Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction

Generalized dKP: Manak o v-San tini hierarc h y and its w aterbag reduction L.V. Bogdanov ∗ , Jen-Hsu Chang ‡ † and Y u-T ung Chen † Octob er 2 5, 201 8 Abstract W e study Ma nak o v-Santini equation, starting from Lax-Sato form of as- so cia ted hierarc h y . The w aterbag r e duction for M anak o v-Santini hierarc h y is in tro duced. Equations of reduced hierarc h y are deriv ed. W e construct new co o rdinates transforming non-h ydro dynamic ev olution of w aterbag reduction to non -homogeneous Riemann inv aria n ts form of hydro dyn amic t yp e. Keyw ords: Manak o v-Santini hierarch y , L ax r epresen tation, W a terbag redu c- tion, Non-homogeneous sy stems of hydro dynamic t yp e, Riemann inv a rian ts P ACS: 02.30.Ik 1 In tro d uction In this pap er w e study a n integrable system in tro duced recen tly by Manak ov and San tini [1] (see also [2, 3]). This system is connected with commutation of general 2- dimensional v ector fields (con taining deriv ativ e on sp ectral v ariable). Reduction to Hamiltonian vec tor fields leads to the w ell-know n disp ersionless KP (or Khokhlo v- Zab olo t sk ay a) equation. Alternativ ely , a natural reduction to 1-dimensional v ector fields reduces Manak ov-San tini system to the equation intro duced b y Pa vlo v [4] (see also [5, 6, 7 ]). Using general construction of the w orks [8, 11], w e in tro duce the hierarc hy fo r Manak o v-Sa n t ini system in Lax-Sato form and generating equation for it (the hierarc h y in terms of recursion op erator was in tro duced in [2]). W e in tro duce w aterbag ansatz for Manak o v-San tini hierar ch y and deriv e equations of the reduced hierarc hy . Using ratio nal form of the G function (see b elow), one can intro duce new co ordinates suc h that the non-hydrodynamic ev o lution of w aterbag reduction ∗ L.D. Landau ITP , Kosygin str. 2, Moscow 119334, Russia † Department o f Computer Science, Nationa l Defense U niversit y , T a oyuan, T aiw a n, E- mail: jhc hang@ndu.edu.tw, ‡ cor resp onding author . 1 transforms t o non- ho mogeneous Riemann inv arian ts form of hydro-dynamic ty p e. This pap er is organized as follows . In section 2, GdKP hierarc h y is describ ed, connection to Manako v-Santini system is demonstrated. In section 3, waterbag re- duction for Manak ov-San tini hierarc h y is introduced, equations of reduced hierar ch y are deriv ed (in non-hydrodynamic f orm). In se ction 4, w e in tro duce new co ordinates transforming the ev olution of w aterbag reduction to non-homogeneous Riemann in- v arian ts for m of hydro-dynamic type. The examples are giv en. Section 5 is dev oted to the concluding remarks. 2 Generalize d dKP hierarc h y T o in tro duce generalize d dKP (Manak ov-San tini) hierarc hy , w e use general construc- tion of the w orks [8, 11]. The hierarc hy is described b y the Lax-Sato equations ∂ ψ ∂ t n = A n ∂ ψ ∂ x − B n ∂ ψ ∂ p , ψ =  L M  , (1) or, equiv alen tly , b y the generating equation ( J − 1 0 d L ∧ d M ) − = 0 , (2) where A n ≡ ( J − 1 0 ∂ L n /∂ p ) + , B n ≡ ( J − 1 0 ∂ L n /∂ x ) + with the Lax a nd O rlo v op erat o rs L ( p ) , M ( p ) b eing the Lauren t series L = p + ∞ X n =1 u n ( x ) p − n , (3) M = ∞ X n =1 nt n L n − 1 + ∞ X n =1 v n ( x ) L − n . (4) Here ( · · · ) + (( · · · ) − ) denote resp ectiv ely the pro jection on the p olynomial part ( neg- ativ e pow ers), and J 0 is defined b y J 0 = ∂ L ∂ p ∂ M ∂ x − ∂ L ∂ x ∂ M ∂ p = ∂ L ∂ p ∂ M ∂ L     t n ,v n fixed ∂ L ∂ x + ∂ M ∂ x     L fixed ! − ∂ L ∂ x ∂ M ∂ L     t n ,v n fixed ∂ L ∂ p = ∂ L ∂ p ∂ M ∂ x     L fixed = 1 + v 1 x p − 1 + ( v 2 x − u 1 ) p − 2 + · · · . W e list some o f A n and B n as follows A 1 = 1 , 2 A 2 = 2 p − 2 v 1 x , A 3 = 3 p 2 − 3 v 1 x p + 6 u 1 + 3( v 1 x ) 2 − 3 v 2 x , A 4 = 4 p 3 − 4 v 1 x p 2 + (12 u 1 + 4( v 1 x ) 2 − 4 v 2 x ) p +12 u 2 − 4 v 3 x + 8 v 1 x v 2 x − 4( v 1 x ) 3 − 8 u 1 v 1 x , (5) and B 1 = 0 , B 2 = 2 u 1 x , B 3 = 3 u 1 x p − 3 u 1 x v 1 x + 3 u 2 x , B 4 = 4 u 1 x p 2 + (4 u 2 x − 4 u 1 x v 1 x ) p +4 u 1 x (4 u 1 + ( v 1 x ) 2 − v 2 x ) − 4 u 2 x v 1 x + 4 u 3 x . (6) The t 1 flo w of the generalized dKP hierarc h y (1) say s that the dep endence on t 1 and x app ear in the linear com bination t 1 + x . Prop osition 2.1 The c omp atibility of the c ommuting flow [ ∂ t m , ∂ t n ] ψ = 0 r e quir es A n , B n to satisfy ∂ t m A n − ∂ t n A m = h A m , A n i x + B n A mp − B m A np , ∂ t m B n − ∂ t n B m = h B n , B m i p + A m B nx − A n B mx , (7) wher e h U, V i i := U ( ∂ i V ) − ( ∂ i U ) V . Pr o of. Substituting (1) in to ∂ t m ∂ t n ψ = ∂ t n ∂ t m ψ , and comparing the co efficien ts of indep enden t v ariables ψ x and ψ p resp ectiv ely to the b oth sides, w e obtain (7).  The ev olution o f L , M with resp ect to t 2 = y in (1) are giv en by 1 2 ∂ L ∂ y = ( p − v 1 x ) ∂ L ∂ x − u 1 x ∂ L ∂ p , (8) 1 2 ∂ M ∂ y = ( p − v 1 x ) ∂ M ∂ x − u 1 x ∂ M ∂ p . (9) Using the con ven tion ( P n a n p n ) [ s ] = a s for a formal Lauren t series, then from Eq.(8) w e hav e 1 2 u 1 y =  ( p − v 1 x ) L x − u 1 x L p  [ − 1] = u 2 x − v 1 x u 1 x , (10) 1 2 u 2 y =  ( p − v 1 x ) L x − u 1 x L p  [ − 2] = u 3 x − v 1 x u 2 x + u 1 u 1 x , (11) On the other hand, t he expression of Eq.(9) tog ether with (8) giv es L + 1 2 ∞ X n =1 v ny L − n = ( p − v 1 x ) 1 + ∞ X n =1 v nx L − n ! . 3 Comparing the co efficien ts o f p o w ers p − 1 and p − 2 to the ab ov e, w e ha v e v 2 x = u 1 + v 2 1 x + 1 2 v 1 y , (12) v 3 x = u 2 + 1 2 v 2 y + u 1 v 1 x + v 1 x v 2 x . (13) Similarly , the ev o lution of L , M w.r.t. t 3 = t are g iv en by 1 3 ∂ L ∂ t =  p 2 − v 1 x p + u 1 − 1 2 v 1 y  ∂ L ∂ x −  u 1 x p + 1 2 u 1 y  ∂ L ∂ p , (14) 1 3 ∂ M ∂ t =  p 2 − v 1 x p + u 1 − 1 2 v 1 y  ∂ M ∂ x −  u 1 x p + 1 2 u 1 y  ∂ M ∂ p , (15) Then the t -flow of u 1 can b e read by Eq.(14) b y ta king the co efficien t of p − 1 : 1 3 u 1 t = u 3 x − v 1 x u 2 x + ( u 1 − 1 2 v 1 y ) u 1 x + u 1 u 1 x , = 1 2 u 2 y − 1 2 u 1 x v 1 y + u 1 u 1 x , (16) where w e ha v e used (11) to reach the second line. Also, the expression of Eq.(15) together with (14) giv es L 2 + 1 3 ∞ X n =1 v nt L − n =  p 2 − v 1 x p + u 1 − 1 2 v 1 y  1 + ∞ X n =1 v nx L − n ! , in which the co efficien t of p − 1 giv es 1 3 v 1 t = − u 2 + 1 2 v 2 y + u 1 v 1 x − 1 2 v 1 x v 1 y , (17) where w e hav e used Eq.(13). No w differen tiating Eqs.(16), (17) resp ectiv ely with resp ect to x and eliminating u 2 x and v 2 x b y Eqs.(10) and (12), w e obtain the followin g t w o coupled equations for u 1 := u and v 1 = v : 1 3 u xt = 1 4 u y y + ( uu x ) x + 1 2 v x u xy − 1 2 u xx v y , 1 3 v xt = 1 4 v y y + uv xx + 1 2 v x v xy − 1 2 v xx v y . (18) Eq.(18) is the so called Manakov-Santini e quation [1, 2, 3]. The La x pair for t his equation is defined b y linear equations (8,9) and (14,15). Notice t ha t for v = 0 reduction, the system reduces to t he dKP equation 1 3 u xt = 1 4 u y y + ( uu x ) x . (19) Resp ectiv ely , u = 0 reduction giv es an equation [4] (see also [5, 6, 7]) 1 3 v xt = 1 4 v y y + 1 2 v x v xy − 1 2 v xx v y . (20) 4 Prop osition 2.2 Equation (1) c an b e written i n Hamilton-Jac obi typ e e q uation ∂ p ( L ) ∂ t n     L fixed = A n ( p ( L )) ∂ p ( L ) ∂ x     L fixed + B n ( p ( L )) , (21) wher e A n ( p ) = ( J − 1 0 ∂ L n /∂ p ) + and B n ( p ) = ( J − 1 0 ∂ L n /∂ x ) + . Pr o of. By taking into accoun t t he partial deriv a t ives with resp ect to t n for fixed p or L , it is easy to sho w that ∂ p ∂ t n = 0 = ∂ p ( L ) ∂ t n     L + ∂ p ( L ) ∂ L ∂ L ∂ t n , or ∂ p ( L ) ∂ t n     L = − ