (2) y 2 t + [ w (2) y 2 , w (1) x 1 ] + [ w (1) y 1 , w (2) x 2 ] + [ w (1) x 2 , w (1) y 2 ] w (1) 2 +  w (1) x 1 y 2 w (1) − w (1) x 2 y 1 w (1) + w (2) x 1 y 2 − w (2) x 2 y 1  w (1) + w (1) x 2  w (1) τ − w (1) w (1) y 1 + [ w (2) y 2 , w (1) ]  − w (1) y 2  w (1) t − w (1) w (1) x 1 + [ w (2) x 2 , w (1) ]  + [ w (2) y 2 , w (2) x 2 ] = 0 , (111) reducing to (103) for w (1) if the fields do not dep end on x 2 , y 2 . T o deriv e the Lax pa ir o f (1 11), w e consider again the dual of syste m (99): ˜ Ψ w = Λ ˜ Ψ , (112) ˜ Ψ t + ˜ Ψ x 1 w + ˜ Ψ x 2 w 2 = 0 , ˜ Ψ τ + ˜ Ψ y 1 w + ˜ Ψ y 2 w 2 = 0 . 18 Using (13) a nd (64 ), the first row s of equations (112) app ear in the form: ˜ Ψ (11) w ( n ) + ˜ Ψ (1( n +1)) = E ˜ Ψ (1 n ) , (113) ˜ Ψ (1 n ) t + ˜ Ψ (11) x 1 w ( n ) + ˜ Ψ (1( n +1)) x 1 + ˜ Ψ (11) x 2 ( w (1) w ( n ) + w ( n +1) ) + ˜ Ψ (12) x 2 w ( n ) + ˜ Ψ (1( n +2)) x 2 = 0 , (114) ˜ Ψ (1 n ) τ + ˜ Ψ (11) y 1 w ( n ) + ˜ Ψ (1( n +1)) y 1 + ˜ Ψ (11) y 2 ( w (1) w ( n ) + w ( n +1) ) + ˜ Ψ (12) y 2 w ( n ) + ˜ Ψ (1( n +2)) y 2 = 0 . (115) Setting n = 1 in equations (114,115) , and eliminating ˜ Ψ (12) , ˜ Ψ (13) using eqs.(113 ) fo r n = 1 , 2, one obtains the dual of the Lax pair o f (1 11) for the sp ectral function ˜ ψ = ˜ Ψ (11) : ˜ ψ t + 2 X j =1 ( E j ˜ ψ ) x j = ˜ ψ  w (1) x 1 + w (2) x 2 + w (1) x 2 w (1)  + E ˜ ψ w (1) x 2 , (116) ˜ ψ t + 2 X j =1 ( E j ˜ ψ ) y j = ˜ ψ  w (1) y 1 + w (2) y 2 + w (1) y 2 w (1)  + E ˜ ψ w (1) y 2 . Then the Lax pair of (111) reads: ψ t + ψ x 1 E + ψ x 2 E 2 +  w (2) x 2 + w (1) x 2 w (1) + w (1) x 1  ψ + w (1) x 2 ψ E = 0 , (117) ψ τ + ψ y 1 E + ψ y 2 E 2 +  w (2) y 2 + w (1) y 2 w (1) + w (1) y 1  ψ + w (1) y 2 ψ E = 0 . As b efore, it is easy to verify that equation (10 8) holds, namely that the co efficien ts of the E large expansion of ψ in (11 7) are the elemen ts of the c hain (110). Therefore one reconstructs equations (1 09) from the Lax pair (111) of the six dimensional generalization (1 11) of the SD YM equation. 3.3.3 Multidimension al generalization of the S D Y M equation Motiv ated b y the ab ov e form ulae for the simplest cases N = 1 , 2, here w e discuss the general N situation. If w is the blo c k F rob enius matrix (13), the pow er w j exhibits the following structure w j =          ˜ w ( j ; 11) ˜ w ( j ; 12) ˜ w ( j ; 13) · · · · · · · · · · · · · · · ˜ w ( j ; j 1) ˜ w ( j ; j 2) ˜ w ( j ; j 3) · · · I M 0 M 0 M · · · 0 M I M 0 M · · · . . . . . . . . . . . .          , (118) where the ma t r ix blo c ks ˜ w are defined b y the equations ˜ w ( j ; 1 n ) = j − 1 X i =1 w ( i ) ˜ w ( j − i ; 1 n ) + w ( j + n − 1) , n, j ≥ 1 , ( ˜ w (1;1 n ) = w ( n ) ) , (119) ˜ w ( j ; k n ) = ˜ w ( j − k +1; 1 n ) , 2 ≤ k ≤ j, ( ˜ w ( j ; j n ) = w ( n ) ) (120) in terms of the basic blo ck s w ( j ) , j ≥ 1. The first few examples read: ˜ w (2;1 n ) = w (1) w ( n ) + w ( n +1) , (121) ˜ w (3;1 n ) = ( w (1) ) 2 w ( n ) + w (1) w ( n +1) + w (2) w ( n ) + w ( n +2) , ˜ w (4;1 n ) = w (1)  ( w (1) ) 2 w ( n ) + w (1) w ( n +1) + w (2) w ( n ) + w ( n +2)  + w (2)  w (1) w ( n ) + w ( n +1)  + w (3) w ( n ) + w ( n +3) . 19 F urthermore, ev alua t ing the (1 n )-blo c k of w j +1 , written as ( w j w ), w e o btain the additional form ula ˜ w ( j − 1;1( n +1)) = ˜ w ( j ; 1 n ) − ˜ w ( j − 1;11) w ( n ) , j ≥ 1 , n > 1 , (122) implying ˜ w ( j ; 1 n ) = ˜ w ( j + n − 1;11) − n − 1 X l =1 ˜ w ( j + l − 1;11) w ( n − l ) , j, n ≥ 1 . (123) Eq.(123) reduces, for j = 1, to the follow ing equation: ˜ w ( j ; 11) = j − 1 X i =1 ˜ w ( j − i ; 11) w ( i ) + w ( j ) , j ≥ 1 (1 24) useful later on (compare it with equation (119) for n = 1). Using equation (1 3 ), the system (100) generates the discrete chains w ( n ) t + N X j =1  j X i =1 w ( i ) x j ˜ w ( j − i + 1; 1 n ) + w ( j + n ) x j  = 0 , (125) w ( n ) τ + N X j =1  j X i =1 w ( i ) y j ˜ w ( j − i + 1+; in ) + w ( j + n ) y j  = 0 . Setting n = 1 , . . . , N in equations (125), one o btains a determined system of 2 N equations for the fields w ( i ) , i = 1 , . . . , 2 N . A s in the previous tw o illustrativ e examples for N = 1 , 2, it is p ossible to eliminate t he N fields w ( i ) , i = N + 1 , . . . , 2 N , obtaining a system on N equations in (2 N + 2) dimensions for the r emaining fields w ( i ) , i = 1 , . . . , N . Such system, whic h pr ovides the natural m ultidimensional generalization of the SDYM equation (1 03), is con v enien tly written as follo ws p ( N , 0) τ − q ( N , 0) t + [ q (0) , p (0) ] = 0 , p ( N ,n ) τ − q ( N ,n ) t + n P j =1  p ( N ,n − j ) y j − q ( N ,n − j ) x j  + n P j =0  q ( N ,j ) , p ( N ,n − j )  = 0 , 1 ≤ n ≤ N − 1 , (126) where the fields p ( N ,j ) , q ( N ,j ) are suitable com binations of the N fields w ( n ) , n = 1 , . . . , N : p ( N ,j ) = N P s = j +1  s − j − 1 P l =1 w ( l ) x s ˜ w ( s − j − l ;11) + w ( s − j ) x s  , q ( N ,j ) = N P s = j +1  s − j − 1 P l =1 w ( l ) y s ˜ w ( s − j − l ;11) + w ( s − j ) y s  ; (127) the first few r ead as follow s: p ( N ,N − 1) = w (1) x N , q ( N ,N − 1) = w (1) y N , p ( N ,N − 2) = w (1) x N − 1 + w (2) x N + w (1) x N w (1) , q ( N ,N − 2) = w (1) y N − 1 + w (2) y N + w (1) y N w (1) , p ( N ,N − 3) = w (1) x N − 2 + w (2) x N − 1 + w (3) x N +  w (1) x N − 1 + w (2) x N  w (1) + w (1) x N  w (1) 2 + w (2)  , q ( N ,N − 3) = w (1) y N − 2 + w (2) y N − 1 + w (3) y N +  w (1) y N − 1 + w (2) y N  w (1) + w (1) y N  w (1) 2 + w (2)  . (128) T o sho w it, it is more con v enien t to go through the La x pair deriv at ion. 20 Lax pair. The system dual of (99) reads ˜ Ψ w = Λ ˜ Ψ , (129) ˜ Ψ t + N X j =1 ˜ Ψ x j w j = 0 , ˜ Ψ τ + N X j =1 ˜ Ψ y j w j = 0 . and it is conv enien tly rewritten in the equiv a len t form ˜ Ψ w = Λ ˜ Ψ , (130) ˜ Ψ t + N X j =1 (Λ j ˜ Ψ) x j − N X j =1 ˜ Ψ( w j ) x j = ˜ Ψ t + N X j =1 (Λ j ˜ Ψ) x j − N − 1 X s =0 Λ s ˜ Ψ N X j = s +1 w x j w j − s − 1 = 0 , (131) ˜ Ψ τ + N X j =1 (Λ j ˜ Ψ) y j − N X j =1 ˜ Ψ( w j ) y j = ˜ Ψ t + N X j =1 (Λ j ˜ Ψ) y j − N − 1 X s =0 Λ s ˜ Ψ N X j = s +1 w y j w j − s − 1 = 0 . (132) As b efore, the blo c k (1 , 1 ) of the matrix equations (1 31),(132) leads to the dual of the Lax pa ir ( E = Λ (1) ) of t he multidime nsional SDYM equations: ˜ ψ t + N X j =1 ( E j ˜ ψ ) x j = N − 1 X j =0 E j ˜ ψ p ( N ,j ) , ( 1 33) ˜ ψ τ + N X j =1 ( E j ˜ ψ ) y j = N − 1 X j =0 E j ˜ ψ q ( N ,j ) , (134) where p ( N ,j ) and q ( N ,j ) are defined in terms of w ( i ) and their deriv ativ es in (127). Then one deriv es t he corresp onding Lax pair ψ t + N X j =1 ψ x j E j + N − 1 X j =0 p ( N ,j ) ψ E j = 0 , (135) ψ τ + N X j =1 ψ y j E j + N − 1 X j =0 q ( N ,j ) ψ E j = 0 , (136) together with its compatibility condition, the following determined system of 2 N equation in (2 N + 2) v ariables for the fields p ( N ,j ) , q ( N ,j ) , j = 0 , . . . , N − 1: p ( N , 0) τ − q ( N , 0) t + [ q (0) , p (0) ] = 0 , (137) p ( N ,n ) τ − q ( N ,n ) t + n X j =1  p ( N ,n − j ) y j − q ( N ,n − j ) x j  + n X j =0  q ( N ,j ) , p ( N ,n − j )  = 0 , 1 ≤ n ≤ N − 1 , (138) N X j = n − N +1  p ( N ,n − j ) y j − q ( N ,n − j ) x j  + N − 1 X j = n − N +1  q ( N ,j ) , p ( N ,n − j )  = 0 , N ≤ n ≤ 2 N − 2 , (139) p ( N ,N − 1) y N − q ( N ,N − 1) x N = 0 . (140) W e remark that only the first N equations (137),(138) in v olv e deriv ativ es with resp ect to the time v ariables t, τ ; the remaining N equations (13 9),(140), providing a set of relations among 21 the 2 N fields p ( N ,j ) , q ( N ,j ) , are remark a bly parametrized b y equations (127) in terms o f t he N fields w ( j ) , j = 1 , . . . , N . Therefore one is left with equations (12 6),(127). W e also remark that the generalization (137-140) of t he SDYM equation is known in the literature [18, 48], to b e generated b y the Lax pair ψ t + N X j =1 λ j ψ x j + N − 1 X j =0 λ j p ( N ,j ) ψ = 0 , (141) ψ τ + N X j =1 λ j ψ y j + N − 1 X j =0 λ j q ( N ,j ) ψ = 0 , (142) differing from (13 5,136 ) b y the fact that here λ is j ust a constan t and scalar spect ral parameter . Therefore the remark able deriv ation of (13 7-1 4 0) from the mat r ix equations (100) and its in tegration sc heme has allo w ed one to unco ver the follow ing t w o impo rtan t prop erties of the system (137-140). • Half of the equations of the sys tem (13 7-1 4 0) (the non ev olutionary part ) can be parametrized in terms of the blo c ks w ( j ) , j = 1 , .., N of the F rob enius matrix w , reducing b y half the n um b er of equations. • Equations (137-1 4 0) t urn o ut t o b e asso ciated with the no v el La x pair (135,136), in whic h the diagonal ma t r ix E satisfies, from (58), the in tegrable quasi-linear equations E t + N P j =1 E x j E j = 0 , E τ + N P j =1 E y j E j = 0 . (143) Therefore, as it w as alr eady observ ed in [37] in the case of the SDYM equation (103), the in tegration sche me associated with suc h a nov el Lax pair makes clear the ex istence of a ric h solution space exhibiting in teresting phenomena of m ultidimensional w av e breaking. A detailed study of these solutions is po stp oned t o