BKP and CKP revisited: The odd KP system

BKP and CKP r evisited: The odd KP system ∗ Aristophanes Dimakis Department of Financial and Management Engineering, Univ ersity of the Ae gean, 31 Fostini S tr ., GR-82100 Chios, Greece dimakis@aegean.gr Folker t M ¨ uller-Hoissen Max-Planck-Institute for Dynamics and Self-Or ganizatio n Bunsenstrasse 10, D-37073 G ¨ ottingen, Germany folkert.mueller-hoissen@ds.mpg.de Abstract Restricting a linear system for the KP hierarchy to those indepen dent variables t n with odd n , its compatibility (Zakharov-Sh abat condition s) leads to the “odd KP hierarchy”. The latter consists of pairs of equations for two depen dent v ariab les, taking values in a (typ ically non commuta ti ve) associative algebra. If the algebra is com mutative, the odd KP hierarch y is known to admit reductions to the BKP and the CKP hierarchy . W e approach the odd KP hierarchy and its relation to BKP and CKP in different ways, and addr ess the question whether non commutative versions of the BKP and the CKP equatio n (and some of their reductions) exist. In particular , we derive a functio nal represen tation of a linear system for the o dd KP hierarchy , which in the commutative case produc es functional represen tations of the BKP and CKP hierarch ies in terms of a tau fun ction. Furthermo re, we consider a function al representatio n of the KP hierarch y that in volves a second (auxiliary) dependen t variable and features the odd KP hierarchy directly as a subh ierarchy . A method to g enerate large classes of exact solutions to the KP hierarchy fr om solutions to a li near matrix ODE system, via a hierarchy of matrix Riccati equations, t hen also applies to the odd KP h ierarchy , and this in turn can be exploited, in particu lar , to ob tain solutions to the BKP and CKP hierar chies. 1 Introd uction Many (e.g. in the sense of the in verse scattering method) “inte grable” partial dif ferential (or dif ference) equati ons (PDE s) admit gene ralizat ions to ve rsions where the depend ent va riable tak es va lues in an arbi- trary associati ve and typically noncommutat ive algebra (prov ided that dif ferentiabili ty with respect to the indepe ndent varia bles can be deﬁned ) (see e.g. [1–4]). This fact can be exploited to generate lar ge classes of exact solutions to a scalar integrab le PD E via simple solutio ns to the correspon ding matrix PD E (see also [5, 6]). In particul ar , the existence of families of solutions like multi-solito ns is then a consequen ce of the exis tence of certain solutions to the matrix PDE uni vers ally for arbitrar y m atrix size. There are, ho weve r , integra ble equations that do not admit a direct noncommutati ve generaliza tion in the abov e sense. The Sawada-K otera equation [7 ] belongs to these exception s [3]. This equation is a reduct ion of the BK P equat ion, the ﬁrst member of the BKP hierarchy [8–14] (see also [15–56]), w hich also lacks a nonc ommutati ve ver sion (the latter should not be conf used w ith the multi-co mponent version of BKP). The BK P hierarc hy and also the C KP hierarch y [9, 10] (see also [15, 31, 41, 43–45 , 50, 55, 57 – ∗ c  2008 by A. Dimakis and F . M ¨ uller-Hoissen 1 60]) originat e from the “commutati ve” KP hierar chy in the Gelfan d-Dick ey-Sat o (GDS) formalism (see sectio n 2.6) by ﬁrst restricting the Lax equations to only odd-numbere d varia bles t 1 , t 3 , t 5 , . . . , and then imposing addition al reductio n condition s. T he ﬁrst step clearly also works in the nonco mmutati ve case. It leads to the (nonco mmutati ve) “odd KP hierarchy” . The GDS formulatio n of the KP hierarch y i n vo lve s an inﬁnite number of dependent variab les. All beside s one can be eliminated, resultin g in PDEs for a single dependent var iable. In the same way , the odd K P hierar chy (in the GDS formali sm) leav es us with PDEs for two dependent varia bles. These PDEs admit symmetries by means of which t he full KP hierar chy can be r estored ( and the tw o d epend ent va riables reduce d to a single one). This sho ws that the odd KP hierarchy is a part (sub hierarc hy) of the KP hierarchy , somethin g that is obvio us in its GDS form. So why should we deal with a subh ierarch y if w e could treat the full hierarchy ? The crucia l point is that the BKP and CKP reductio ns of the odd KP hierarch y are not compatib le with the abov ementioned KP-restoring symmetries. The genera l message is that a subhierarch y can admit a reduction that does not ext end to a reduc tion of the full hierar chy . And this is the reaso n why BKP and CKP retain their indi viduality , despi te their KP origin. In sectio n 2 w e deriv e the ﬁ rst member of the odd KP hierarchy in an elementary way . This “odd KP system” is a system of two PDEs for the KP varia ble and one addition al depen dent variab le. 1 W ithin this system we can then look for noncommuta ti ve version s of reduct ions known in the commutati ve case, and this is done in some subs ection s of section 2. BK P and CKP posses s a certain noncommut ati ve exten sion with a single depen dent var iable, but sev erely constrain ed. It turns out, in particular , that these exten sions are solved by any solution to the ﬁrst two equat ions of the “noncommutati ve” (potential) KdV hier archy , and this result remains true in the commutati ve case (where the constra ints disappear ). F urther more, there is a natural nonco mmutati ve generaliza tion of the CK P equati on, thoug h as a system with two dependent v ariabl es. Nothing similar is found in the BKP case. In section 3 we deri ve a linear system, in functiona l form, for the whole odd KP hiera rchy and deduc e corres pondin g result s for the BKP and CKP hierarc hies. Section 4 takes a diff erent route, startin g from a functi onal representati on of the KP hierarchy that in v olv es an auxiliary dependent var iable [61]. In this for - mulation , the odd KP hierarchy appear s as the subhiera rchy tha t consists o f equations containing on ly partial deri vati ves with respect to the odd-n umbered v ariable s, t 1 , t 3 , t 5 , . . . . The aux iliary dependen t vari able then tak es the role of the se cond depen dent variab le of the odd KP syst em. A certain symmetry reduct ion for the (odd) KP hi erarchy is then intr oduced , which plays a crucia l role in th e step from odd KP to BKP and CKP . Sev eral classes of solutio ns to the matrix KP hierarch y and, if a r ank one c onditi on holds (see e.g. [62]), then also the scalar KP hierarch y , can be obtained from solutions to a system of linear matrix ordinary dif ferential equation s, via a system of matrix Riccati equations [61, 63–65]. This is a ﬁnite-dimens ional ver sion of the famou s Sato theory for the KP hierarchy . Using the abo vementione d formulatio n of the KP hierarc hy that exhib its the odd KP hierarchy directly as a subhierarc hy , this immediately also generates soluti ons to the odd KP hierarchy . This is elaborated in section 5. Furthermore , we show ho w solut ions to the B KP and CKP hierarchi es can be obtained from solu tions to the matrix odd KP hierarchy . Some ﬁnal remarks are collec ted in section 6. 2 The odd KP system The K adomtse v-Petviash vili (KP) hierarchy (see e.g. [66]) is gi ven by the inte grability (or zero curv ature) condit ions B m,t n − B n,t m + [ B m , B n ] = 0 (2.1) 1 Throughout we will work with a potential φ related to the KP variable u by u = φ t 1 , hence this system may rather be called “potential odd KP system”. 2 of the linear system [67] ψ t n = B n ψ , n = 1 , 2 , 3 , . . . , (2.2) where B n = ∂ n + n − 2 X k =0 b n,k ∂ k . (2.3) Here ∂ is the operator of partial dif ferentiation w ith respect to the variab le t 1 (hence the ﬁrst of equat ions (2.2) is tri vially satisﬁed), and ψ t n denote s the parti al deriv ati ve of ψ with respect to the varia ble t n . The object s b n,k are diffe rentiab le 2 functi ons of t = ( t 1 , t 2 , . . . ) with value s in some associat i ve algebra A , and ψ is an element of a left A -module. C orresp onding ly , the depende nt varia ble of the “noncommutati ve” K P hierarc hy is an A -v alued function. If A is commutati ve, restrictin g (2.2) to only odd values of n , setti ng b n, 0 = 0 for n = 1 , 3 , 5 , . . . , and “freezi ng” the v ariable s t 2 , t 4 , . . . , leads to the BKP hierarchy [9, 10]. In th e follo wing we also restric t (2.2) to on ly odd v alues of n , but do not impose fur ther condit ions right awa y (see also [10] for the commutati ve case ). Section 2.1 deri ves the “odd KP system” from (2.2) w ith n = 3 , 5 in a direc t way . Sect ion 2.6 identiﬁes it as the ﬁ rst non-tri vial m ember of the GDS formulation of the KP hierarch y , restric ted to odd-numbered e v olutio n vari ables. In sections 2.2 -2.5 we consi der some reduct ions of the odd KP system. 2.1 Elementary deriv ation of the odd KP system Let us cons ider the ﬁrst two non -tri vial equa tions of the abov e linear system with odd n , i.e. ψ t 3 = ( ∂ 3 + b 3 , 1 ∂ + b 3 , 0 ) ψ , (2.4) ψ t 5 = ( ∂ 5 + b 5 , 3 ∂ 3 + b 5 , 2 ∂ 2 + b 5 , 1 ∂ + b 5 , 0 ) ψ . (2.5) By exp loiting the integra bility conditi on and introdu cing potentia ls φ and θ via 3 b 3 , 0 = 3 θ t 1 + 3 2 φ t 1 t 1 , b 3 , 1 = 3 φ t 1 , (2.6) the coef ﬁcients of the linear system are ﬁxed in terms of φ and θ , ψ t 3 = ( ∂ 3 + 3 φ t 1 ∂ + 3 θ t 1 + 3 2 φ t 1 t 1 ) ψ , (2.7) ψ t 5 =  ∂ 5 + 5 φ t 1 ∂ 3 + 5 ( θ t 1 + 3 2 φ t 1 t 1 ) ∂ 2 + 5 ( θ t 1 t 1 + 1 3 φ t 3 + 7 6 φ t 1 t 1 t 1 + φ t 1 2 ) ∂ + b 5 , 0  ψ , (2.8) where b 5 , 0 = 5 3 θ t 3 + 10 3 θ t 1 t 1 t 1 + 5 6 φ t 1 t 3 + 5 3 φ t 1 t 1 t 1 t 1 + 5 { θ t 1 , φ t 1 } + 5 2 ( φ t 1 2 ) t 1 + 5 3 [ φ t 1 , φ t 1 t 1 ] + 5 3 Z [ φ t 3 , φ t 1 ] t . 1 . (2.9) 2 This requires some additional structure that we need not specify here. If A is an algebra of real or comple x matri ces, the usual differe ntial structure will be assumed. 