Interpolating Dispersionless Integrable System

D AMTP-2008-27 In terp olating Disp ersionless In tegrable System Maciej Duna jski ∗ Department of Applied Mathema tics and T heoreti cal Ph ysi cs Universit y of Cambridge Wilb er force Ro ad, Cambridge CB3 0W A, UK Abstract W e in tr o duce a disp ers ionless in tegrable system which in terp olates b et wee n the d isp er- sionless Kadom tsev–Pe tviash vili equation and the h yp er–CR equation. T he in terp olating system arises as a sym m etry reduction of the an ti–self–dual Eins tein equations in (2, 2) signature b y a conformal Killing v ector whose self–dual deriv ativ e is null. It also arises as a sp ecial case of the Manak o v–Santini inte grable system. W e discuss the corresp onding Einstein–W eyl structures. ∗ email m.duna jski@damtp.cam.ac.uk 1 1 In tro duct ion It has b een kno wn for more than 20 y ears that man y in tegrable systems admitting soliton solutions arise as symmetry reductions of anti–se lf–dual Y ang Mills (ASDYM) equations in four dimensions [25]. The Riemann–Hilb ert factor isatio n problem underlies this a ppro ac h to in tegrability: it a pp ears in classical solution generating tec hniques lik e dressing transformations [18], as w ell as in t he t wistor treatmen t o f ASDYM [24]. The disp ersionless in tegrable systems in 2+ 1 dimensions do not fit in to this framework: they do not admit soliton solutions and there is no asso ciated Riemann–Hilb ert problem where the corresp onding Lie group is finite dimensional. These systems can neve rtheless b e describ ed in terms of a n ti–self–duality (ASD) conditions on a four–dimensional conformal structure. In this case the ‘unkno wn’ in the equations is not a gauge field, but rather a metric (up to scale) on some four–manifold [2 1 ]. This make s the disp ersionless systems more geometric tha n their solitonic cousins. This p oint of view ma y b e of deep significance in description of sho c k formations: a recen t b eautiful analysis of Manak o v and Santini [16 ] deduced the gradien t catastrophe of the lo calised solutio ns to the dKP equation using the in ve rse scattering transform. It may b e ho wev er, that this catastrophe is o nly an artifact of a c ho sen co ordinate system, and the underlying conformal structure is regular, but needs to b e co vered b y more than one co ordinate patc h. It remains to b e seen whether this is indeed the case. T o classify existing 2+1 disp ersionless inte grable systems, and p erhaps discov er some new ones one needs to classify the symmetry reductions of the conformal ASD equations. If the Ricci–flat condition is imp osed on top of the an ti–self–dualit y , the w ork of Pleba ´ nski [22] implies the existence of a lo cal coor dinate system ( X, Y , W , Z ) and a function Θ on an o p en set M ⊂ R 4 suc h than any ASD Ricci–flat metric is lo cally of the form g = 2(d Z d Y + d W d X − Θ X X d Z 2 − Θ Y Y d W 2 + 2Θ X Y d W d Z ) , (1.1) where Θ X Y = ∂ X ∂ Y Θ etc, and Θ satisfies the second hea venly equation Θ Z Y + Θ W X + Θ X X Θ Y Y − Θ 2 X Y = 0 . (1.2) This metric has signature (+ + −− ) but this is precisely what need: Giv en a non–null symmetry , the conforma l structure on a space of orbits will ha ve signature (2 , 1) and will b e describ ed by a hy p erb olic in tegrable equation. W e are thus led t o study ( 1 .1), (1.2) sub j ect to the existence of a conformal Killing v ector K L K g = cg , g ( K, K ) 6 = 0 where c is some function. The classification of reductions is based