a subsequen t pap er, together with the comparison with t he finite gap solutions of the SDYM equation constructed in [49], and asso ciat ed with a Riemann surface with branch p oints satisfying equations (143) for N = 1. F rom the Lax pair of t he m ultidimensiona l SD YM to the in tegrable chains (125) As for the pa r t icular cases N = 1 , 2, in this section we sho w that the E -large limit of the La x pair (135,136) yields the expansion (108) for the eigenfunction ψ . Therefore the co efficien ts of the E -large expansion of the sp ectral function asso ciated with the S -integrable m ultidimensional generalization of the SDYM equations are solutions of the nonlinear c hains (125), pro viding the sp ectral meaning to suc h nonlinear chains . In addition, r ecompiling the matrices w ( j ) in to the blo c k F rob enius matrix w , via (13), one establishes a remark able relation b et w een the Lax pair of the m ultidimensional SDYM and the basic mat r ix equations (100), solv able b y the metho d of characteristics . T o sho w the v alidit y of the expansion (108), w e substitute it into the Lax pair (135,13 6), 22 obtaining the follo wing pairs of equations p ( N ,i ) = N − 1 P j = i +1  w ( j − i ) x j + p ( N ,j ) w ( j − i )  + w ( N − i ) x N , 0 ≤ i ≤ N − 1 , q ( N ,i ) = N − 1 P j = i +1  w ( j − i ) y j + q ( N ,j ) w ( j − i )  + w ( N − i ) y N , 0 ≤ i ≤ N − 1 , (144) w ( i ) t + N − 1 P j =1  w ( j + i ) x j + p ( N ,j ) w ( j + i )  + w ( N + i ) x N + p ( N ;0) w ( i ) = 0 , i ≥ 1 , w ( i ) τ + N − 1 P j =1  w ( j + i ) y j + q ( N ,j ) w ( j + i )  + w ( N + i ) y N + q ( N ;0) w ( i ) = 0 , i ≥ 1 , (145) corresp onding respectiv ely to the condition that the co efficien ts of the p ositiv e and negativ e p o w ers of E are zero in all o rders. Equations (144) are iden tically satisfied using the definitions (127) and (119) of p ( N ,j ) , q ( N ,j ) and ˜ w ( j ; 11) . T o sho w that also equations (1 45) are iden tically satisfied, w e compare them with equations (125) and use again (127), (119), to finally deriv e the equation N X k =1 k X l =1 w ( l ) x k k − l X j =1 w ( j ) ˜ w ( k − l − j +1;1 n ) − k − l − 1 X j =0 ˜ w ( k − l − j ;11) w ( j + n ) ! = 0 , n ≥ 1 (146) iden tically satisfied, due to (119) and (124). 3.4 Deriv ation of Calogero systems In the L a x pair of the S-integrable Calogero systems [42, 43, 44, 45, 46], the sp ectral problem is 1-dimensional ( like that of, sa y , KdV and NLS), while the equation describing the evolution of the eigenfunction is a m ultidimensional PD E in whic h a n arbitrary n um b er of additional indep enden t v ariables are graded by p ow ers of the sp ectral parameter; in addition, suc h sp ectral parameter satisfies a quasilinear PDE. It is therefore clear that Calogero systems com bine prop erties o f the S - in tegrable PDEs in (1+1)-dimensions of § 3.2 with prop erties of the m ultidimensional generalizations of the SDY M equation of § 3.3 and, to generate them from equations (59), w e hav e