3 The shift by 3 2 φ t 1 t 1 leads to a more ‘symmetric’ form of the resulting equations (2.10) and (2.11). 3 Here [ , ] and { , } mean commutator and anti-c ommutator , respecti vely . The remaining inte grabil ity condi- tions then result in the follo wing pair of equati ons,  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 2 { φ t 1 , φ t 3 − φ t 1 t 1 t 1 − φ t 1 2 } + 45 4 ( φ t 1 t 1 2 − 4 θ t 1 2 )  t 1 − 5 φ t 3 t 3 + 15  [ φ t 1 , θ t 3 − θ t 1 t 1 t 1 ] + [ θ t 1 , φ t 3 + 1 2 φ t 1 t 1 t 1 ] + 3 2 [ θ t 1 t 1 , φ t 1 t 1 ] + [ φ t 1 , Z [ φ t 3 , φ t 1 ] t . 1 ]  = 0 , (2.10) and h 9 θ t 5 − 5 θ t 1 t 1 t 3 + θ t 1 t 1 t 1 t 1 t 1 + 15 2  − { φ t 3 , θ t 1 } + { θ t 1 t 1 t 1 , φ t 1 } + 1 2 { θ t 1 , φ t 1 t 1 } t 1 +6 φ t 1 θ t 1 φ t 1 + 1 6 [ φ t 3 , φ t 1 t 1 ] + 1 6 [ φ t 1 , φ t 1 t 1 ] t 1 t 1 − 1 4 [ φ t 1 t 1 , φ t 1 t 1 t 1 ]  i t 1 − 5 θ t 3 t 3 − 15 2 { θ t 1 , φ t 1 } t 3 + 15 [ θ t 1 , θ t 3 + 1 2 θ t 1 t 1 t 1 + Z [ φ t 3 , φ t 1 ] t . 1 ] +45 [( θ t 1 ) 2 , φ t 1 ] + 15 [ φ t 1 t 1 , [ φ t 1 , θ t 1 ] ] + 15 2 [ [ θ t 1 , φ t 1 t 1 ] , φ t 1 ] + 25 4 [ φ t 1 t 1 t 3 , φ t 1 ] − 5  Z [ φ t 3 , φ t 1 ] t . 1  t 3 + 15 2 [ φ t 3 − φ t 1 t 1 t 1 , ( φ t 1 ) 2 ] + 45 4 [ φ t 1 , ( φ t 1 t 1 ) 2 ] = 0 . (2.11) In the follo wing w e refer to (2.10) and (2.11) as the “odd KP system”. W e no te that by introduc ing ˜ θ := θ + 1 2 Z [ φ , φ t 1 ] t . 1 , (2.12) which implies θ t 3 = ˜ θ t 3 − 1 2 [ φ , φ t 3 ] − R [ φ t 3 , φ t 1 ] t . 1 , the result ing equatio ns no longer in volv e integral s, see also sectio n 4. Remark 2 .1 Switching on “ev en ﬂo ws”, we hav e in particular ψ t 2 = ( ∂ 2 + b 2 , 0 ) ψ . Compatibility with (2.4) (using (2.6)) then leads to b 2 , 0 = 2 φ t 1 , θ t 1 = 1 2 φ t 2 , and the (pote ntial) KP equation for φ .  2.2 Recove ring BKP and CKP in the commutative case If A is commutati ve, then the abov e pair of equation s reduces to  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 φ t 3 φ t 1 + 15 φ t 1 φ t 1 t 1 t 1 + 15 φ t 1 3 + 45 4 φ t 1 t 1 2 − 45 θ t 1 2  t 1 − 5 φ t 3 t 3 = 0 , (2.13)  9 θ t 5 − 5 θ t 1 t 1 t 3 + θ t 1 t 1 t 1 t 1 t 1 − 15 φ t 3 θ t 1 + 15 φ t 1 θ t 1 t 1 t 1 + 15 2 ( φ t 1 t 1 θ t 1 ) t 1 +45 φ t 1 2 θ t 1  t 1 − 5 ( θ t 3 + 3 φ t 1 θ t 1 ) t 3 = 0 . (2.14) Setting θ = k φ t 1 , (2.15) it turns out that the second equatio n is a conseq uence of the ﬁrst if k = 0 , ± 1 2 . (2.16) 4 If k = ± 1 / 2 , (2.13) becomes the BKP equation  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 φ t 1 φ t 3 + 15 φ t 1 φ t 1 t 1 t 1 + 15 φ t 1 3  t 1 − 5 φ t 3 t 3 = 0 . (2.17) Setting φ t 3 = 0 reduces (2.17) to the (potenti al) Sawada-K otera equat ion [7, 9, 47] 9 φ t 5 + φ t 1 t 1 t 1 t 1 t 1 + 15 ( φ t 1 φ t 1 t 1 t 1 + φ t 1 3 ) = 0 , (2.18) which is kn o wn not to posse ss a non commutat i ve (e.g. matrix) version [3] . S etting φ t 5 = 0 in (2.17), yields the Ramani equatio n [9, 68] (also called (potentia l) bidirecti onal Sawada -K otera (bSK) equat ion [50, 69– 71]),  − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 φ t 1 φ t 3 + 15 φ t 1 φ t 1 t 1 t 1 + 15 φ t 1 3  t 1 − 5 φ t 3 t 3 = 0 . (2.19) If k = − 1 / 2 w e ha ve b 3 , 0 = b 5 , 0 = 0 and thus the familia r linear syste m for the BKP equatio n [9, 10], ψ t 3 = ( ∂ 3 + 3 φ t 1 ∂ ) ψ , (2.20) ψ t 5 =  ∂ 5 + 5 ∂ φ t 1 ∂ 2 + 5 3 ( φ t 3 + 2 φ t 1 t 1 t 1 + 3 φ t 1 2 ) ∂  ψ . (2.21) If k = 1 / 2 , we obtain another linear system for the BKP equation: ψ t 3 = 3 φ t 1 t 1 ψ + 3 φ t 1 ψ t 1 + ψ t 1 t 1 t 1 = ( ∂ 3 + 3 ∂ φ t 1 ) ψ , (2.22) ψ t 5 = 5 3 ( φ t 3 + 2 φ t 1 t 1 t 1 + 3 φ t 1 2 ) t 1 ψ + 5 3 ( φ t 3 + 5 φ t 1 t 1 t 1 + 3 φ t 1 2 ) ψ t 1 +10 φ t 1 t 1 ψ t 1 t 1 + 5 φ t 1 ψ t 1 t 1 t 1 + ψ t 1 t 1 t 1 t 1 t 1 =  ∂ 5 + 5 ∂ 2 φ t 1 ∂ + 5 3 ∂ ( φ t 3 + 2 φ t 1 t 1 t 1 + 3 φ t 1 2 )  ψ , (2.23) which is thus simply an adjoi nt of the ﬁrst linear system. If k = 0 (i.e. θ = 0 ), (2.13) beco mes the C KP equati on [9]  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 φ t 1 φ t 3 + 15 φ t 1 φ t 1 t 1 t 1 + 15 φ t 1 3 + 45 4 φ t 1 t 1 2  t 1 − 5 φ t 3 t 3 = 0 . (2.24) The linear system in this case turns out to be giv en by half the sum of the respecti ve equations of the abov e two BKP linea r systems. Setting φ t 3 = 0 reduces (2.24) to the (potenti al) Kaup-Kupers hmidt equation [72] 9 φ t 5 + φ t 1 t 1 t 1 t 1 t 1 + 15 ( φ t 1 φ t 1 t 1 t 1 + 3 4 φ t 1 t 1 2 + φ t 1 3 ) = 0 . (2.25) Setting φ t 5 = 0 in (2.24), yields the bidirec tional Kaup-Kuper shmidt (bKK) equation [50, 69 –71, 73]. 2.3 BKP and the noncommutativ e KdV hierar chy Imposing the BKP conditi on θ = − 1 2 φ t 1 (2.26) 5 (i.e. k = − 1 2 in (2.15)) in the non c ommutati ve case, we hav e b 3 , 0 = 0 and b 5 , 0 = 5 3 Z [ φ t 3 − φ t 1 t 1 t 1 , φ t 1 ] t . 1 . (2.27) Then (2.10) reduce s to  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 ( φ t 1 ( φ t 3 − φ t 1 t 1 t 1 ) − φ t 1 3 )  t 1 − 5 φ t 3 t 3 +15 [ φ t 1 , Z [ φ t 3 , φ t 1 ] t . 1 ] = 0 , (2.2 8) and (2.11), after use of (2.28), becomes [ φ t 3 − φ t 1 t 1 t 1 , φ t 1 ] t 1 t 1 −  Z [ φ t 3 − φ t 1 t 1 t 1 , φ t 1 ] t . 1  t 3 + 3 φ t 1 [ φ t 3 − φ t 1 t 1 t 1 , φ t 1 ] = 0 . (2.29) The latter equation repre sents a constra int which, howe ver , is not in general preserve d under the ﬂo w with e v olutio n var iable t 5 , gi ven by (2.28). 4 No w we observ e that (2.29) is obviou sly solv ed if φ t 3 = φ t 1 t 1 t 1 + f ( φ t 1 ) , (2.30) where f is an arbitrary polynomial in φ t 1 with coef ﬁ cients in the center of A . But only for a special choice of f , the equation (2.30) has a chance to be compatible with (2.28 ). Addressing inte grabili ty , ev olution equati ons like (2.30), with the restri ction that the right hand side is a homogeneou s differ ential polyno mial, appear to be distinguish ed. This reduces (2.30 ) to the potential KdV equati on, w here f ( φ t 1 ) = a φ t 1 2 with a constant a , or the mKdV equation, where f ( φ t 1 ) = a φ t 1 3 . But only in the KdV case the weightin g of terms is compatib le with (2.28). Using the KdV equation in (2.28), yields  9 φ t 5 − 9 φ t 1 t 1 t 1 t 1 t 1 − 15 a ( φ t 1 2 ) t 1 t 1 + 15 a φ t 1 t 1 2 − 5 ( a + 3) a φ t 1 3  t 1 +5 (9 − a 2 ) φ t 1 φ t 1 t 1 φ t 1 = 0 . (2.31) Choosin g a = 3 , this can be integra ted to φ t 5 − φ t 1 t 1 t 1 t 1 t 1 − 5 ( φ t 1 2 ) t 1 t 1 + 5 φ t 1 t 1 2 − 10 φ t 1 3 = 0 , (2.32) and (2.30) reads φ t 3 = φ t 1 t 1 t 1 + 3 φ t 1 2 . (2.33) (2.33) and (2.32) are the ﬁ rst two equations of the noncommutat iv e potential KdV (ncpKdV) hierar chy . 5 Hence, any solution to the ﬁrst two ncpKdV hiera rchy equations (2.33) and (2.32) also solves the abo ve nonco mmutati ve exte nsion o f th e BKP equatio n. 6 This relatio n the n also holds for the “commut ati ve” scalar equati ons, of course. B ut to ﬁ nd this result the step into the noncommutati ve framew ork was extremely helpfu l. Remark 2 .2 If we i mpose the conditions b 3 , 0 = b 5 , 0 = 0 on the n oncommutat i ve odd KP system, we obtain (2.26) and [ φ t 3 − φ t 1 t 1 t 1 , φ t 1 ] = 0 , which leads more directly to (2.30).  Another noncommutati ve ext ension of t he BKP equation is obtai ned for θ = 1 2 φ t 1 (i.e. k = 1 2 in (2.15)), and one ﬁnds corresp onding results. 4 T aking a look at this problem in th e Saw ada-K otera case, where φ t 3 = 0 simpliﬁes the eq uations a lot, we hav e to compute the deri v ativ e of (2.29) with respect to t 5 and then eliminate φ t 5 by use of (2.28). Already the resulting terms quadratic in (deriv atives of) φ do not cancel as a consequence of (2.29) and its deriv ativ es with respect to t 1 . 5 W ith u = − φ x where x = t 1 we o btain from (2.33) and (2.32), respecti vely , the p otential v ersions of (3.46) and (3.47) in [74]. 6 An analogous relati on exists between the ﬁrst two equations of the (noncommutati ve) Burgers hierarchy and the KP equation [61, 75]. 6 2.4 CKP and the noncommutativ e KdV hierar chy Imposing the CKP conditi on θ = 0 , (2.10) reduces to  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 2 { φ t 1 , φ t 3 − φ t 1 t 1 t 1 − φ t 1 2 } + 45 4 ( φ t 1 t 1 ) 2  t 1 − 5 φ t 3 t 3 + 15 [ φ t 1 , Z [ φ t 3 , φ t 1 ] t . 1 ] = 0 (2.34) and ( 2.11) yie lds a co nstrai nt, in v olving only commutators , w hich is no t in genera l preserv ed under the ﬂow of (2.34). The constrai nt turns out to be satisﬁed as a consequ ence of the ncpKdV equation in the form φ t 3 = 1 4 φ t 1 t 1 t 1 + 3 2 ( φ t 1 ) 2 , (2.35) and (2.34) then integ rates to φ t 5 = 1 16 φ t 1 t 1 t 1 t 1 t 1 + 5 8 ( φ t 1 2 ) t 1 t 1 − 5 8 ( φ t 1 t 1 ) 2 + 5 2 ( φ t 1 ) 3 , (2.36) which is the second equation of the ncpKdV hierarchy . 7 As a conse quence , any solu tion to the ﬁrst two equati ons of the ncpKdV hierar chy (with coefﬁcie nts as giv en abov e) is also a solutio n to the constr ained nonco mmutati ve extensio n of the CKP equation. In the commutati ve case, the correspon ding statement then also hold s, of course, i.e. any solution to the ﬁrst two equations of the potential KdV hierarchy (with coef ﬁcients as gi ven abo ve) is also a solution to the CKP equation. 2.5 Further red uctions of the odd KP system in the noncommutative case Imposing φ t 3 = θ t 3 = 0 , we obtain from (2.10) and (2.11) the follo wing noncommutati ve generali zation of the (poten tial) Sawad a-K otera (2.18) and Kaup-Kupers hmidt equation (2.25), 9 φ t 5 + φ t 1 t 1 t 1 t 1 t 1 + 15 2 { φ t 1 , φ t 1 t 1 t 1 } + 45 4 φ t 1 t 1 2 + 15 φ t 1 3 + 15 [ θ t 1 t 1 , φ t 1 ] + 15 2 [ θ t 1 , φ t 1 t 1 ] − 45 θ t 1 2 = 0 (2.37) and 9 θ t 1 t 5 + θ t 1 t 1 t 1 t 1 t 1 t 1 + 15 2  { θ t 1 t 1 t 1 , φ t 1 } + 1 2 { θ t 1 , φ t 1 t 1 } t 1 + 6 φ t 1 θ t 1 φ t 1 + 1 6 [ φ t 1 , φ t 1 t 1 ] t 1 t 1 − 1 4 [ φ t 1 t 1 , φ t 1 t 1 t 1 ] + [ θ t 1 , θ t 1 t 1 t 1 ]  t 1 + 45 [( θ t 1 ) 2 − 1 4 ( φ t 1 t 1 ) 2 , φ t 1 ] + 15 [ φ t 1 t 1 , [ φ t 1 , θ t 1 ] ] + 15 2 [ φ t 1 , [ φ t 1 t 1 , θ t 1 ] ] − 15 2 [ φ t 1 t 1 t 1 , ( φ t 1 ) 2 ] = 0 . (2.38) In the commutati ve case, the last equati on can be inte grated with respec t to t 1 , and we rec ov er an inte grable system that appea red in [76, 77] (see also [78]), 9 u t 5 + u t 1 t 1 t 1 t 1 t 1 + 10 u u t 1 t 1 t 1 + 25 u t 1 u t 1 t 1 + 20 u 2 u t 1 − 135 θ t 1 θ t 1 t 1 = 0 , (2.39) 9 θ t 5 + θ t 1 t 1 t 1 t 1 t 1 + 10 u θ t 1 t 1 t 1 + 5 ( u t 1 θ t 1 ) t 1 + 20 u 2 θ t 1 = 0 , (2.40) where u := 3 2 φ t 1 . In [79] an attempt was made to ﬁnd a nonco mmutati ve version of “couple d systems of Kaup-Ku pershmid t and Sawad a-K otera type”, b ut without success. The above equations (2.37) and (2.38) consti tute a soluti on to this problem. 