on studying the action of K on the bundle of self–dual t wo–forms o ver M . Assume t ha t M is oriented and recall that giv en a metric o f (+ + −− ) signature on M , the Ho dge ∗ op erator is an inv olution o n t wo–forms and induces a decomp osition Λ 2 = Λ 2 + ⊕ Λ 2 − of t w o–fo rms in t o self–dual a nd anti–se lf–dual comp onen ts. Moreo ve r there exist real tw o- dimensional v ector bundles S and S ′ (called spin bundles) ov er M suc h that T M ∼ = S ⊗ S ′ and Λ 2 + ∼ = S ⊙ S . Therefore the self–dual deriv ativ e of ( t he one–form metric dual to) K d K + := 1 2 (d K + ∗ d K ) 2 corresp onds to a symmetric 2 b y 2 matrix φ explicitly giv en b y d K + = φ AB Σ AB , A, B = 0 , 1 , where the self–dual t wo forms 1 (Σ 00 , Σ 01 , Σ 11 ) span Λ 2 + . The r a nk of the matrix φ do es not dep end on the c hoice of the basis Σ AB and the classification is based on this rank. In the Riemannian signature this can only b e 0 or 2 (the former case is called the tri–holomorphic symmetry), but in the (+ + −− ) signature w e can also hav e rank( φ ) = 1, in whic h case (d K ) + ∧ (d K ) + = 0 so the self–dual deriv ativ e of K is null. This classification pro gramme has almost b een completed and one aim o f this pap er is to remo ve the question mark from the follow ing table summarising the reductions of the second hea ve nly equation (1.2) c = 0 c 6 = 0 rank( φ ) = 0 2+1 Linear w a ve equation [11, 9] Hyp er–CR equation [3, 4] rank( φ ) = 1 Disp ersionless KP equation [5] ? rank( φ ) = 2 SU( ∞ ) T o da equation [9] Inte grable equation studied in [6] The case where rank( φ ) = 1 and c 6 = 0 has not y et b een inv estigated, and the resulting in tegrable system is the sub ject of the presen t pap er. This system is giv en b y u y + w x = 0 , u t + w y − c ( uw x − w u x ) + buu x = 0 , (1.3) where b and c are constan ts and u, w a re smo oth functions of ( x, y , t ). W e prop ose to call (1 .3) an interpola t ing system as it contains t w o well known disp ersionless equations as the limiting cases: Setting b = 0 , c = − 1 g iv es the hyper–CR equation [3, 20, 8, 4, 17 , 14, 19] and setting c = 0 , b = 1 giv es the dKP equation. In fact one constan t can a lwa ys b e eliminated f rom (1.3) b y redefining the co ordinates and it is only the ra tio of b/c whic h remains 2 . W e prefer to k eep b oth constan t s as it mak es v arious limits more transparen t. In the next Section w e shall g iv e the disp ersionless Lax pair f or (1.3). The Lie group underlying the Lax f orm ulat io n is D iff (Σ 2 ) – an infinite–dimensional gro up of diffeomorphisms of some t wo–dimen sional manifold Σ 2 . T he interpolating system correspo nds to Lo r entzian Einstein–W eyl structures in thr ee dimensions. This giv es an in trinsic geometric in terpretation without the need of going to four–dimensions. This will b e describ ed in Section 3. In Section 4 w e shall sho w that (1.3) is a sp ecial case of the Manak ov–San tini in tegrable system [15, 14] and find the Manako v–San tini Einstein–W eyl structure. The explicit reduction of the second heav enly equation (1.2) to the interpolating system (1.3) will b e presen ted in the App endix. In particular w e shall show tha t t he most general 1 The ASD Ricci–flat equations are equiv alent to dΣ AB = 0 , Σ ( AB ∧ Σ C D ) = 0 , A, B , C , D = 0 , 1 and the seco nd heavenly equation (1 .2) arises by using the Darb oux theorem to in tro duce co o rdinates such that Σ 00 = d W ∧ d Z , Σ 01 = d W ∧ d X + d Z ∧ d Y , and deducing the existence of the function Θ from the remaining conditions on Σ AB . 