to star t with the matrix equation (56) and with ev olutionary system (57) in whic h ρ ( mj 1) = σ ( m 2) = 0: w Ψ = ΨΛ , (147) Ψ t m 1 − B ( m 1) Ψ σ ( m 1) (Λ) = 0 , m ∈ N + , Ψ t m 2 + P N j =1 Ψ x j 2 ρ ( mj 2) (Λ) = 0 . (148) Here w e illustrate t he construction of the simplest example of Calogero system, corresp onding to N = m = 1 and σ (11) (Λ) = ρ (112) (Λ) = Λ, B = B (11) , using the notation ˜ B = ˜ B (11) , τ = t 11 , t = t 12 , x = x 12 . Then the compatibility b etw een the tw o equations (14 8) yields the following quasi-linear PDEs Λ t + Λ x Λ = 0 , Λ τ = 0 (149) for the matrix of eigen v alues; t he compatibility b et w een equations (147) and (148) yields instead the follo wing matrix equations for field w w τ + [ w , B ] w = 0 , (150) w t + w x w = 0 . 23 Using the F rob enius structure (13) of w , w e get the followin g discrete c hains, for n ∈ N + : w ( n ) τ + [ w ( n +1) , ˜ B ] + [ w (1) , ˜ B ] w ( n ) = 0 , (151) w ( n ) t + w (1) x w ( n ) + w ( n +1) x = 0 , coinciding with eqs.(68) for k = 1, and with (102a). Fixing n = 1 in (151), and eliminating w (2) , one gets the following matrix PDE [ w (1) t , ˜ B ] + [ w (1) x w (1) , ˜ B ] = w (1) τ x +  [ w (1) , ˜ B ] w (1)  x . (152) Let M = 2 and ˜ B = β diag(1 , − 1); then eq.(152) yields − β q t − 1 2 q xτ − 4 β 2 q ∂ − 1 τ ( q r ) x = 0 , (153) β r t − 1 2 r xτ − 4 β 2 r ∂ − 1 τ ( q r ) x = 0 . If r = ¯ q , β = i , then the a b o v e system becomes the follow ing (2+1)-dimensional in tegrable v ariant of NLS: iq t + 1 2 q xx − 4 q ∂ − 1 τ ( q ¯ q ) x = 0 , (154) studied in [50, 51 ]. Using the dual of equations (82) with m = k = 1, and equations (104a,b), w e deriv e the dual Lax pair ((8 7 ) fo r k = 1 and (106a)) fo r (1 52): ˜ ψ τ + E ˜ ψ ˜ B + ˜ ψ [ ˜ B , w (1) ] = 0 , (155) ˜ ψ t + ( E ˜ ψ ) x = ˜ ψ w (1) x and the corresponding Lax pair ((90) f or k = 1 and (107a)): ψ τ − ˜ B ψ E − [ ˜ B , w (1) ] ψ = 0 , (156) ψ t + ψ x E + w (1) x ψ = 0 for the system (1 52). W e end this section remarking that in tegrable PDEs asso ciated with Lax pairs with v arying sp ectral parameter ha v e b een studied also elsewhere, see, for instance, [5 2]. 3.5 Construction of solutions The construction of solutions f or the three classes of S -in tegrable PDEs deriv ed in § 3.2, 3.3 and 3.4 from eq.(59), is based on the solution of the algebraic equations (60,61), taking in to accoun t the blo c k-matrix structure (64) of Ψ. Then t he indep enden t blo c ks Ψ ( ii ) , i = 1 , 2 , . . . are c haracterized b y the explicit form ula: Ψ ( ii ) αβ = F ( ii ) αβ x 11 − X m ≥ 1 ρ ( m 11) (Λ ( i ) β ) t m 1 , . . . , x N 1 − X m ≥ 1 ρ ( mN 1) (Λ ( i ) β ) t m 1 ; (157) x 12 − X m ≥ 1 ρ ( m 12) (Λ ( i ) β ) t m 2 , . . . , x N 2 − X m ≥ 1 ρ ( mN 2) (Λ ( i ) β ) t m 2 ! × e 2 P k =1 P m ≥ 1 ˜ B ( mk ) α σ ( mk ) (Λ ( i ) β ) t mk , α , β = 1 , . . . , M , i = 1 , 2 , . . . , 24 where F ( ii ) αβ are arbitrary scalar f unctions of 2 N arguments (suc h that F ( ii ) αα = 1), while the re- maining blo c ks are given by the equations Ψ ( ij ) = Ψ ( j j ) (Λ ( j ) ) j − i . Once Λ and Ψ are constructed in this wa y , the blo c ks w ( i ) are obtained, from eq.