7 W e note that (2.35) and (2.36) can be obtained from (2.33) and (2.32) via t n 7→ 2 t n . 7 Setting φ t 5 = θ t 5 = 0 in (2.10) and (2.11), we obtain a system that m ay be regard ed as a noncommuta- ti ve general ization of the Ramani (or bSK) equatio n (2.19) and the bidirection al Kaup-K upersh midt (bKK) equati on. Remark 2 .3 The sys tem (2.37) and (2.38) pos sesses the s ymmetry φ t 2 = 2 θ t 1 (see als o remark 2.1), by use of which we obtain from it the ﬁrst and the thir d member of the (noncommutati ve) Bo ussines q hierarchy . The latter is the 3-r eduction of the (noncommutati ve) KP hierarch y (also called third Gelfand-Dick ey hierarc hy [66]). This means that the syste m (2.37) and (2.38) can also be obta ined as a reduction of the KP hierarchy , and not just as a reduction of the odd KP hierarchy . The crucial point is that the reduc tion condition is compatib le with the equation s (like φ t 2 = 2 θ t 1 ) that are needed to complete the odd KP hierarc hy to the KP hierarchy (cf section 4). This is not so for the reductio ns o f odd KP to BKP or CKP . In the same way , the noncommutat i ve gener alizatio n of the bSK and bKK equatio ns is related to the 5-reductio n of the (nonco mmutati ve) KP hierarchy (ﬁfth Gelfand-Dic ke y hierar chy).  2.6 Gelfand-Dickey-S ato formulation of the odd KP hierar chy The odd KP system can be e xtende d to a hierarch y by restric ting the GDS formulatio n (see e.g. [66]) of the KP hierarc hy , L t n = [ B n , L ] , (2.41) where B n = ( L n ) ≥ 0 , L = ∂ + u 2 ∂ − 1 + u 3 ∂ − 2 + . . . , (2.42) to odd-numbe red vari ables t n . Here ∂ − 1 is the formal in verse of ∂ an d ( ) ≥ 0 means the projectio n of a pseud odif ferential operat or to its dif ferential operato r part (see e.g. [66]). W e ha ve in particu lar B 3 = ( L 3 ) ≥ 0 = ∂ 3 + 3 u 2 ∂ + 3 ( u 3 + u 2 ,t 1 ) , (2.43) B 5 = ( L 5 ) ≥ 0 = ∂ 5 + 5 u 2 ∂ 3 + 5 ( u 3 + 2 u 2 ,t 1 ) ∂ 2 + 5 ( u 4 + 2 u 3 ,t 1 + 2 u 2 ,t 1 t 1 + 2 u 2 2 ) ∂ +5 ( u 5 + 2 u 4 ,t 1 + 2 u 3 ,t 1 t 1 + u 2 ,t 1 t 1 t 1 + 2 { u 2 , u 3 } + 2 ( u 2 2 ) t 1 ) . (2.44) (2.41) is known to be equiv alent to the zero curv ature conditi ons (2 .1), with B n deﬁned in (2.42). By comparis on with B 3 and B 5 compute d in secti on 2.1, we ﬁnd u 2 = φ t 1 , u 3 = θ t 1 − 1 2 φ t 1 t 1 , u 4 = − θ t 1 t 1 + 1 3 φ t 3 + 1 6 φ t 1 t 1 t 1 − ( φ t 1 ) 2 , u 5 = 1 3 θ t 3 + 2 3 θ t 1 t 1 t 1 − 1 2 φ t 1 t 3 − { θ t 1 , φ t 1 } + 3 2 ( φ t 1 2 ) t 1 + 1 3 [ φ t 1 , φ t 1 t 1 ] + 1 3 Z [ φ t 3 , φ t 1 ] t . 1 . (2.45) If A is commutativ e , the CKP reduct ion of the KP hiera rchy is determined by L + L ∗ = 0 , and the BKP reducti ons by ∂ L + L ∗ ∂ = 0 , respecti vely L ∂ + ∂ L ∗ = 0 [10]. Here L ∗ denote s the adjoint of the pseud odif ferential operator L (see e.g. [6 6]). W e summarize these well-k no wn relations together with those found in sectio n 2.2 in the follo wing table. BKP ∂ L + L ∗ ∂ = 0 θ = − 1 2 φ t 1 BKP L ∂ + ∂ L ∗ = 0 θ = 1 2 φ t 1 CKP L + L ∗ = 0 θ = 0 8 If A is matrix algebra over R or C , we can genera lize the adjoi nt by settin g ( A∂ ) ∗ := − ∂ A ⊺ , where A ∈ A with transpose A ⊺ . 8 The CKP cond ition then genera lizes to matrix CKP L + L ∗ = 0 φ ⊺ = φ , θ ⊺ = − θ The condition s for φ and θ indeed yield a consi stent reduct ion of the odd KP syste m, which may thus be reg arded as a noncommuta ti ve version of the CKP equat ion. For m > 1 , it is a pair of equatio ns for two depen dent (matrix) v ariables, howe ver . The correspond ing hiera rchy will be called matrix CKP hierar chy . In the fo llo wing, “CKP equation ” or “CKP h ierarch y” thr ougho ut refers to th e familiar scalar (commutati ve) case, i.e. m = 1 , and we w ill add “matrix” whene ver we mean the matrix general ization . I n contrast to the CKP case, the above BK P reduction co nditio n for L does not cons istentl y generaliz e to th e n oncommuta ti ve case. The formulation (2.41), w ith n ∈ N , of the KP hierarchy depen ds on an inﬁnite number of dependent v ariabl es. Elimination of u 3 , u 4 , . . . leads to PDE s that only in volv e the vari able u 2 ( = φ t 1 ). Omitting some of the e quatio ns (2.41), it will no lo nger be pos sible to eliminate all the aux iliary v ariable s u 3 , u 4 , . . . . In the step to the odd KP hierarchy , w here all equations (2.41) in volvi ng deri vat iv es with respe ct to ev en- numbere d variab les are dropped, one of the additional var iables is retained , namely u 3 , wh ich leads to the appearance of θ . It would be desirable to ﬁ nd a way to ex plicitly eliminate all the remaining auxilia ry v ariabl es u 4 , u 5 , . . . from the seq uence of equat ions (2.41) with odd n . In section 3 we solve this problem on the lev el of the corresp ondin g linear system. The odd KP hierarchy expre ssed in terms of φ and θ (without auxili ary v ariable s) then arises from the integ rabilit y conditions. Also in case of the full KP system, (2.41) w ith n ∈ N , we may think of eliminatin g only u 4 , u 5 , . . . . The resulting equations then depend on u 2 and u 3 , and further elimination of u 3 would lead to the KP equati on and its companion s. The m ore interestin g aspect, howe ver , is that in such a formulati on of the KP hierarc hy , we should expe ct the odd KP system (and its hierarch y companions ) to form a subhierarchy . In fact , in section 4 , we start from a functional f orm o f the KP hierarch y that in volv es one additio nal (aux iliary) v ariabl e that, by no w not surpr isingly , turns out to be relate d to θ . In thi s rep resenta tion of the KP hierarc hy , the odd KP hierarchy is indeed nicely described as a subhierar chy . W e note that in this picture a solutio n to the odd KP hier archy in genera l still depen ds on the ev en-numbered varia bles t 2 n , which are con stants with respec t to the odd KP hierar chy . 3 A linear system f or t he odd KP hierar chy in functional f orm In this section w e present a linear system for the whole nonco mmutati ve odd KP hierarch y in functional form. This extends the linear syste m for the odd KP system obtaine d in section 2.1. The bilinear identity for the KP hiera rchy (see e.g. [66]), restricted to odd-nu mbered va riables , is res[ ψ ( s o , z ) ˜ ψ ( t o , z )] = 0 , (3.1) where t o = ( t 1 , t 3 , t 5 , . . . ) , ψ ( t o , z ) = w ( t o , z ) e ˜ ξ ( t o ,z ) , ˜ ψ ( t o , z ) = ˜ w ( t o , z ) e − ˜ ξ ( t o ,z ) , (3.2) with ˜ ξ ( t o , z ) = P n ≥ 1 t 2 n − 1 z 2 n − 1 and w ( t o , z ) = I + X n ≥ 1 w n ( t o ) z − n , ˜ w ( t o , z ) = I + X n ≥ 1 ˜ w n ( t o ) z − n . (3.3 ) 8 More generally , we may consider an algebra A with an in volution ∗ , and deﬁne ( A∂ ) ∗ := − ∂ A ∗ . 9 W e will often omit the argume nt t o , for simplici ty . Insertin g (3.2) in (3.1), the bilinear identity reads res  w ( s o , z ) ˜ w ( t o , z ) e ˜ ξ ( s o − t o ,z )  = 0 . (3.4) The residu e res f ( z ) of a formal series f ( z ) = P + ∞ n = −∞ f n z − n is the coefﬁcie nt f 1 . In particular , setting s o = t o , (3.4) implies ˜ w 1 = − w 1 =: φ . (3.5) W e write w 2 = − ˜ θ + 1 2 ( φ t 1 + φ 2 ) , (3.6) with a variab le ˜ θ . W e shall see that φ can be identiﬁed with the va riable of the same name introd uced in sectio n 2.1 , and that ˜ θ coinc ides w ith the v ariable deﬁned in (2.12). B elo w we use the Miwa shift notation φ [ λ ] ( t o ) = φ ( t o + [ λ ]) , [ λ ] = ( λ, λ 3 / 3 , λ 5 / 5 , . . . ) . The proof of the follo w ing theore m is presented in Appendi x A. Theor em 3.1 The bilinear identit y implies 1 λ F ( λ ) ( ψ 2[ λ ] − ψ ) − ( ψ 2[ λ ] + ψ ) t 1 = λ 2  ˜ θ 2[ λ ] − ˜ θ + 1 2 ( φ 2[ λ ] − φ ) t 1 − 1 2 [ φ, φ 2[ λ ] ]  F ( λ ) − 1 ( ψ 2[ λ ] + ψ ) , (3.7) where F ( λ ) := I − λ 2  φ 2[ λ ] − φ  . (3.8)  (3.7) is a functio nal re presen tation o f the linear system for the odd KP hierarch y . B y ex pansio n in po w ers of th e indete rminate λ , we rec ov er from the lo west order s the line ar system of t he od d KP sys tem deri ved in sectio n 2.1. Indeed, at order λ 2 we obta in ψ t 3 =  ∂ 3 + 3 φ t 1 ∂ + 3 2 (2 ˜ θ t 1 + φ t 1 t 1 + [ φ t 1 , φ ])  ψ , (3.9) which i s ( 2.7) by u se of (2.1 2). At or der λ 3 we obtain the d eri vati ve of the ab ov e equation wit h resp ect to t 1 . At order λ 4 we get an equation that contains ψ t 3 , which can be replaced with the help of (3.9). This results in ψ t 5 =  ∂ 5 + 5 φ t 1 ∂ 3 + 5 2 (2 ˜ θ t 1 + 3 φ t 1 t 1 + [ φ t 1 , φ ]) ∂ 2 + 5 6 (6 ˜ θ t 1 t 1 + 7 φ t 1 t 1 t 1 + 2 φ t 3 +6 φ t 1 2 + 3 [ φ t 1 t 1 , φ ]) ∂ + 5 3 ( ˜ θ t 3 + 2 ˜ θ t 1 t 1 t 1 + φ t 1 t 1 t 1 t 1 + 1 2 φ t 1 t 3 ) + 5 { ˜ θ t 1 , φ t 1 } + 5 2 ( φ t 1 2 ) t 1 + 5 6 [ φ t 3 , φ ] + 5 3 [ φ t 1 t 1 t 1 , φ ] + 5 2 [ φ t 1 2 , φ ]  ψ , (3.10) which by use of (2.12) becomes (2.8). 