2 Jeny a F er a p onto v has po int ed out that if b 6 = 0 the hydrody na mic reductions of (1.3) coincide with the hydrodynamic reductions of the dKP equation, despite the fa c t that dKP and (1.3) are not p oint or c o ntact equiv alent unless c = 0. 3 (+ + −− ) ASD Ricci flat metric with a conformal Killing ve ctor whose self–dual deriv ativ e is n ull is of the for m g = e cφ ( V h − V − 1 (d φ + A ) 2 ) , (1.4) where φ parametrises the or bit s of the conformal Killing vec tor K = ∂ /∂ φ and h = (d y + cu d t ) 2 − 4(d x + cu d y − ( cw + bu )d t )d t, A = − 1 2 u x d y +  c 2 uu x − u y  d t, V = 1 2 u x . This r eduction explains the origin of the tw o parameters ( b, c ) in (1.3 ) as in this case the conformal symmetry is K = c × (dilatation) + b × (rot a tion with n ull SD deriv ativ e). 2 Lax pair and Diff (Σ 2 ) h i e rarc hies The system (1.3) admits a disp ersionless Lax pa ir L 0 = ∂ ∂ t + ( cw + bu − λ cu − λ 2 ) ∂ ∂ x + b ( w x − λu x ) ∂ ∂ λ , L 1 = ∂ ∂ y − ( cu + λ ) ∂ ∂ x − bu x ∂ ∂ λ (2.1) with a sp ectral parameter λ ∈ C P 1 . The ov erdetermined system of linear equations L 0 Φ = L 1 Φ = 0, where Φ = Φ( x, y , t, λ ) admits solutions b ecause equations (1.3) are equiv alen t to [ L 0 , L 1 ] = 0. In general, consider the v ector fields of the fo r m L i = ∂ ∂ t i + A i ∂ ∂ x + B i ∂ ∂ λ , (2.2) where A i , B i are p olynomials in λ with co efficien ts dep ending on ( t 0 = x, t i ). The flo ws o f the Diff(Σ 2 ) hierarc h y are defined by [ L i , L j ] = 0 , i, j = 1 , . . . , n. (2.3) T o ach iev e a dual formulation, generalising Kric heve r’s approach to disp ersionless in tegr a ble systems [13 ], complexify the hierarc hy (so that ( t 0 , t i ) ∈ C n +1 ) a nd define a t wo–form Ω on C n +1 × CP 1 b y Ω( X, Y ) = d t 1 ∧ . . . ∧ d t n ∧ d x ∧ d λ ( L 1 , . . . , L n , X , Y ) so that Ω = d x ∧ d λ + X i ( A i d λ − B i d x ) ∧ d t i + X i,j ( A i B j − B j A i )d t i ∧ d t j . The tw o–form Ω is simple and satisfies the F rob enius integrabilit y conditions Ω ∧ Ω = 0 , dΩ = Ω ∧ β for some one–form β . W e reco ver v arious disp ersionless hierarc hies as special cases of this form ula tion 4 • SDiff (Σ 2 ) hierarc hy . The group Diff(Σ 2 ) r educes to SDiff(Σ 2 ) g enerated by Hamiltonian v ector fields. The corresp onding Lie algebra is ho momorphic with the P o isson brack et algebra. The v ector fields L i preserv e the t w o–f orm d x ∧ d λ and A i = ∂ H i ∂ λ , B i = − ∂ H i ∂ x . The tw o–form is give n b y Ω = d x ∧ d λ + P i d H i ∧ d t i (where we ha ve used ( 2.3)), and the SDiff(Σ 2 ) hierarc h y is giv en by Ω ∧ Ω = 0 , dΩ = 0 , whic h is the orig ina l Kric hev er’s form ula tion [13]. The Darb oux theorem implies the existence of functions P , Q suc h tha t Ω = d P ∧ d Q . These tw o functions a re lo cal co ordinates on the t wistor space, whic h is a quotien t of C n +1 × CP 1 b y the in tegrable distribution { L i } . Both the dKP and SU( ∞ ) T o da hierarchie s fit in to this category [13, 23, 5]. • Diff ( S 1 ) hierarc h y . The group Diff(Σ 2 ) reduces to Diff( S 1 ), where Σ 2 = T S 1 . This case corresp onds to B i = 0 . The underlying Lie algebra is that of a W ronskian with a Lie brac ke t < f , g > = f x g − f g x . In t his case Ω = e ∧ d λ , where e = d x − P i A i d t i . This one–form is integrable in the F rob enius sense e ∧ d e = 0 , where here d kee ps λ =const. The t wistor space fibres holo morphically ov er CP 1 . T he h yp er-CR