(62 ), through the compact for mula: w ( i ) αβ = (ΨΛΨ − 1 ) α ( iM − M + β ) , α , β = 1 . . . , M . (1 58) In the case of the (1+1)-dimensional S -in tegrable mo dels of § 3.2, corresp onding to ρ ( ij k ) = 0, Λ is an arbitra r y constant diagonal matrix and form ula (157) reduces to: Ψ ( ii ) αβ = F ( ii ) αβ e 2 P k =1 P m ≥ 1 ˜ B ( mk ) α σ ( mk ) (Λ ( i ) β ) t mk , α , β = 1 , . . . , M , i = 1 , 2 , . . . , (159) where F ( ii ) αβ are constant amplitudes. Then the solution (158) is a rational com bination of exp o nen tials. It w ould b e interesting t o compare, in this case, the solution space generated b y (158) with that generated by Sato theory . In the case of the Calogero systems of § 3.4, when ρ ( mj 1) = σ ( m 2) = 0, form ula (157) reduces to: Ψ ( ii ) αβ = F ( ii ) αβ x 12 − X m ≥ 1 ρ ( m 12) (Λ ( i ) β ) t m 2 , . . . , x N 2 − X m ≥ 1 ρ ( mN 2) (Λ ( i ) β ) t m 2 ! × (160) e P m ≥ 1 ˜ B ( m 1) α σ ( m 1) (Λ ( i ) β ) t m 1 , α, β = 1 , . . . , M , i = 1 , 2 , . . . . where F ( ii ) αβ are now arbitrar y functions of N argumen ts and Λ is the implicit solution of the nondifferen tial equation Λ = E x 12 I − X m ≥ 1 ρ ( m 12) (Λ) t m 2 , . . . , x N 2 I − X m ≥ 1 ρ ( mN 2) (Λ) t m 2 ! , (161) follo wing from the eq.(60), where E is arbitra r y diago na l matrix function of N arguments . 4 Generalizati o ns The blo c k F rob enius matrix (13) is not the only p ossible structure of w allo wing one to generate new types of integrable nonlinear PDEs, starting with C -in tegrable a nd C h -in tegrable equations (6) and ( 5 9). A more general represen t a tion is giv en by the follo wing blo ck matrix: w =    W (11) W (12) · · · W (21) W (22) · · · . . . . . . . . .    , (162) where eac h blo ck W ( ij ) has one of the following t w o blo c k matrix forms: W ij = S ( ij ) =      w ( ij ; 1) w ( ij ; 2) w ( ij ; 3) · · · I M 0 M 0 M · · · 0 M I M 0 M · · · . . . . . . . . . . . .      , (163) W ij = C ( ij ) =      w ( ij ; 1) w ( ij ; 2) w ( ij ; 3) · · · 0 M I M 0 M · · · 0 M 0 M I M · · · . . . . . . . . . . . .      , 25 and the blo c ks w ( ij ; k ) are M × M matrices. T o provide consistency o f t his structure with the L a x pairs (4,5) or (56,57), we must tak e appropriate structure of matrices B ( nm ) and Ψ. Consider the simplest example of eqs.(4,5) with n = 1. In this case w satisfies the N-w av e equations (8). Let w =  S (11) C (12) C (21) C (22)  . (164) Then B (1 m ) m ust b e tak en in the form B (1 m ) = diag( ˆ B ( m 1) , ˆ B ( m 2) ) , ˆ B ( mj ) = diag( ˜ B ( mj ) , ˜ B ( mj ) . . . ) , j = 1 , 2 , ( 1 65) where ˜ B ( mj ) are diag o nal matr ices, and Ψ m ust hav e the following blo ck structure: Ψ =  Ψ (11) Ψ (12) Ψ (21) Ψ (22)  , (166) where Ψ ( ij ) m ust satisfy the followin g system of linear PDEs (consequence of eq.