10 3.1 The commutative case If A is commutati ve, imposing the red uction condition (2.15), i.e. ˜ θ = θ = k φ t 1 with a consta nt k , the linear system (3.7) take s the form 1 λ ( ψ 2[ λ ] − ψ ) = F ( λ ) k − 1 2  F ( λ ) − k − 1 2 ( ψ 2[ λ ] + ψ )  t 1 . (3.11) Hence 1 λ ( ψ 2[ λ ] − ψ ) =        F ( λ ) − 1 ( ψ 2[ λ ] + ψ ) t 1 k = − 1 2 (BKP)  F ( λ ) − 1 ( ψ 2[ λ ] + ψ )  t 1 for k = 1 2 (BKP) F ( λ ) − 1 2  F ( λ ) − 1 2 ( ψ 2[ λ ] + ψ )  t 1 k = 0 (CKP) (3.12) The CKP function al linear equatio n is half of the sum of the two BK P functi onal linear equa tions. In the remainde r of this section we conside r the case where φ is a C -valu ed functio n and write φ = (ln τ 2 ) t 1 = 2 (ln τ ) t 1 , ( 3.13) with a funct ion τ . (In sections 5.1 and 5.2 we use a differ ent function τ giv en by φ = (ln τ ) t 1 .) Lemma 3.1 The biline ar identity (3.1) with the reduct ion θ = k φ t 1 implies w ( λ − 1 ) = τ − 2[ λ ] τ F ( − λ ) k + 1 2 , ˜ w ( λ − 1 ) = τ 2[ λ ] τ F ( λ ) − k + 1 2 , (3.14) where F ( λ ) no w takes the form F ( λ ) = 1 − λ  ln τ 2[ λ ] τ  t 1 . (3.15) Pr oof: W e re fer to some conseq uences of (3.1) deri ved in Appendix A. (A.3) can be written as ˜ w ( λ − 1 ) = F ( λ ) w 2[ λ ] ( λ − 1 ) . From (A.4) we get w ′ 2[ λ ] ( λ − 1 ) ˜ w ( λ − 1 ) = 1 λ ( F ( λ ) − 1) F ( λ ) + 1 2 F ′ ( λ ) − λ 2 ( θ 2[ λ ] − θ ) , where a prime indicat es a partial deri vati ve with respect to t 1 , and thus (ln w 2[ λ ] ( λ − 1 )) ′ = − 1 2 ( φ 2[ λ ] − φ ) + 1 2 (ln F ( λ )) ′ − λ 2 F ( λ ) ( θ 2[ λ ] − θ ) . Using (3.13), the preced ing equatio n can be inte grated to w 2[ λ ] ( λ − 1 ) = τ τ 2[ λ ] F ( λ ) k + 1 2 , which is equi val ent to the ﬁrst equation in (3.14). W ith its help, the equat ion we starte d with becomes the second of (3.14).  11 By use of the lemma, and setting z = λ − 1 , we ﬁnd ˜ w ( z ) =    w ( − z ) − z − 1 w ( − z ) t 1 k = − 1 2 (BKP) ˜ w ( − z ) + z − 1 ˜ w ( − z ) t 1 for k = 1 2 (BKP) w ( − z ) k = 0 (CKP) (3.16) and thus the follo wing relatio ns for the Baker -Akhiezer functio n ψ and its adjoin t ˜ ψ , ˜ ψ ( z ) = − z − 1 ψ ( − z ) t 1 k = − 1 2 (BKP) ψ ( z ) = z − 1 ˜ ψ ( − z ) t 1 for k = 1 2 (BKP) ˜ ψ ( z ) = ψ ( − z ) k = 0 (CKP) (3.17) Pro position 3.1 The bilinear identity (3.1) with the reduction θ = k φ t 1 implies the “dif ferential Fay iden- tity” λ + µ λ − µ τ 2[ λ ] τ 2[ µ ]  λ F 2[ λ ] ( µ ) k + 1 2 F ( µ ) − k + 1 2 − µ F 2[ µ ] ( λ ) k + 1 2 F ( λ ) − k + 1 2  = ( λ + µ ) τ τ 2[ λ ]+2[ µ ] − λµ  ( τ 2[ λ ]+2[ µ ] ) t 1 τ − τ 2[ λ ]+2[ µ ] τ t 1  . (3.1 8) Pr oof: This is obtained from (A.5) using (3.14) and (A.7).  In the BKP case ( k = ± 1 / 2 ), the dif ferenti al Fay identity (3.18) is bilinear , ( λ − 1 + µ − 1 )( τ 2[ λ ]+2[ µ ] τ − τ 2[ λ ] τ 2[ µ ] ) = ( τ 2[ λ ]+2[ µ ] ) t 1 τ − τ 2[ λ ]+2[ µ ] τ t 1 + λ + µ λ − µ  ( τ 2[ λ ] ) t 1 τ 2[ µ ] − τ 2[ λ ] ( τ 2[ µ ] ) t 1  , (3 .19) whereas in the CKP case ( k = 0 ) it is not 9 , λ + µ λ − µ τ 2[ λ ] τ 2[ µ ]  λ F 2[ λ ] ( µ ) 1 2 F ( µ ) 1 2 − µ F 2[ µ ] ( λ ) 1 2 F ( λ ) 1 2  = ( λ + µ ) τ τ 2[ λ ]+2[ µ ] − λµ  ( τ 2[ λ ]+2[ µ ] ) t 1 τ − τ 2[ λ ]+2[ µ ] τ t 1  . (3.20) Expansio n in powers of the indetermina tes λ and µ generates the BKP , respecti vely C KP , hierarch y equa- tions. 4 Fr om a functional rep re sentation of the KP hierarch y to o dd KP A funct ional represen tation of the m × m matrix KP hierarch y is determine d by [61] λ − 1 ( φ − φ − [ λ ] ) − φ t 1 − ( φ − φ − [ λ ] ) φ = ˆ θ − ˆ θ − [ λ ] (4.1) with an additional depend ent variab le ˆ θ , and φ [ λ ] ( t ) = φ ( t + [ λ ]) where t = ( t 1 , t 2 , t 3 , . . . ) and [ λ ] = ( λ, λ 2 / 2 , λ 3 / 3 , . . . ) . By expans ion in powers of the indeterminate λ and elimina tion of ˆ θ one reco vers the equati ons of the KP hie rarchy . W e n ote that alth ough (4.1) conta ins a “bare” φ besides deri vati ves of it with respec t to t n , after eliminatio n of ˆ θ the resulting equations do not. Writing ˆ θ = ˜ θ − 1 2 ( φ t 1 + φ 2 ) , (4.2) 9 This is in agreement with the fact that the CKP hierarchy cannot be expressed in Hirota bilinear form with a single τ -fu nction [10]. 12 (4.1) take s the follo w ing form, after a Miwa shift, λ − 1 ( φ [ λ ] − φ ) − 1 2 ( φ [ λ ] + φ ) t 1 − 1 2 ( φ [ λ ] − φ ) 2 + 1 2 [ φ, φ [ λ ] ] = ˜ θ [ λ ] − ˜ θ . (4.3) Clearly , no w one recov ers the equations of the matrix KP hierarchy by exp ansion in po wers of λ and elimi- nation of ˜ θ . T he ﬁrst four equ ations from expansio n of (4.3) can be written as 10 φ t 2 = 2 ˜ θ t 1 − [ φ, φ t 1 ] , (4.4) ˜ θ t 2 = 2 3 φ t 3 − 1 6 φ t 1 t 1 t 1 − ( φ t 1 ) 2 − 1 2 [ φ, [ φ, φ t 1 ]] + [ φ, ˜ θ t 1 ] , (4.5) φ t 4 = 4 3 ˜ θ t 3 + 2 3 ˜ θ t 1 t 1 t 1 + 2 { φ t 1 , ˜ θ t 1 } − 1 3 [ φ, 2 φ t 3 + φ t 1 t 1 t 1 ] − [ φ, ( φ t 1 ) 2 ] , (4.6) ˜ θ t 4 = 4 5 φ t 5 − 1 3 φ t 1 t 1 t 3 + 1 30 φ t 1 t 1 t 1 t 1 t 1 − 1 6 { φ t 1 , 4 φ t 3 − φ t 1 t 1 t 1 } + 1 2 ( φ t 1 t 1 ) 2 − 2 ( ˜ θ t 1 ) 2 − [ φ t 1 , ˜ θ t 1 t 1 ] + 1 3 [ φ, 2 ˜ θ t 3 + ˜ θ t 1 t 1 t 1 ] − 1 6 [ φ, [ φ, 2 φ t 3 + φ t 1 t 1 t 1 ]] +[ φ, { φ t 1 , ˜ θ t 1 } ] + { ˜ θ t 1 , [ φ, φ t 1 ] } + 1 2 [ φ t 1 , [ φ, φ t 1 t 1 ]] − 1 2 [ φ, φ t 1 ] 2 − 1 2 [ φ, [ φ, ( φ t 1 ) 2 ]] . (4.7) Solving the ﬁrst equation for ˜ θ t 1 and using the resultin g exp ressio n to eliminat e ˜ θ from the secon d, results in the (pote ntial) KP equation 4 φ t 3 − φ t 1 t 1 t 1 − 6 ( φ t 1 ) 2 − 3 Z φ t 3 t 3 t . 1 + 6 Z [ φ t 1 , φ t 2 ] t . 1 = 0 . (4.8) Instea d o f eliminatin g ˜ θ from (4.3), which yield s t he matrix KP hierarchy , we can eliminate the deri v ativ es of φ and ˜ θ with respect to the e ven-n umbered vari ables, t 2 n . This means we solve the equatio ns resulting from (4.3) for the deri vati ves of φ and ˜ θ with respec t to t 2 n , as in (4.4), (4.5), etc., compute their inte grability condit ions, and further use them to eliminate in the latter all deri vati ves w ith respect to ev en-nu mbered v ariabl es. In partic ular , φ t 2 t 4 = φ t 4 t 2 yields , after eliminatio n of “ev en” deriv ati ves ,  9 φ t 5 − 5 φ t 1 t 1 t 3 + φ t 1 t 1 t 1 t 1 t 1 − 15 2 { φ t 1 , φ t 3 − φ t 1 t 1 t 1 } + 15 ( φ t 1 ) 3 + 45 4  ( φ t 1 t 1 ) 2 − 4 ( ˜ θ t 1 ) 2  + 45 2 { ˜ θ t 1 , [ φ, φ t 1 ] } − 15 4 [[ φ, φ t 1 ] , φ t 1 t 1 ] − 15 2 [[ φ, φ t 1 t 1 ] , φ t 1 ] − 45 4 [ φ, φ t 1 ] 2  t 1 − 5 φ t 3 t 3 +15  [ φ t 1 , ˜ θ t 3 − ˜ θ t 1 t 1 t 1 ] + [ ˜ θ t 1 , φ t 3 + 1 2 φ t 1 t 1 t 1 ] + 3 2 [ ˜ θ t 1 t 1 , φ t 1 t 1 ] − 1 2 [ φ, [ φ t 1 , φ t 3 ]]  = 0 , (4.9) and from ˜ θ t 2 t 4 = ˜ θ t 4 t 2 one obta ins another quit e lengthy equation for the two dependent v ariable s φ and ˜ θ , i n v olving only deri vati ves with respect to t 1 , t 3 , t 5 . W e veriﬁed indepen dently with FORM [80] and Mathematic a [81] that via (2.12) these two equations are equiv alent to (2.10) and (2.11), which is our odd KP system. The structure displaye d in (4.4)-(4.7) in fact exte nds to the whole hierarchy , since the expans ion of (4.3) in po wers of λ has the follo wing leading deri vati ves (which do not appear in the remaining terms, repres ented by dots), λ 2 n − 1 : 1 2 n φ t 2 n = 1 2 n − 1 ˜ θ t 2 n − 1 + . . . , λ 2 n : 1 2 n ˜ θ t 2 n = 1 2 n + 1 φ t 2 n +1 + . . . , (4.10) 10 Here we used e.g. the ﬁ rst equation to eliminate φ t 2 from the second. By use of (2.12), (4.4) simpliﬁes to φ t 2 = 2 θ t 1 (see also remark 2.1). 13 where n = 1 , 2 , . . . . Hence the method of computin g the integ rabilit y conditio ns φ t 2 m t 2 n = φ t 2 n t 2 m and ˜ θ t 2 m t 2 n = ˜ θ t 2 n t 2 m , and then eliminati ng all deri v ati ves of φ and ˜ θ with respect to ev en-numbe red variab les, ext ends to the w hole KP hierarc hy . This yields a hierarchy of equations in volv ing only deri vati ves with respec t to odd-numb ered variab les and we ha ve sho w n that its ﬁrst m ember is our odd KP system. Because of the hierar chy property , it should then coincid e w ith the od d KP hierarchy as formul ated in section 2.6, or genera ted by the linear system deri ved in section 3. Abov e we started with a formulati on of the KP hierarchy in terms of two dependen t varia bles, φ and ˜ θ (or equi v alently θ ). ˜ θ entered the stage a s an au xiliary v ariable and its el imination leads to an expres sion for the KP hierarch y in terms of a sin gle d epende nt v ariable, which is φ . In this formulation of the KP hierarchy , the odd KP hierar chy is directly described as a subhierarc hy (without further auxiliary varia bles as in the GDS formulatio n of section 2.6). A particula r conseq uence is that any method to construct ex act solutio ns to the KP hierar chy in the formulation using the auxiliary dependent va riable θ (or ˜ θ ) automatical ly yields soluti ons to the odd KP hierar chy . This fact will be us ed in section 5. W e note that (4.4), (4.5), etc ., are symmetries of the o dd KP hierar chy equati ons, with the help of w hich one reco ver s the whole KP hierarch y . The next result will turn out to be crucial for establ ishing a relation between solutions to the (noncom- mutati ve) odd KP hierarchy and solution s to the BKP and CK P hierarchies. From now on we consider matrices ov er R or C . Pro position 4.1 The function al represen tation (4.3) of the m × m matrix KP hierarchy is in vari ant under φ 7→ φ ⊺ ◦ ε , ˜ θ 7→ − ˜ θ ⊺ ◦ ε , (4.11) where ε ( t 1 , t 2 , t 3 , t 4 , . . . ) := ( t 1 , − t 2 , t 3 , − t 4 , . . . ) , and φ ⊺ is the transp ose of φ . Pr oof: W e consider (4.1) with φ and ˜ θ replaced by φ ⊺ ◦ ε and − ˜ θ ⊺ ◦ ε , res pecti vely . T aking the t ranspo se of the resulting equatio n, noting that ( f ◦ ε ) [ λ ] = ( f − [ − λ ] ) ◦ ε , and composing with ε (which has the propert y ε ◦ ε = id ), leads to λ − 1 ( φ − [ − λ ] − φ ) − 1 2 ( φ − [ − λ ] + φ ) t 1 − 1 2 ( φ − [ − λ ] − φ ) 2 − 1 2 [ φ, φ − [ − λ ] ] = − ˜ θ − [ − λ ] + ˜ θ . W ith the substituti on λ → − λ and a Miwa shift with [ λ ] , thi s becomes (4.1).  