hierarc hy [4] and the univers al hierar c hy [17] are of this type. 3 In terp olati n g Einstei n –W eyl st ructure A three–dimensional Loren tzian W eyl structure ( M , D , [ h ]) consists o f a 3–manifold M , a torsion-free connection D a nd a conf o rmal metric [ h ] of Lorentz ian signature suc h tha t the n ull g eo desics of [ h ] are also geo desics fo r D . This condition is equiv alen t to D h = ω ⊗ h for some one form ω . Here h is a represen tativ e metric in the confo r mal class. If w e c hange this represen tativ e by h − → γ 2 h , then ω − → ω + 2 d ln γ . A W eyl structure is called Einstein–W eyl if the conformally inv a rian t equations R ( ab ) = Λ h ab , a, b, = 1 , . . . , 3 (3.1) hold for some function Λ. Here R ( ab ) is the symmetrised Ricci tensor of D and h ab is a repre- sen tativ e metric in a conformal class [ h ]. In practice the Einstein–W eyl structure is giv en b y sp ecifying the metric h ∈ [ h ], and the one–form ω whic h measures the difference b etw een the W eyl connection D and the Levi–Civita connection of h . 5 The t hr ee–dimensional Einstein–W eyl condition is a disp ersionless in tegrable system [5]: Let Z , W, f W b e indep enden t v ector fields on M suc h tha t a con t r a v ariant metric in [ h ] is h = h ab ∂ ∂ x a ⊗ ∂ ∂ x b = Z ⊗ Z − 2( W ⊗ f W + f W ⊗ W ) . Then there exists a connection D suc h that ( M , [ h ] , D ) is Einstein–W eyl if the disp ersionless Lax pair L 0 = W − λZ + f 0 ∂ ∂ λ , L 1 = Z − λ f W + f 1 ∂ ∂ λ , (3.2) satisfies the in tegrabilit y condition [ L 0 , L 1 ] = 0 mo dulo L 0 , L 1 for some functions ( f 0 , f 1 ) whic h are cubic p olynomials in λ ∈ CP 1 . Conv ersely , eve ry Einstein– W eyl structure arises from some Lax pair (3.2). The corr espo nding o ne form ω can b e read off from the Levi–Civita connection of h and the co efficien ts of ( f 0 , f 1 ). This Lax form ulation has a geometric origin whic h go es bac k t o E. Cartan [2] • Einstein–W eyl condition is equiv alen t to the existe nce of a t w o para meter family of totally geo desic n ull surfaces in M . Comparing this with the L a x pa ir ( 2 .1) fo r the in terp ola ting system, and ta king linear com- binations to put (2.1) in the form (3.2) w e find the corresp onding Einstein–W eyl structure to b e h = (d y + cu d t ) 2 − 4(d x + cu d y − ( cw + bu )d t )d t, (3.3) ω = − cu x d y + (4 bu x + c 2 uu x − 2 cu y )d t. T aking the limits w e reco ve r v arious kno wn Einstein–W eyl structures fro m (3.3 ). These struc- tures can b e c haracterised by the prop erties null shear- f ree and geo desic congruence d t (this elegan t framew ork is describ ed, in the Riemannian case, in [1]) • Setting b = 0 , c = − 1 gives the hy p er–CR Einstein–W eyl structure [4]. The congruence is dive rgence–free. This is a Lorentzian analo gue Einstein–W eyl structures studied in [10, 1, 6]. • Setting c = 0 , b = 1 giv es t he dKP Einstein–W eyl spaces [5 ]. The congruence d t is now t wist-free, a nd the dual v ector ∂ /∂ x is parallel with a w eigh t − 1 / 2 with resp ect to the W eyl connection. W e remark that the Einstein–W eyl (3.3) structure can also b e read off from the ASD Ricci–flat metric (1.4) using the Jones–T o d corresp ondence [12]. 