(4)), for i ≥ 2: Ψ (11; ij ) x = Ψ (11;( i − 1) j ) + Ψ (21; ij ) , (167) Ψ (12; ij ) x = Ψ (12;( i − 1) j ) + Ψ (22; ij ) , (168) Ψ (21; ij ) x = Ψ (11; ij ) + Ψ (21; ij ) , (169) Ψ (22; ij ) x = Ψ (12; ij ) + Ψ (22; ij ) . (170) In view of (164), eq.(8) reads ( t i = t 1 i ): S (11) t i − ˆ B ( i 1) S (11) x + [ S (11) , ˆ B ( i 1) ] S (11) + ( C (12) ˆ B ( i 2) − ˆ B ( i 1) C (12) ) C (21) = 0 , (171) C (12) t i − ˆ B ( i 1) C (12) x + [ S (11) , ˆ B ( i 1) ] C (12) + ( C (12) ˆ B ( i 2) − ˆ B ( i 1) C (12) ) C (22) = 0 , C (21) t i − ˆ B ( i 2) C (21) x + ( C (21) ˆ B ( i 1) − ˆ B ( i 2) C (21) ) S (11) + [ C (22) , ˆ B ( i 2) ] C (21) = 0 , C (22) t i − ˆ B ( i 2) C (22) x + ( C (21) ˆ B ( i 1) − ˆ B ( i 2) C (21) ) C (12) + [ C (22) , ˆ B ( i 2) ] C (22) = 0 , where i = 1 , 2. W riting t he blo c k (11) of eac h of these equations, one o btains w (11;1) t i − ˜ B ( i 1) w (11;1) x + [ w (11;1) , ˜ B ( i 1) ] w (11;1) + [ w (11;2) , ˜ B ( i 1) ] + ( w (12;1) ˜ B ( i 2) − (172) ˜ B ( i 1) w (12;1) ) w (21;1) = 0 , w (12;1) t i − ˜ B ( i 1) w (12;1) x + [ w (11;1) , ˜ B ( i 1) ] w (12;1) + ( w (12;1) ˜ B ( i 2) − ˜ B ( i 1) w (12;1) ) w (22;1) = 0 , w (21;1) t i − ˜ B ( i 2) w (21;1) x + ( w (21;1) ˜ B ( i 1) − ˜ B ( i 2) w (21;1) ) w (11;1) + ( w (21;2) ˜ B ( i 1) − ˜ B ( i 2) w (21;2) ) + [ w (22;1) , ˜ B ( i 2) ] w (21;1) = 0 , w (22;1) t i − ˜ B ( i 2) w (22;1) x + ( w (21;1) ˜ B ( i 1) − ˜ B ( i 2) w (21;1) ) w (12;1) + [ w (22;1) , ˜ B ( i 2) ] w (22;1) = 0 . Eliminating w (11;2) and w (21;2) from t he eq s.(172a ,c) with i = 1 , 2 and t a king eqs.(172 b,d) with i = 2, one obtains t he following (2+1)- dimensional ev olutionary system of PDEs in the time v ariable t 2 : [ E (1) , ˜ B (21) ] − [ E (2) , ˜ B (11) ] = 0 , (1 73) q t 2 − ˜ B (21) q x + [ v , ˜ B (21) ] q + ( q ˜ B (22) − ˜ B (21) q ) u = 0 , (174) E (1) ˜ B (21) − ˜ B (22) E (1) − E (2) ˜ B (11) + ˜ B (12) E (2) = 0 , (175) u t 2 − ˜ B (22) u x + ( w ˜ B (21) − ˜ B (22) w ) q + [ u, ˜ B (22) ] u = 0 , (176) 26 where E ( i ) = v t i − ˜ B ( i 1) v x + [ v , ˜ B ( i 1) ] v + ( q ˜ B ( i 2) − ˜ B ( i 1) q ) w , i = 1 , 2 , E ( i ) = w t i − ˜ B ( i 2) w x + ( w ˜ B ( i 1) − ˜ B ( i 2) w ) v + [ u, ˜ B ( i 2) ] w , i = 1 , 2 , (177) and v = w (11;1) , q = w (12;1) , w = w (21;1) , u = w (22;1) , (178) supplemen ted by eqs.(172b,d) with i = 1: q t 1 − ˜ B (11) q x + [ v , ˜ B (11) ] q + ( q ˜ B (12) − ˜ B (11) q ) u = 0 , (179) u t 1 − ˜ B (12) u x + ( w ˜ B (11) − ˜ B (12) w ) q + [ u, ˜ B (12) ] u = 0 , (180) that can b e considered as compatible constraints f or the ev olutionary system (173)-(176). The Lax pair, as w ell as the solution space fo r this system, can b e obta ined following the pro cedures describ ed in § 2.3 and § 2.5. If w = q = u = 0, one obtains the S -in tegrable (2+1)-dimensional N -w av e equation (173) for v (see eq.(20)). Instead, if v = w = q = 0, o ne obtains the C -integrable (1+1)-dimensional N - w a v e type equation (176) or (180 ) fo r u (see eq.