As a conseq uence, the (matrix) KP hierarc hy admits the symmetry reduct ion φ = φ ⊺ ◦ ε , ˜ θ = − ˜ θ ⊺ ◦ ε . (4.12 ) Restricti ng to the odd KP hierarchy , and setting t 2 n = 0 , n = 1 , 2 , . . . , we hav e φ ◦ ε = φ and ˜ θ ◦ ε = ˜ θ , hence the last conditi ons simplify to φ = φ ⊺ , ˜ θ = − ˜ θ ⊺ . (4.13) In particu lar , for m = 1 we obtain ˜ θ = 0 , hence θ = 0 by ( 2.12), and thus the CKP hierarchy . The conditio ns (4.13) are equi val ent to those that determine the matrix CKP hierarch y , see sectio n 2.6 . Obvio usly the reduc tion (4.13) is not compatible w ith the symmetries (the ﬂo w s associated with t 2 n ) that extend the odd KP to the KP hierarc hy . This e xample s ho ws that a su bhiera rchy can admit a (symmetry) reduct ion that is not a reduction of the complete hierarc hy . Remark 4.1 A functio nal represe ntation of the (noncommuta ti ve) disc r ete KP hierarch y is gi ven by [6 4] λ − 1 ( φ − φ − [ λ ] ) − ( φ + − φ − [ λ ] ) φ = ˆ θ + − ˆ θ − [ λ ] , (4.14) 14 where n ∈ Z and ( φ + ) n = φ n +1 . T o or der λ 0 , we obtain φ t 1 − ( φ + − φ ) φ = ˆ θ + − ˆ θ . (4.15) Subtracti ng this from (4.14 ) yields (4.1), hence each φ n , n ∈ Z , has to satisfy the KP hiera rchy , thus also φ + . 11 The transf ormation (4.2) con ve rts the discrete KP hierarc hy into λ − 1 ( φ [ λ ] − φ ) − 1 2 ( φ [ λ ] + φ ) t 1 − 1 2 ( φ [ λ ] − φ ) 2 + 1 2 [ φ, φ [ λ ] ] = ˜ θ [ λ ] − ˜ θ , (4.16) 1 2 ( φ + + φ ) t 1 + 1 2 ( φ + − φ ) 2 + 1 2 [ φ, φ + ] = ˜ θ + − ˜ θ . (4.17) Accordin g to propositio n 4.1, φ + = φ ⊺ ◦ ε and ˜ θ + = − ˜ θ ⊺ ◦ ε solve (4.16) if φ and ˜ θ do. Restrict ing the KP hierarc hy (in the form pres ented in this section) to the odd KP hierarchy , in the scalar case ( m = 1 ) these condit ions read φ + = φ , ˜ θ + = − ˜ θ , (4.18) and (4.17 ) becomes θ = ˜ θ = − 1 2 φ t 1 , which is the BKP re duction ! W e also refe r t o [ 82] (p. 969) for a relate d result.  5 Solutions to t he odd KP system a nd some of its reductions via a matrix Riccati system W e consider the matrix linear system Z t n = H n Z n = 1 , 2 , . . . , H =  R Q S L  , Z =  X Y  , (5.1) where L, Q, R, S are, respecti vely , cons tant M × M , N × M , N × N and M × N matrices over C , X is an N × N and Y an M × N m atrix. W ith suitable technica l assumptions, the size of th e matrices may also be inﬁnite. The solutio n to the abov e linear system is giv en by Z = exp  ξ ( t , H )  Z 0 where ξ ( t , H ) := ∞ X k =1 t k H k . (5.2) For the n e w va riable Φ := Y X − 1 , (5.3) assuming that X possesses an in ver se, (5.1) implies the follo wing hierarchy of matrix Riccati equatio ns Φ t n = S n + L n Φ − Φ R n − Φ Q n Φ n = 1 , 2 , . . . , (5.4) where  R n Q n S n L n  := H n (5.5) 11 By eliminating ˆ θ and ˆ θ + , one obtains the modiﬁed K P (mKP ) hierarchy for v , w here v t 1 = φ + − φ , and the Miura t ransfor- mation. 15 (see [61, 63 –65]). Using its funct ional representat ion λ − 1 (Φ − Φ − [ λ ] ) = S + L Φ − Φ − [ λ ] R − Φ − [ λ ] Q Φ , (5.6) it turns out (see [61] for details ) that Φ together with ˆ Θ = Φ R (5.7) solv es the M × N matrix KP Q hierarc hy , w hich is determine d by λ − 1 (Φ − Φ − [ λ ] ) − Φ t 1 − (Φ − Φ − [ λ ] ) Q Φ = ˆ Θ − ˆ Θ − [ λ ] . (5.8) If rank( Q ) = m , hence Q = V U ⊺ (5.9) with an M × m matrix U (with transpose U ⊺ ) and an N × m matrix V , then φ := U ⊺ Φ V (5.10) solv es the m × m matrix KP hierarchy (4.1). B y us e of the ﬁ rst Riccati equ ation Φ t 1 = S + L Φ − Φ R − Φ Q Φ (5.11) in ˆ Θ = ˜ Θ − 1 2 (Φ t 1 + Φ Q Φ) (cf (4.2)), and using (5.7), we obtain ˜ Θ = 1 2 ( S + L Φ + Φ R ) . (5.12) Here w e shall drop S since it cancels out in ˜ Θ [ λ ] − ˜ Θ . It follo ws that the Q -modiﬁed version of (4.3) is satisﬁed as a conse quence of the Riccati system. Recalling (2.12), which now tak es the form ˜ Θ = Θ + 1 2 Z (Φ Q Φ t 1 − Φ t 1 Q Φ) t . 1 , (5.13) we arri ve at the follo w ing conclusio n. Pro position 5.1 An y solution Φ to the odd Riccati hierar ch y , i.e. th e R iccati hierarch y (5.4) restricted to odd n , togeth er with 12 Θ = 1 2  L Φ + Φ R − Z (Φ Q Φ t 1 − Φ t 1 Q Φ) t . 1  , (5.14) solv es the odd KP Q hierarc hy . 13 Furthermor e, if (5.9) holds, then φ = U ⊺ Φ V and θ = U ⊺ Θ V (5.15) solv e the m × m matrix odd KP hierarch y (hence in particula r the odd KP system (2.10) and (2.11)). If m = 1 , then φ = U ⊺ Φ V and θ = 1 2 U ⊺ ( L Φ + Φ R ) V (5.16) solv e the scalar odd KP hierarchy (thus in particu lar (2.13) and (2.14)).  12 By use of the Riccati system (5.4), this can also be writ ten as Θ = 1 2 R ( S 2 + L 2 Φ − Φ R 2 − Φ Q 2 Φ) t . 1 . The integrand is the right hand side of t he Riccati equation for the variable t 2 (which, howe ver , is prohibited in proposition 5.1), so that Θ t 1 = 1 2 Φ t 2 , a symmetry of the odd KP (here odd KP Q ) hierarchy which we already met in remark 2.1. 13 Hence it solves i n particular (2.10) and (2.11) with φ and θ replaced by matrices Φ and Θ , and wi th the product modiﬁed by the constant matrix Q . 16 Remark 5.1 For some ﬁx ed r ∈ N , r > 1 , let us impose the conditio n H r Z 0 = Z 0 P , (5.17) with an N × N matrix P , on the solution (5.2) of the linear matrix system (5.1). This implies H nr Z 0 = Z 0 P n and thus H nr Z = Z P n for n ∈ N . Hence R nr X + Q nr Y = X P n and S nr X + L nr Y = Y P n , which leads to the algebra ic Riccati equatio ns S nr + L nr Φ = Y P n X − 1 = Φ X P n X − 1 = Φ ( R nr + Q nr Φ) n ∈ N . (5.18) The corresp ondin g equat ions of the R iccati hiera rchy then imply Φ t nr = 0 , for all n ∈ N . The conditio n (5.17) thus ensures that Φ solv es the r -reduct ion of the KP hierarc hy ( r th Gelfand- Dicke y hierarc hy). If r is odd, this also yields a reducti on of the odd KP hierarchy . H ence, adding the conditio n (5.17) to the assumpti ons of propo sition 5.1, (5.15) co nstitut es a solution to th e r -reduction of the m × m matrix odd KP hierarc hy . For r = 3 this is the hierarchy w ith the pair (2.37), (2.38) as its ﬁ rst member , for r = 5 it starts with the nonco mmutati ve generaliz ation of the bSK and bKK equations , see section 2.5.  In propo sition 5.1 “odd KP hierarchy” more directl y refers to the form in section 4 , where it has been descri bed as a subhi erarchy of the KP hierar chy , in the formulatio n of the latter in vol ving the auxiliary v ariabl e θ . In the scalar case, th is hierarch y then admits reductio ns to the CKP and BKP hierarchy by imposing θ = 0 , respecti vely θ = − 1 2 φ t 1 (see section 2). In the follo wing we sho w how the precedi ng propo sition generat es solution s to the BKP and the (matrix) CKP hierarchy . Lemma 5.1 Let M = N . The trans formation giv en by L 7→ − R ⊺ , R 7→ − L ⊺ , Q 7→ ± Q ⊺ , S 7→ ± S ⊺ , Φ 7→ ± Φ ⊺ ◦ ε , (5.19) with ε deﬁned in propos ition 4.1, lea ves the Riccati hierar chy (5.4) in v ariant. Pr oof: The ﬁrst four replacement rules in (5.19) can be combined into H 7→ −T H ⊺ T − 1 with T =  0 ∓ I N I N 0  . (5.20) This implies H n 7→ ( − 1) n T ( H n ) ⊺ T − 1 , and thus L n 7→ ( − 1) n L ⊺ n R n 7→ ( − 1) n R ⊺ n , Q n 7→ ∓ ( − 1) n Q ⊺ n , S n 7→ ∓ ( − 1) n S n . Applying the m ap to the Riccati hierarchy (5 .4), takin g the transpo se and us ing (Φ ◦ ε ) t n = ( − 1) n +1 Φ t n ◦ ε , reprod uces (5.4).  As a consequen ce of the preceding lemma, w e ha ve the follo wing symmetry reduc tion of the Riccati hierarc hy (5.4), R = − L ⊺ , Q ⊺ = ± Q , S ⊺ = ± S , (5.21) togeth er with Φ = ± Φ ⊺ ◦ ε . (5.22) Restricti ng to the odd Riccati hierar chy , we are allo wed to set t 2 n = 0 , n = 1 , 2 , . . . . Then Φ giv en by (5.3) solv es the od d Riccat i hier archy and has t he proper ty Φ ◦ ε = Φ . Furthermo re, (5.21) and ( 5.22), which now reads Φ = ± Φ ⊺ , constitu te a symmetry reduct ion of the odd Riccati hierarch y . 17 Pro position 5.2 Let M = N and Φ a solution to the odd Riccati hierarchy with R = − L ⊺ , S = S ⊺ , Q = V V ⊺ , (5.23) where V is a constant N × m matrix. If Φ ⊺ = Φ , (5.24) then φ = V ⊺ Φ V and θ = V ⊺ Θ V (5.25) with Θ gi ven in (5 .14) solve the m × m m atrix CKP hierarc hy (see section 2.6). Pr oof: The conditi ons (5.23 ) and (5.24) correspon d to the upper signs in (5.21). A ccordi ng to proposi- tion 5.1, φ and θ solv e the m × m matrix odd KP hierarchy . Using (5.14), (5.24) and Q ⊺ = Q , one easily ver iﬁes that θ ⊺ = − θ holds, which is the reduction to the matrix CKP hierarchy .  Cor ollary 5.1 Let M = N and Φ a solution to the odd Riccati hierarchy with (5.23), where V is a constant N -component vector . If Φ ⊺ = Φ , then φ = V ⊺ Φ V solve s the CKP hiera rchy . Pr oof: The assertio n follo ws from the last propositio n, with Θ deﬁned in (5.14) and m = 1 , in which case the CKP reduc tion condition θ = 0 holds.  T o obtain BK P sol utions via propositio n 5.1 is a bit less direct. Pro position 5.3 Let M = N and Φ a solution to the odd Riccati hierarc hy subje ct to the condition s (5.23) with a cons tant N -component vector V . If Φ satisﬁes S + L Φ + Φ ⊺ L ⊺ − Φ ⊺ Q Φ = 0 , (5.26) then φ = V ⊺ Φ V solves the BKP hierarc hy . Pr oof: First we note th at the fra ctional linear transfo rmation Φ 7→ Φ + := ( S + L Φ)( R + Q Φ ) − 1 (pro vided the in ver se exist s) leav es the R iccati hierarch y (5.4) in v ariant. This is so becaus e this transfo rmation is induce d by Z 7→ H Z , which lea ves the linear m atrix system (5.1) in varian t. W e m ay then impose the symmetry reduc tion Φ ⊺ = Φ + , i.e. Φ ⊺ = ( S + L Φ)( R + Q Φ ) − 1 , which is (5.26). U sing the deﬁnitions (5.16) with U = V , the ﬁrst Riccati equation (5.11), and then the last equati on, we sho w that the BKP reductio n conditio n is satisﬁed, 2 ( θ + 1 2 φ t 1 ) = V ⊺ ( L Φ − Φ L ⊺ + Φ t 1 ) V = V ⊺ ( S + 2 L Φ − Φ Q Φ ) V = V ⊺ ( L Φ − Φ ⊺ L ⊺ + (Φ ⊺ − Φ) V V ⊺ Φ) V = V ⊺ L Φ V − V ⊺ Φ ⊺ L ⊺ V = 0 . (One also ﬁnds φ + = φ and θ + = − θ , cf (4.18).)  Remark 5.2 The discr ete KP Q hierarc hy is solved by a sequen ce Φ = (Φ n ) n ∈ Z of solutio ns to the Riccati hierarc hy (5.4) i f L Φ − Φ + R − Φ + Q Φ = 0 , where Φ + n = Φ n +1 . This is the fractiona l line ar transformatio n appear ing in the proof of proposit ion 5.3, with S = 0 . It follows from (4.15) by use of (5.7) and (5.11).  18 Remark 5.3 The case with the lower signs in lemma 5.1 might be expect ed to be related to BKP . B ut it requir es a ske w-symmetric Q and thus does not quite ﬁt togeth er with proposition 5.1. Howe ver , writing Q = ˜ QL − L ⊺ ˜ Q with a rank one matrix ˜ Q = V V ⊺ , it turns out that φ = V ⊺ ( L Φ − Φ L ⊺ ) V = 2 V ⊺ L Φ V solv es the BKP equatio n (and its hierar chy), if Φ satisﬁes the condition s of lemma 5.1 with the lo wer signs. W e shall elaborate on the underlyi ng structure elsewhe re.  