6 4 The Manak o v–San tin i s ystem The system (1.3) is a sp ecial case of the Mana ko v–San t ini system [15, 14] U xt − U y y + ( U U x ) x + V x U xy − V y U xx = 0 (4.1) V xt − V y y + U V xx + V x V xy − V y V xx = 0 , where U = U ( x, y , t ) , V = V ( x, y , t ). T o see it, notice that the first equation in (1.3) implies the existence of v ( x, y ) suc h that u = v x , w = − v y , and v satisfies v xt − v y y + c ( v x v xy − v y v xx ) + buv xx = 0 . (4.2) Differen tiat ing the second equation in (1.3), and eliminating w yields u xt − u y y − c ( v y u xx − v x u xy ) + b ( uu x ) x = 0 . (4.3) No w assume the generic case when the constan ts c, b are non zero, and set U = bu, V = cv . Then the system (4.2) and (4.3) is equiv alen t to (4.1) with a n a dditio nal constrain t cU − bV x = 0 . (4.4) The Mana ko v–San t ini system is more general than (1.3) : Regarding the second equation in (4.1) as the definition of U , a nd substituting U t o the first equation in (4 .1) yields a f o urth order scalar PDE for V . Thu s the naiv e counting suggest that the general solutio n to (4.1) dep ends on 4 functions of 2 v ariables (a caution is needed as the r esulting PDE is not in the Cauc h y–Ko w alewsk a form). In the sp ecial case when the constrain t (4.4) holds b oth equations in (4.1 ) r educe to a single second order PDE fo r V . The solution dep ends o n t w o functions o f t wo v ariables, and a constan t (the ratio b/c ). The Manak o v–Santini system also corresp onds to an Einstein–W eyl structure h = (d y − V x d t ) 2 − 4(d x − ( U − V y )d t )d t, (4.5) ω = − V xx d y + (4 U x − 2 V xy + V x V xx )d t. T o v erify it set x a = ( y , x, t ). The (11 ) , (12) , (22) , (2 3) comp onen t s of the Einstein–W eyl equa- tions hold iden tically . The (13) comp onen t v anishes if the second equation in (4.1 ) holds, and finally the (33) comp onent v anishes if b oth equations in (4 . 1 ) are satisfied. Con ve rsely , consider the general conformal structure in (2 + 1) dimensions given in lo cal co ordinates x a b y a represen ta tiv e metric h =   h 11 h 12 h 13 h 12 h 22 h 23 h 13 h 23 h 33   . Using the diffeomorphism freedom, a nd the conformal rescaling w e can imp ose fo ur constraint on six functions h ab ( x c ) a s long as the resulting quadratic fo rm is non–degenerate. W e c ho ose to set h 11 = h 12 = 0 , h 13 = − 2 , h 23 = − A, h 33 = A 2 + 4 B , where A and B are some functions of ( x, y , t ), so that 3 h = (d y − A d t ) 2 − 4(d x − B d t )d t. 3 This is a nalogous to the existence of or thogonal co o rdinates in three dimensio ns. The pro of is relatively straightforward in the real– analytic category , and more subtle in the smo oth categor y . 7 No w given A, B we can alw ays find t wo functions U, V such that A = V x , B = U − V y so that the metric is in the fo r m (4.5) No w we find the corresp onding dual basis, and construct the Lax pair (3.2). Before imp o sing the in tegra bility conditions it is conv enien t to ta k e a linear combination of the v ectors in this distribution, so t ha t t he resulting pair of ve ctors commute s exactly . This yields L 0 = ∂ y − ( λ + A ) ∂ x + f 0 ∂ λ , L 1 = ∂ t − ( λ 2 + λA − B ) ∂ x + f 1 ∂ λ , where the p olynomials f 0 and f 1 are resp ectiv ely cubic and quartic in λ . There is some addi- tional freedom which will preserv e the ab ov e fo r m of the Lax pa ir . W e can translate the fibres b y λ → λ + κ ( x, y , t ). W e use this freedom to set the linear term in f 0 to zero. It is p o ssible that some further co ordinate freedom can used to set t he quadratic term in f 0 to zero whic h would imply that ω is giv en by (4.5). This w ould imply that that ev ery Einstein–W eyl structure is equiv alen t to the Manak ov–San tini EW structure (4.5). So far w e hav e b een