(8)). Therefore equations (1 73)-(176) and (179,180) can b e view ed as nonlinear PDEs sharing prop erties of S- and C- in tegrable systems. One can show that this prop erty is shared b y a ll the PDEs generated b y reductions o f the t yp e (162,163); reco v ering, in particular, the ( n +1)-dimensional ( n > 2) nonlinear PDEs constructed in [34] by a generalization of the dressing metho d. 5 Summary and future p ersp ectiv es W e hav e established deep and r emark able connections a mong PD Es in tegrable b y the inv erse sp ectral transform metho d, the metho d of characteristics and the Hopf-Cole transformation. These relations can be used effectiv ely to construct, for the generated S-in tegrable PDEs, the asso ciated compatible systems of linear operato rs, their comm uting flo ws and large classe s of solutions. These results o p en sev eral researc h p erspectiv es. 1. Use of the ab o v e deriv ation of the S - in tegrable system s to in v estigate the corresp onding space of analytic solutions generated from t he seed solutions of the original C - and Ch - in tegrable PDEs. In particular, (a) the connections b et w een suc h solution space and that generated b y Sato theory . In the KP case, the t w o solution spaces coincide; in other cases the connec tion is, at the momen t, less clear. (b) The use of the quasi-linear PDEs for the eigen v alues to study in detail the w av e breaking phenomena asso ciat ed with solutions of the SDY M equation and of its m ultidimensional generalizations. 2. Searc h for the integrable system s that should generate, through a suitable matrix reduc- tion, the in t egr a ble PDEs equiv a lent to the comm utation of vector fields, lik e equation (3), a class of S - in tegrable systems not fitting ye t in to the general picture illustrated in this pap er. 27 3. Generalization of the tec hniques presen t ed in this pap er to g enerate nov el in tegrable systems , p ossibly in m ultidimensions. In particular, a systematic use of group theory to ols to explore reductions different fr o m the blo c k F rob enius matrix one. 4. Construction of the discrete analog ue of the results of this pap er. In this resp ect, we remark that, while the discretization of the r esults o f § 2 do es not presen t, in principle, an y concep tual problem, and will b e the sub ject of a subsequen t pap er, the discretization of the results of § 3, if ρ ( ij k ) 6 = 0, is not clear, since a satisfactory discretization of the metho d o f c haracteristics and of equation (1) for ρ ( i ) 6 = 0, in the scalar and matrix cases, are, at the momen t, unkno wn. Ac kno wledgmen ts . AIZ w as supp orted by the R FBR gran ts 07-01 - 00446, 06-01- 92053, 06-01- 90840, b y the grant NS 7550.20 0 6.2 and by the INFN g ran t 200 7. AIZ t ha nks Prof. A.B.Shabat for useful discuss ion. References [1] J. B. Whitham, Line ar and Nonline ar Waves , Wiley , NY, 1974 [2] P . M. Santini and A. I. Zenc h uk, Ph ysics Letters A 368 (20 0 7) 48-52, arXiv:nlin.SI/0612036 [3] S.P .Tsarev, Soviet Math. 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