Remark 5.4 As a consequenc e o f (5.1) and (5.21), which implies H = −T H ⊺ T − 1 with T deﬁned in (5.20), we ha ve ( Z ⊺ T H k Z ) t n = 0 for all odd n (5 .27) and k = 0 , 1 , . . . . Choosin g T with the minus sign, our CKP condition Φ ⊺ = Φ originat es fro m Z ⊺ T Z = 0 , and the BKP conditi on (5 .26) corresponds t o Z ⊺ T H Z = 0 . These conditions ar e the ﬁrst two in a sequence that of fers addition al possibil ities, Z ⊺ T H k Z = 0 k = 0 , 1 , 2 , . . . . (5.28) W e note that ( T H k ) ⊺ = ( − 1) k +1 T H k , so that the left hand side of (5.28) is a symmetric bilinear form if k is odd, and ske w -symmetric if k is e ven. In va riance unde r a transformation Z 7→ GZ , with a constant in ver tible matrix G , requir es G ⊺ T H k G = T H k . If the bilinear form is non-de g enerate 14 , this means that G has to be (complex ) orthogonal if k is odd, and symplectic if k is ev en. This connects with original work like [10]. It should be noticed, howe ver , that the abo ve method to construct solution s to the BKP hierarchy also works if th e bilinear form is de gener ate .  Remark 5.5 Adding the r -redu ction condition (5.17) to the assumptions of corolla ry 5.1, respecti vely propo- sition 5.3, the y g enerat e solutio ns to the r -reduction of the CKP , respecti vely BKP , hierarchy . For r = 3 , this yields soluti ons to the K aup-K upersh midt, respecti vely the S awa da-K otera equation. For r = 5 , we obtain soluti ons to the bKK , respecti vely the bSK equat ion (see section 2.2 ). W e will not elaborate this further in this work, b ut a comparis on with the results in [50, 60 , 69–71] would certainly be of interest.  In the follo wing subsection s we elabo rate some classes of solut ions more exp licitly . W e consid er the odd Riccati hierarchy with M = N , impose the condit ions (5.23) with S = 0 , and treat the rank one case where Q = V V ⊺ with a vector V . The choices (5.29) and (5.53) belo w ha ve their origin in certain normal forms of the matrix H , see [64]. 5.1 A class of BKP and CKP solutions Setting Q = R K − K L = − ( L ⊺ K + K L ) (5.29) with a symmetric matrix K (i.e. K ⊺ = K ), (5.2) can be computed expl icitly (cf [64]) and we ﬁnd the follo wing solutio n to the odd Riccati hierarc hy , Φ = e ˜ ξ ( t o ,L ) Φ 0  e − ˜ ξ ( t o ,L ⊺ ) ( I N + K Φ 0 ) − K e ˜ ξ ( t o ,L ) Φ 0  − 1 , (5.30) where Φ 0 = Y 0 X − 1 0 and ˜ ξ ( t o , L ) = ∞ X k =0 t 2 k + 1 L 2 k + 1 . (5.31) 14 In the CKP case ( k = 0 ) this is fulﬁlled. In the BKP case ( and more generally for k > 0 ), and if S = 0 and R = − L ⊺ , which is the case we address in more detail belo w , the bilinear form is non-degen erate iff det( L ) 6 = 0 . 19 Assuming Φ 0 in ver tible, this simpliﬁes to Φ =  e − ˜ ξ ( t o ,L ⊺ ) (Φ − 1 0 + K ) e − ˜ ξ ( t o ,L ) − K  − 1 . (5.32) Using Q = V V ⊺ , the cy clicity of the trace, and tr ln = ln det , we obtain φ = V ⊺ Φ V = tr( Q Φ) = − tr(( L ⊺ K + K L ) Φ) = (ln τ ) t 1 with τ = det(Φ − 1 0 + K − e ˜ ξ ( t o ,L ⊺ ) K e ˜ ξ ( t o ,L ) ) . (5.33) Here K, L, V hav e to solve the rank one conditio n L ⊺ K + K L = − V V ⊺ . (5.34) In or der that (5.33 ) s olve s the CKP or the BKP hierarchy , (5.24), re specti vely (5.26), still h as to be satisﬁed. CKP . I f Φ 0 is symmetric, i.e. Φ ⊺ 0 = Φ 0 , then also Φ gi ven in (5.32). W e can thus e xpress τ in (5.33) as τ = det( C − e ˜ ξ ( t o ,L ⊺ ) K e ˜ ξ ( t o ,L ) ) (5.35) with an arbitrary constant symmetric N × N matrix C , i.e. C ⊺ = C . Accordin g to corollary 5.1, this determin es a solution φ = (ln τ ) t 1 to the CKP hierarc hy , prov ided that K and L satisfy (5.34 ). BKP . W e ha ve to elaborate the BKP condition (5.26) (with S = 0 ). Using (5.29), it can be ex presse d as L ⊺ (Φ − 1 + K ) = − (Φ − 1 + K ) ⊺ L . (5.36) Inserti ng (5.32), written in the form Φ − 1 + K = e − ˜ ξ ( t o ,L ⊺ ) (Φ − 1 0 + K ) e − ˜ ξ ( t o ,L ) , (5.37 ) this reduce s to 1 2 C := L ⊺ (Φ − 1 0 + K ) = − (Φ − 1 0 + K ) ⊺ L = − 1 2 C ⊺ , (5.38) i.e. C has to be a ske w-symmetric matrix. It is known that BK P τ -fun ctions can be expresse d as the square of a Pfaf ﬁ an. In the follo wing we transla te (5.33) into such a form, assuming that L is in vert ible. W e may replace τ giv en in (5.33) by τ = det( C − 2 e ˜ ξ ( t o ,L ⊺ ) L ⊺ K e ˜ ξ ( t o ,L ) ) , (5.39) since the two expressio ns dif fer only by a cons tant factor that drops out in φ = (ln τ ) t 1 . Using L ⊺ K = − K L − V V ⊺ , this becomes τ = det( e ˜ ξ ( t o ,L ) ) 2 det( A + V V ⊺ ) where A := e − ˜ ξ ( t o ,L ⊺ ) C e − ˜ ξ ( t o ,L ) − L ⊺ K + K L . (5 .40) If the siz e N of the matrices is e ven, then det( A + V V ⊺ ) = det( A ) (for ske w-symmetric A , see e.g. (2.92) in [47]) leads to τ = det  C − e ˜ ξ ( t o ,L ⊺ ) ( L ⊺ K − K L ) e ˜ ξ ( t o ,L )  . ( 5.41) This is the determinant of a ske w-symmetric m atrix, hence τ can be express ed as the squa re of the Pfaf ﬁan of this matrix. If N is odd, then det( A ) = 0 , but (5.40) with a suitab le choice of V can still lead to non-tri vial soluti ons. In this case we can use the ident ity det( A + V V ⊺ ) = det  0 V ⊺ − V A  =  Pf  0 V ⊺ − V A  2 (5.42) 20 (see Appendi x B ) to ex press τ as the square of a Pfaf ﬁan. 15 A subclass of solutio ns is obtaine d by choosing L = diag ( p 1 , . . . , p N ) (5.43) with const ants p i , i = 1 , . . . , N . The solution to (5.34) is then giv en by K ij = − v i v j p i + p j i, j = 1 , . . . , N , (5.44) assuming p i + p j 6 = 0 for all i, j and w riting V ⊺ = ( v 1 , . . . , v N ) . From this one recov ers in particular B KP and CKP multi-soli ton solutio ns (see also [8, 10, 8 2] for differe nt approaches ). 5.1.1 Examples Example 5.1 W e consider the CKP case with (5.43). For N = 1 , (5.35) becomes τ = 1 + b e 2 ˜ ξ ( t o ,p ) with b = v 2 2 cp , dropping an irrele vant factor c . This yields a regular solution if b > 0 , and u = φ t 1 then describe s a single line solito n. For N = 2 and C = diag ( c 1 , c 2 ) we obtain, droppin g an irrele v ant factor c 1 c 2 , τ = 1 + b 1 e 2 ˜ ξ ( t o ,p 1 ) + b 2 e 2 ˜ ξ ( t o ,p 2 ) + b 1 b 2  p 1 − p 2 p 1 + p 2  2 e 2 ˜ ξ ( t o ,p 1 )+2 ˜ ξ ( t o ,p 2 ) , b i := v 2 i 2 c i p i , (5.45) assuming p 1 , p 2 , c 1 , c 2 6 = 0 and p 2 6 = − p 1 . If the paramete rs are real and such that b 1 , b 2 > 0 , this yields a regular CKP solution φ , and u = φ t 1 generi cally descri bes two obliqu e line soliton s. In this case we can simplify the abov e exp ression by writing b i = exp(2 a i ) with constant s a i , i = 1 , 2 .  Example 5.2 In the BKP case with (5.43), we consid er N = 2 , hence L =  p 1 0 0 p 2  , C =  0 c − c 0  , V =  v 1 v 2  . (5.46) (5.41) leads to τ = p 2 with p = c + v 1 v 2 p 1 − p 2 p 1 + p 2 e ˜ ξ ( t o ,p 1 )+ ˜ ξ ( t o ,p 2 ) , (5.47) if p 1 + p 2 6 = 0 . W ithout restriction of gener ality we can set v 1 = v 2 = 1 . For rea l c, p 1 , p 2 , th e functio n φ is then reg ular (for all t 1 , t 2 , . . . ) iff c ( p 2 1 − p 2 2 ) > 0 , and u = φ t 1 descri bes a single line soliton .  Solution s can be supe rposed as follows. If ( L i , V i , K i , C i ) , i = 1 , 2 , are two sets of matrix data that determin e (BKP or CKP) soluti ons, then L =  L 1 0 0 L 2  , V =  V 1 V 2  , K =  K 1 K 12 K ⊺ 12 K 2  , C =  C 1 0 0 C 2  (5.48) determin e a ne w solutio n, provid ed that a solution K 12 exi sts to L ⊺ 1 K 12 + K 12 L 2 = − V 1 V ⊺ 2 . (5.49) 15 The factor det( e ˜ ξ ( t o ,L ) ) 2 in (5.40) can be dropped since it does not inﬂuence φ t n . 21 Figure 1: Plot of u = φ t 1 = 2 (ln | p | ) t 1 t 1 (BKP) at t 5 , t 7 , . . . = 0 with p giv en by (5.50), where p 1 = 1 / 2 , p 2 = − 1 / 4 , p 3 = 1 , p 4 = − 3 / 4 and c 1 = c 2 = 1 . Figure 2: Plot of u = φ t 1 = 2 (ln | p | ) t 1 t 1 at t 5 , t 7 , . . . = 0 with p gi ven by (5.52), where p 1 = 1 + 3i / 2 , p 2 = 1 + i and c = 1 . The array of BKP solit ons extends periodic ally to inﬁnity . Example 5.3 W e consid er the BKP case. B y supe rpositi on of tw o solut ions of the form gi ven in exa mple 5.2, setting V 1 = V 2 = (1 , 1) ⊺ , one obtai ns τ = p 2 with p = b  ˜ c 1 ˜ c 2 + ˜ c 1 e ˜ ξ ( t o ,p 3 )+ ˜ ξ ( t o ,p 4 ) + ˜ c 2 e ˜ ξ ( t o ,p 1 )+ ˜ ξ ( t o ,p 1 ) + a e ˜ ξ ( t o ,p 1 )+ ˜ ξ ( t o ,p 2 )+ ˜ ξ ( t o ,p 3 )+ ˜ ξ ( t o ,p 4 )  , (5.50) where a = ( p 1 − p 3 )( p 2 − p 3 )( p 1 − p 4 )( p 2 − p 4 ) ( p 1 + p 3 )( p 2 + p 3 )( p 1 + p 4 )( p 2 + p 4 ) , b = ( p 1 − p 2 )( p 3 − p 4 ) ( p 1 + p 2 )( p 3 + p 4 ) , (5.51) and ˜ c 1 = c 1 ( p 1 + p 2 ) / ( p 1 − p 2 ) , ˜ c 2 = c 2 ( p 3 + p 4 ) / ( p 3 − p 4 ) . If p 1 6 = p 2 and p 3 6 = p 4 , we may drop the facto r b . W ith real para meters and ˜ c 1 , ˜ c 2 > 0 , one reco ver s a well-kno wn exp ression for the 2-solito n soluti on ( a > 0 ) to the BKP hierarchy [16, 47], see also Fig. 1. Allowing the parame ters to be comple x, w e can superp ose the solutio n data (5.46) and the comple x conjugate data, so that p =    c + p 1 − p 2 p 1 + p 2 e ˜ ξ ( t o ,p 1 )+ ˜ ξ ( t o ,p 2 )    2 +  Im( p 1 ) Im( p 2 ) Re( p 1 ) Re( p 2 ) −    p 1 − p ∗ 2 p 1 + p ∗ 2    2  e 2 Re( ˜ ξ ( t o ,p 1 )+ ˜ ξ ( t o ,p 2 )) . (5.52 ) A regul ar solution from this family is plotted in Fig. 2. See also Appendix C for a general receipe to obtain real soluti ons from complex matrix dat a.  5.2 Another class of BKP and CKP solutions No w we set R = L , so th at L is ske w-symmetric, i.e. L ⊺ = − L , and Q = I N + [ L, K ] (5.53) 22 with a symmetric matrix K . Assuming Φ 0 in ver tible, computat ion of (5.2) (cf [64]) leads to the followin g soluti on to the odd Riccati hierar chy , Φ =  e ˜ ξ ( t o ,L ) (Φ − 1 0 + K ) e − ˜ ξ ( t o ,L ) + ˜ ξ ′ ( t o , L ) − K  − 1 , (5.54) where ˜ ξ ′ ( t o , L ) := ∞ X n =0 (2 n + 1) t 2 n +1 L 2 n . ( 5.55) If also Q = V V ⊺ with a vecto r V , hence K, L, V hav