unable to find the righ t tr ansformation, and w e need to imp ose ∂ 2 λ f 0 = 0 as an a dditio nal condition. Then the in tegrability condition [ L 0 , L 1 ] = 0 implies (4.1). Ac kno wledge men ts I thank D a vid Calderbank, Jen ya F eraponto v and Luis Mart ´ ınez–Alonso for useful comments . This w ork is partly supp ort ed b y the Europ ean net work Metho ds o f In tegrable Systems, G e- ometry and Applied Mathematics (MISGAM). App endix. Reduc tion of the s e cond hea venly equatio n W e shall pro ve that the system (1.3) arises as the most general symmetry reduction of the second hea venly equation (1.2) by a confo r ma l Killing v ector with null self-dual deriv ativ e. Let Θ = Θ( W, Z , X , Y ) satisfy the second heav enly equation and let the corresp onding metric b e giv en by (1 .1). Let K b e a conformal Killing v ector for (1.1). Using P enrose’s tw o- comp onen t spinor f o rmalism we can sho w t hat the confor mal Killing equations a nd the Ricci iden tity imply that ∇ AA ′ K A B ′ is co v arian tly constan t, or ot herwise g is o f P etrov–P enrose ty p e N an can b e found explicitly . In the spin frame of the hea v enly metric (1.1 ) the connection on the spin bundle S v anishes, so ∇ A ′ A K A ′ B is in fact constan t. W e are in terested in the case where the self–dual deriv ative φ AB = ∇ A ′ ( A K A ′ B ) is of rank one. Therefore w e need to integrate the linear system ∇ A ′ A K A ′ B =  0 c − c b  . The constan t c app ears b ecause K is a conformal Killing v ector. F ollow ing the metho d of Finley and Pleba ´ nski [9] and using a freedom in the hea ve nly p oten tial Θ as we ll a s in the c hoice of co ordinates w e find t he general solution to b e K = ( cZ + b ) ∂ Z + ( cX − 2 bZ ) ∂ X . The conformal Killing equations L K g = cg no w yield L K (Θ X X ) = − c Θ X X , L K (Θ X Y ) = b, L K (Θ Y Y ) = c Θ X X . 8 Let U and T b e functions suc h that K = ∂ T and L K ( U ) = 0. F o r c 6 = 0 w e can take T = ln ( cZ + b ) c , U = 2 b c T + 2 b 2 + X c 2 c 2 ( cZ + b ) . The compatibilit y conditions for the Killing equations imply the existence of ˜ G = ˜ G ( Y , W, U ) suc h that Θ X X = e − cT ˜ G U U , Θ X Y = ˜ G Y U + bT , Θ Y Y = e cT ˜ G Y Y . The hea v enly equation (1.2) b ecomes bU + 2 b c ˜ G Y U + c ( ˜ G Y − U ˜ G Y U ) + ˜ G U W + ˜ G Y Y ˜ G U U − ˜ G 2 Y U = 0 . T o obtain a simplified form define G ( Y , W, U ) = ˜ G ( Y , W, U ) + b c U Y + b 2 c 2 U W so that bU + c ( G Y − U G Y U ) + G U W + G Y Y G U U − G 2 Y U = 0 . (A1) No w r ewrite (A1) in terms of differen tial forms bU d Y ∧ d U ∧ d W + c ( G Y d Y ∧ d U ∧ d W − U d G U ∧ d U ∧ d W ) +d G U ∧ d Y ∧ d U + d G Y ∧ d G U ∧ d W = 0 . (A2) Define x = G U , y = Y , t = − W , H ( x, y , t ) = xU ( x, y , t ) − G ( Y , W , U ( x, y , t )) , and preform a Legendre tra nsform d H = d( xU − G ) = U d x − G Y d Y − G W d W = H x d x + H y d y + H t d t. Therefore U = H x , G Y = − H y , G W = H t . Differen tiat ing these relations we find G U U = 1 H xx , G Y U = − H xy H xx , G Y Y = − H y y + H 2 xy H xx . The differen tia l equation f or H ( x, y , t ) is obtained fro m (A2) H xt + bH x H xx + c ( H xy H x − H y H xx ) = H y y . (A3) Setting u = H x , w = − H y w e reco ve r ( 1.4), where ( u, w ) solv e (1.3). 9 References [1] Calderbank, D. M. J. P edersen, H. (2000) Selfdual spaces with complex structures, Einstein-W eyl geometry and geo desics. Ann. Inst. F ourier (Grenoble) 50 , 9 21–963. [2] Carta n, E. 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