e to satisfy the rank one condition I N + [ L, K ] = V V ⊺ , (5.56) then we obtain φ = V ⊺ Φ V = tr(( I N + [ L, K ]) Φ) = (ln τ ) t 1 with τ = det  e ˜ ξ ( t o ,L ) (Φ − 1 0 + K ) e − ˜ ξ ( t o ,L ) + ˜ ξ ′ ( t o , L ) − K  . (5.57) In order that (5.57) solv es the CKP or the B KP hierarc hy , the conditi on (5.24), respecti vely (5.26), still has to be elabor ated. CKP . I f Φ 0 is symmetric, then also Φ . W e can the n replace the abov e function τ by τ = det  e ˜ ξ ( t o ,L ) C e − ˜ ξ ( t o ,L ) + ˜ ξ ′ ( t o , L ) − K  , (5.58) with an arbitra ry con stant symmetric N × N matrix C . Accordi ng to coroll ary 5 .1, th is determines a solution φ to the CKP hierarch y , if K and L satisfy (5.56). BKP . The c onditi on (5.26) (with S = 0 ) can be w ritten in the form L (Φ − 1 + K ) − (Φ − 1 + K ) ⊺ L = − I N . (5.59) Inserti ng (5.54), re written as Φ − 1 + K = e ˜ ξ ( t o ,L ) (Φ − 1 0 + K ) e − ˜ ξ ( t o ,L ) + ˜ ξ ′ ( t o , L ) , (5.60) leads to L (Φ − 1 0 + K ) − (Φ − 1 0 + K ) ⊺ L = − I N , (5.61) which is C ⊺ = − C where C := 2 L (Φ − 1 0 + K ) + I N , (5.62) i.e. C has to be ske w-symmetric . Next we translate (5.57) in the BKP case into a form, w here τ is the determinant of a ske w-symmetric matrix, under the assump tion that det( L ) 6 = 0 . According to remark 5.4, the latter cond ition correspond s to the gen uine BKP case. A functio n equiv alent to τ giv en in (5.57) is then τ = det  e ˜ ξ ( t o ,L ) ( C − I N ) e − ˜ ξ ( t o ,L ) + 2 L ( ˜ ξ ′ ( t o , L ) − K )  = det( A − V V ⊺ ) where A := e ˜ ξ ( t o ,L ) C e − ˜ ξ ( t o ,L ) − ( K L + LK ) + 2 L ˜ ξ ′ ( t o , L ) . (5.63) 23 This is the determinant of the sum of the ske w-symmetric m atrix A and a rank one matrix. If N is ev en, then det( A − V V ⊺ ) = det( A ) and thus τ = d et  e ˜ ξ ( t o ,L ) C e − ˜ ξ ( t o ,L ) − ( K L + LK ) + 2 L ˜ ξ ′ ( t o , L )  , (5.6 4) which is then the squar e of the Pfaf ﬁ an of A . Remark 5 .6 (5.53) implies tr( L k ( Q − I N )) = 0 , k = 0 , 1 , . . . , N − 1 . 16 These con straints are o bstruc tions to solvi ng (5.53) for K [83]. In p articula r , w e ha ve tr( Q ) = N . Hence V lies on a sphere in N dimensions. Since the (comple x) orthogon al group acts transit iv ely on a (complex iﬁed) sphere (see e.g. L emma 4.1 in [84]), V can be transformed to V = (1 , . . . , 1) ⊺ . Since a similarity transforma tion of the matrices leav es (5.57) in v ariant, this means that without loss of generality w e can set V = (1 , . . . , 1) ⊺ , as long as no restric tions are placed on the antisymmetr ic matrix L .  Choosin g Φ 0 such that [Φ − 1 0 + K , L ] = 0 , (5.65) the abov e solutions become rational functions of t 1 , t 3 , t 5 , . . . . 17 W e conﬁne ourselv es to this case in the follo wing exampl es. For the matrix C (which has to be symmetric in the CKP and ske w-symmetric in the BKP case), (5.65) implies [ C , L ] = 0 . 5.2.1 Examples Example 5.4 Let N = 2 and L =  0 p − p 0  (5.66) with a constant p . A ccordi ng to the last remark w e can set V ⊺ = (1 , 1) without restrictio n of generali ty . The soluti on to (5.56) is then gi ven by K = c I 2 + 1 2 p  − 1 0 0 1  (5.67) with an arbitra ry constan t c . The condition (5.65) leads to C = a I 2 + b L with constan ts a, b . In th e CKP case, b = 0 and the res ulting term in (5.57) in vo lving a can be a bsorbe d by re deﬁnition of c . W e obtain τ = η 2 − 1 4 p 2 where η ( t o , p, c ) = ∞ X n =0 ( − 1) n (2 n + 1) t 2 n +1 p 2 n − c . (5.68) If p is imagina ry , the correspond ing CKP solution is real and reg ular . Tr eating t 5 as a ‘time’ va riable (and freezin g the higher variab les), u = φ t 1 = (ln τ ) t 1 t 1 descri bes a line soliton (with rational decay) moving in t 1 t 3 -space . In the BKP case, ( 5.62) requires b = 1 2 p − 2 , hence C = 2 a L . In (5.64 ), a can b e absorbe d by red eﬁnition of c . Hence we can set C = 0 w ithout loss of gene rality . W e obtain Pf ( A ) = 2 p η ( t o , p, c ) , which cannot pro vide us with a real and regula r BKP solu tion.  16 There are no independent equations for k > N − 1 because of the Cayley-Hamilton theorem. 17 For other approac hes t o rational solutions see [9, 21, 48 ] in the BKP and [10, 43 , 44 ] in the CKP case. 24 Giv en two sets of matrix data ( L i , V i , K i ) , i = 1 , 2 , that determine (B KP or CKP) solu tions, we can superp ose them as follo ws, L =  L 1 0 0 L 2  , V =  V 1 V 2  , K =  K 1 K 12 K ⊺ 12 K 2  . (5.69 ) Then (5.56) is satisﬁed if K 12 solv es L 1 K 12 − K 12 L 2 = V 1 V ⊺ 2 . (5.70) Example 5.5 W e superpos e two solution s of the form gi ven in ex ample 5.4. The solution to (5.70) is then K 12 = − 1 p 1 + p 2 − 1 p 1 − p 2 1 p 1 − p 2 1 p 1 + p 2 ! . (5.71) In the CKP case, (5.57) with C = 0 yields τ =  η 1 η 2 − 1 ( p 1 − p 2 ) 2 − 1 ( p 1 + p 2 ) 2 − 1 4 p 1 p 2  2 − 1 4  η 1 p 2 − η 2 p 1  2 − 1 p 1 p 2 ( p 1 − p 2 ) 2 , (5.72) where η i = η ( t o , p i , c i ) , i = 1 , 2 (see (5.68)). If p 2 = p ∗ 1 (the comple x conjug ate of p 1 ), c 2 = c ∗ 1 , and Re( p 1 ) Im( p 1 ) 6 = 0 , this expr ession is real (see also Appendix C ) and strictly positi ve, and thus determines a reg ular solution to the CKP hierarch y . See also Fig. 3. In the BKP case, we obtain Pf ( A ) = 4 p 1 p 2  η 1 η 2 − 2 p 2 1 + p 2 2 ( p 2 1 − p 2 2 ) 2  , (5.73) where again η i = η ( t o , p i , c i ) , i = 1 , 2 . Choosin g p ∗ 2 = p 1 =: p and c ∗ 2 = c 1 =: c , this takes the form Pf ( A ) = 4 | p | 2  Re( p 2 ) Im( p 2 ) 2 + | η ( t o , p, c ) | 2  , (5.74) which i s s trictly p ositi ve if Re( p 2 ) > 0 , he nce the solution is regular . Writing p = α + i β , the last c onditio n means | α | > | β | . This solutio n appeared in [21] (w ith the opposite inequality | α | < | β | , since our p corres ponds to i p in that work). Fig. 4 shows a plot. The f actor 4 | p | 2 in (5 .74) drops ou t in the pas sage to φ and c an th us be omitted. Setting t 2 n +1 = 0 for n > 2 , the maximum v alue o f u = φ t 1 for the abo ve solutio n is gi ven by u max = 4 Im( p 2 ) 2 / Re( p 2 ) and the maximum mov es, in ‘time’ t 5 , accor ding to t 1 = 5 | p | 4 t 5 + Re( c ) − Re( p 2 ) Im( p 2 ) Im( c ) , t 3 = 10 3 Re( p 2 ) t 5 − Im( c ) 3 Im( p 2 ) . (5.75) The solution has two minima with u min = − I m( p 2 ) / (2Re( p 2 )) , located symmetricall y with respect to the maximum. See also Fig. 4.  Example 5.6 Let L i =     0 p i 0 0 − p i 0 0 0 0 0 0 p ∗ i 0 0 − p ∗ i 0     , V i =     1 1 1 1     i = 1 , 2 . (5.76) 25 Figure 3: Plot of the CKP solutio n φ determine d by (5.72) at t 5 , t 7 , . . . = 0 , with p ∗ 2 = p 1 = 1 + i and c 1 = c 2 = 0 . This conﬁguration simply moves in t he t 1 t 3 -plane with v arying t 5 . Figure 4: A lump solutio n to the BK P equation . Plot of u = φ t 1 at t 5 , t 7 , . . . = 0 for the solutio n gi ven by (5.74) with c = 0 and p = 1 + 9i / 10 . In the precedin g example we hav e seen that these data determine single BKP lumps, and the correspo nding K i are obtaine d from (5.67) and (5.71). The superpos ition condition (5.70) is then solved by K 12 =      − 1 p 1 + p 2 1 p 2 − p 1 − 1 p 1 + p ∗ 2 1 p ∗ 2 − p 1 1 p 1 − p 2 1 p 1 + p 2 1 p 1 − p ∗ 2 1 p 1 + p ∗ 2 − 1 p ∗ 1 + p 2 1 p 2 − p ∗ 1 − 1 p ∗ 1 + p ∗ 2 1 p ∗ 2 − p ∗ 1 1 p ∗ 1 − p 2 1 p ∗ 1 + p 2 1 p ∗ 1 − p ∗ 2 1 p ∗ 1 + p ∗ 2      . (5.77) All this determin es BKP 2-lump solution s via (5.64) and Fig. 5 displays an exampl e.  Example 5.7 Let N = 3 . The general ske w-symmetric 3 × 3 matrix is L =   0 p 1 p 3 − p 1 0 p 2 − p 3 − p 2 0   (5.78) with constan ts p 1 , p 2 , p 3 . W ithout loss of generalit y we may set V ⊺ = (1 , 1 , 1) . F rom tr( L 2 ( Q − I 3 )) = 0 we obtain the constrain t p 1 p 2 − p 1 p 3 − p 2 p 3 = 0 , which we solve for p 3 = p 1 p 2 / ( p 1 + p 2 ) , assuming 26 Figure 5: A 2-lump solution to the BKP equation. Plot of u = φ t 1 at t 5 = − 50 , 0 , 50 (and t 7 , t 9 , . . . = 0 ) for the solution in example 5.6 with C = 0 and p 1 = 1 2 + i 3 , p 2 = 1 3 + i 4 . The two lumps ne ver merge but seem to exc hange their identities at a certain minimal separati on. p 1 + p 2 6 = 0 . The solution to (5.56) is then gi ven by K = k 1 I 3 +    k 2 (1 + p 2 p 1 )( p 1 p 2 − p 2 p 1 ) − 1 p 1 − 1 p 2 k 2 p 2 p 1 − k 2 (1 + p 2 p 1 ) k 2 p 2 p 1 k 2 ( p 1 p 2 + p 1 p 1 + p 2 ) − 1 p 2 k 2 − k 2 (1 + p 2 p 1 ) k 2 0    (5.79) with arbitrary consta nts k 1 , k 2 . In the CKP case, the resulting funct ion τ cannot be real and regula r (since e.g. at t 3 = t 5 = . . . = 0 it is a third o rder pol ynomial in t 1 ). In t he BKP case, it is not really justiﬁed to use (5.63), since it has been deriv ed unde r the conditio n d et( L ) 6 = 0 , b ut here N is odd and thus det( L ) = 0 (becau se L is ske w-symmetric). N e ver theless , (5.63) yields a soluti on, thoug h an uninteresti ng one, since τ = − p 2 with p linear in t 1 , t 3 , . . . . W e sh ould rathe r go back to (5.6 2) and (5.65), b ut it turns out tha t these equati ons cannot both be satisﬁed non-tri vially in the case under conside ration.  6 Conclusions The odd KP syste m studied in this work is a system of two PDE s for two depende nt va riables , φ and θ , taking valu es in any associat i ve (and typically noncommutati ve) algebr a A . W e ha ve sho wn ho w this is embedde d in the K P hierarc hy , if the latter is expre ssed with the help of an auxiliary depend ent v ariable (relate d to θ ). In particu lar , this allo wed to adap t a cons tructio n of exa ct solut ions for the KP hierarchy to the odd K P system (and the corre spond ing hiera rchy). W e furthe r demonstr ated how this can be expl oited to genera te solution s to th e BKP and the C KP equation (and their hierarc hies). In the latter cas es we worked out only compara ti vely simple examples of solution s explicitl y . The general formulae, howe ver , in volv e consta nt matrices of arbitra ry size, with little restrictio ns, and w ith certain choices they may lead to further interes ting solution s. If A is commutati ve, the odd KP system admits re duction s to the BKP and the C KP equat ion. In the n on- commutati ve case, the se reducti ons l ead to se vere ly constra ined e xtensions of these equatio ns. N e ver theless , the y turned out to be helpf ul since they allowed to uncov er some properti es of the commutati ve equation s (see the relatio ns w ith the K dV hiera rchy in sections 2.3 and 2.4) that are hardly recogniza ble without the step into the noncommutati ve realm. Whereas the CK P equation possesse s a natural noncommuta ti ve gen- eraliza tion, though as a system with two depend ent varia bles, nothing comparab le has been found for BKP . W e also cons idered some other reductio ns of the odd KP syste m w ith noncommutati ve A and obtained in particu lar a noncommuta ti ve version of a coupled syste m of Kaup-K upersh midt and Sawada-K otera type. The odd KP syste m, (2.10) and (2.11) with noncommuta ti ve A , and its redu ctions, hav e not been studied pre viously according to our kno wledge. Furthermor e, we present ed dif ferent formulation s of the odd KP hierarchy (with noncommutati ve A ), and deri ved in particular a functiona l repre sentat ion of a linear system for the whole hierarchy . W e ve riﬁed that all these hierarchy formulations possess the odd K P system as their simplest member . Because of 27 the KP hierarc hy origin and the hierarc hy property one then expects the equi valen ce of all these hierarchy formulat ions, b ut a formal proof would nicely complemen t this work. The relation between KP and BKP (CKP) via odd K P sho ws that a subhierar chy can admit a symmetry reduct ion that does not exten d to a symmetry reductio n of the whole hierarchy . This suggests to take a corres pondin g look at other subhierarch ies of KP , and moreo ver s ubhier archies of other hi erarchi es. Besides the odd KP t here is e vidently also an “e ven KP” sub hierarc hy of the KP hier archy . In the GDS formula tion, this means restric ting (2.41 ) to e ven -numbere d varia bles. W e shall report on this else where. A ppendix A: Proof of Theor em 3.1 For the ev aluation of the bilinear identit y (3.1), we will use the residu e formula (which is Lemma 6.3.2 in [66]) res f ( z ) 1 − λz = λ − 1 f < 0 ( λ − 1 ) , (A.1) where f < 0 ( z ) = P + ∞ n =1 f n z − n . In the follo wing, a prime denotes a parti al deri v ati ve with respect to t 1 , hence e.g. φ ′ := φ t 1 . Lemma A.1 The follo wing are conseq uences of the bilinear identit y (3.1). W e ha ve ˜ w 2 = ˜ θ + 1 2 ( φ ′ + φ 2 ) , (A.2) and w 2[ λ ] ( λ − 1 ) ˜ w ( λ − 1 ) = F ( λ ) , (A.3) with F ( λ ) deﬁned in (3.8). Furthermore,  w ′ ( λ − 1 ) + λ − 1 w ( λ − 1 )  2[ λ ] ˜ w ( λ − 1 ) = λ − 1 F ( λ ) 2 − λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) + λ 4 [ φ, φ 2[ λ ] ] , (A.4) µ − 1 w 2[ λ ]+2[ µ ] ( µ − 1 ) ˜ w ( µ − 1 ) − λ − 1 w 2[ λ ]+2[ µ ] ( λ − 1 ) ˜ w ( λ − 1 ) = ( µ − 1 − λ − 1 ) F ( λ, µ ) , (A.5) µ − 1  w ′ ( µ − 1 ) + µ − 1 w ( µ − 1 )  2[ λ ]+2[ µ ] ˜ w ( µ − 1 ) − λ − 1  w ′ ( λ − 1 ) + λ − 1 w ( λ − 1 )  2[ λ ]+2[ µ ] ˜ w ( λ − 1 ) = ( µ − 2 − λ − 2 ) F ( λ, µ ) 2 − 1 2 λ − µ λ + µ ( ˇ θ 2[ λ ]+2[ µ ] − ˇ θ ) + 1 4 λ − µ λ + µ [ φ, φ 2[ λ ]+2[ µ ] ] , (A.6) where ˇ θ := ˜ θ + 1 2 φ ′ , and F ( λ, µ ) := 1 − 1 2 λµ λ + µ ( φ 2[ λ ]+2[ µ ] − φ ) = 1 λ + µ  µ F 2[ µ ] ( λ ) + λ F ( µ )  . (A.7) Pr oof: T ak ing the deri vati ve of (3.4 ) with respe ct to s 1 and then setting s o = t o , leads to 0 = res  w ′ ( z ) ˜ w ( z ) + z w ( z ) ˜ w ( z )  = w ′ 1 + w 2 + w 1 ˜ w 1 + ˜ w 2 . Using (3.5) and (3.6), this becomes (A.2). W ith the help of the identities exp   ± X n ≥ 1 ( λz ) n n   = (1 − λz ) ∓ 1 , hence exp   2 X n ≥ 1 ( λz ) 2 n − 1 2 n − 1   = 1 + λz 1 − λz , 28 (3.4) for s o = t o + 2[ λ ] becomes 0 = res  1 + λz 1 − λz w 2[ λ ] ( z ) ˜ w ( z )  = 2 λ − 1 w 2[ λ ] ( λ − 1 ) ˜ w ( λ − 1 ) − 2 λ − 1 − ( w 1 ) 2[ λ ] − ˜ w 1 , which is (A.3). Next we dif ferentia te (3.4) w ith resp ect to s 1 and then set s o = t o + 2[ λ ] to obtain res  1 + λz 1 − λz  w ′ 2[ λ ] ( z ) ˜ w ( z ) + z w 2[ λ ] ( z ) ˜ w ( z )   = 0 . Elaborate d with th e h elp o f (A.1), and usi ng (3.5), (3.6) an d (A.2), th is res ults in (A.4). Furthermore, setti ng s o = t o + 2[ λ ] + 2[ µ ] in (3.4), we obtain res  1 + z λ 1 − z λ 1 + z µ 1 − z µ w 2[ λ ]+2[ µ ] ( z ) ˜ w ( z )  = 0 . W ith the partial fraction decomposi tion 1 + λz 1 − λz 1 + µz 1 − µz = 1 + 2 λ + µ λ − µ  1 1 − λz − 1 1 − µz  , this results in (A.5). Finally , w e dif ferentiate (3.4) with respect to s 1 , and then set s o = t o + 2[ λ ] + 2[ µ ] to obtain res  1 + z λ 1 − z λ 1 + z µ 1 − z µ  w ′ 2[ λ ]+2[ µ ] ( z ) ˜ w ( z ) + z w 2[ λ ]+2[ µ ] ( z ) ˜ w ( z )   = 0 , which e v aluate s to (A.6).  Pro of of the theore m: W ith the help of (A.3), we can write (A.4) in the form  w ′ ( λ − 1 ) + λ − 1 w ( λ − 1 )  2[ λ ] =  λ − 1 F ( λ ) − λ 2  ˇ θ 2[ λ ] − ˇ θ − 1 2 [ φ, φ 2[ λ ] ]  F ( λ ) − 1  w 2[ λ ] ( λ − 1 ) . No w we apply a Miwa shift with 2[ µ ] and then multiply by ˜ w ( λ − 1 ) from the right to obtain  w ′ ( λ − 1 ) + λ − 1 w ( λ − 1 )  2[ λ ]+2[ µ ] ˜ w ( λ − 1 ) =  λ − 1 F ( λ ) − λ 2  ˇ θ 2[ λ ] − ˇ θ − 1 2 [ φ, φ 2[ λ ] ]  F ( λ ) − 1  2[ µ ] w 2[ λ ]+2[ µ ] ( λ − 1 ) ˜ w ( λ − 1 ) . Inserti ng this in (A.6) leads to µ − 1  w ′ ( µ − 1 ) + µ − 1 w ( µ − 1 )  2[ λ ]+2[ µ ] ˜ w ( µ − 1 ) −  λ − 1 F ( λ ) 2 − λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) + λ 4 [ φ, φ 2[ λ ] ]  2[ µ ] F 2[ µ ] ( λ ) − 1 λ − 1 w 2[ λ ]+2[ µ ] ( λ − 1 ) ˜ w ( λ − 1 ) = λ 2 − µ 2 λ 2 µ 2 F ( λ, µ ) 2 − 1 2 λ − µ λ + µ ( ˇ θ 2[ λ ]+2[ µ ] − ˇ θ ) + 1 4 λ − µ λ + µ [ φ, φ 2[ λ ]+2[ µ ] ] . Next we use (A.5) to el iminate the factor λ − 1 w 2[ λ ]+2[ µ ] ( λ − 1 ) ˜ w ( λ − 1 ) , w ′ 2[ λ ]+2[ µ ] ( µ − 1 ) ˜ w ( µ − 1 ) +  µ − 1 F ( λ ) − λ − 1 F ( λ ) 2 + λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) − λ 4 [ φ, φ 2[ λ ] ]  2[ µ ] F 2[ µ ] ( λ ) − 1 w 2[ λ ]+2[ µ ] ( µ − 1 ) ˜ w ( µ − 1 ) 29 = λ − µ 2  ( ˇ θ 2[ λ ] − ˇ θ ) 2[ µ ] F 2[ µ ] ( λ ) − 1 F ( λ, µ ) − µ λ + µ ( ˇ θ 2[ λ ]+2[ µ ] − ˇ θ )  + µ 4 λ − µ λ + µ [ φ, φ 2[ λ ]+2[ µ ] ] − λ − µ 4 [ φ 2[ µ ] , φ 2[ λ ]+2[ µ ] ] F 2[ µ ] ( λ ) − 1 F ( λ, µ ) + λ − µ λ 2 µ  ( λ + µ ) F ( λ, µ ) − µ F 2[ µ ] ( λ )  F ( λ, µ ) = λ − µ λµ F ( µ ) F ( λ, µ ) + 1 2 λ − µ λ + µ  λ ( ˇ θ 2[ λ ] − ˇ θ ) 2[ µ ] F 2[ µ ] ( λ ) − 1 F ( µ ) − µ ( ˇ θ 2[ µ ] − ˇ θ ) + µ 2 [ φ, φ 2[ λ ]+2[ µ ] ] − λ + µ 2 [ φ 2[ µ ] , φ 2[ λ ]+2[ µ ] ] F 2[ µ ] ( λ ) − 1 F ( λ, µ )  = λ − µ λ + µ  F ( µ )  ( λ − 1 + µ − 1 ) F ( λ, µ ) − µ − 1 F ( µ )  − µ 4 [ φ 2[ µ ] − φ, φ 2[ λ ]+2[ µ ] − φ 2[ µ ] ] + λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) 2[ µ ] F 2[ µ ] ( λ ) − 1 F ( µ ) − λ 4 [ φ 2[ µ ] , φ 2[ λ ]+2[ µ ] ] F 2[ µ ] ( λ ) − 1 F ( µ ) +[ w ′ 2[ µ ] ( µ − 1 ) + µ − 1 w 2[ µ ] ( µ − 1 )] ˜ w ( µ − 1 )  = λ − µ λ + µ  1 λ F ( λ ) + λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) F ( λ ) − 1 − λ 4 [ φ, φ 2[ λ ] ] F ( λ ) − 1  2[ µ ] F ( µ ) + λ − µ λ + µ  w ′ ( µ − 1 ) + µ − 1 w ( µ − 1 )  2[ µ ] ˜ w ( µ − 1 ) , taking accoun t of (A.7), (A.4), ( λ − 1 + µ − 1 ) F ( λ, µ ) − µ − 1 F ( µ ) = λ − 1 F 2[ µ ] ( λ ) , and [ F ( µ ) , F 2[ µ ] ( λ )] = λµ 4 [ φ 2[ µ ] − φ, φ 2[ λ ]+2[ µ ] − φ 2[ µ ] ] . No w we use (A.3) t o replace the f actor F ( µ ) , di vide by ˜ w ( µ − 1 ) , and then a pply a Miwa sh ift with − 2[ µ ] to obtain λ + µ λ − µ h w ′ 2[ λ ] ( µ − 1 ) +  µ − 1 F ( λ ) − λ − 1 F ( λ ) 2 + λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) − λ 4 [ φ, φ 2[ λ ] ]  F ( λ ) − 1 w 2[ λ ] ( µ − 1 ) i = w ′ ( µ − 1 ) + µ − 1 w ( µ − 1 ) +  1 λ F ( λ ) + λ 2 ( ˇ θ 2[ λ ] − ˇ θ ) F ( λ ) − 1 − λ 4 [ φ, φ 2[ λ ] ] F ( λ ) − 1  w ( µ − 1 ) . Setting µ = z − 1 , after some rearra ngements this takes the form 1 + λz 1 − λz  w ′ 2[ λ ] ( z ) + z w 2[ λ ] ( z )  + w ′ ( z ) + z w ( z ) − 1 λ F ( λ )  1 + λz 1 − λz w 2[ λ ] ( z ) − w ( z )  + λ 2  ˇ θ 2[ λ ] − ˇ θ − 1 2 [ φ, φ 2[ λ ] ]  F ( λ ) − 1  1 + λz 1 − λz w 2[ λ ] ( z ) + w ( z )  = 0 . Multiply ing by e ˜ ξ ( t o ,z ) and using ψ 2[ λ ] = w 2[ λ ] ( z ) 1+ λz 1 − λz e ˜ ξ ( t o ,z ) , we arri ve at (3.7).  A ppendix B: A determinant identity Accordin g to (2.90) in [47], we ha ve det  z V ⊺ − V A  = det( A ) z + N X i,j =1 ∆ i,j v i v j , (B.1) 30 where A is an N × N m atrix, ∆ i,j is the cofactor w ith respect to the component A i,j of A , z a parameter , a nd v i , i = 1 , . . . , N , are the components of a vector V . If N is odd and A ske w-symmetric, then d et( A ) = 0 and thus det  z V ⊺ − V A  = N X i,j =1 ∆ i,j v i v j , (B.2) which is thus indepen dent of z . Since det  1 V ⊺ − V A  = det  1 V ⊺ 0 A + V V ⊺  = det( A + V V ⊺ ) , (B.3) we obtain det( A + V V ⊺ ) = det  0 V ⊺ − V A  , (B.4) which is the determin ant of a ske w-symmetric matrix, and thus the square of the Pfafﬁan of th is matrix. A ppendix C: Reality conditions In order to obtain r eal solutions to the BKP or CKP hierarchy from the matrix linea r system in sectio n 5 with comple x matrices, a reality condition is needed. Pro position C.1 Let T be a constant in vertib le N × N matrix with the propert ies T ∗ = T ⊺ = T − 1 (C.1) (where T ∗ denote s the complex conjugate of T ). Let C , K, L be constant comple x N × N matrices and V an N - vect or satisfyin g C ∗ = T C T − 1 , K ∗ = T K T − 1 , L ∗ = T LT − 1 , V ∗ = T V . (C.2) The function τ gi ven b y (5.35),(5.41), (5.58) or (5.64) in terms o f ( C , K, L, V ) (subjec t to the correspon ding rank one condi tion (5.34 ) or (5.56), and C ⊺ = C , resp ecti vely C ⊺ = − C ), is the n r eal . Pr oof: T he assertion is easily ver iﬁed. (C .1) ensures the compatibilit y of (C.2) with C ⊺ = ± C , (5.34 ) and (5.56).  If N = 2 n , then T =  0 I n I n 0  , (C.3) where I n is the n × n unit matrix, satisﬁes the conditio ns (C.1). 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BKP and CKP revisited: The odd KP system

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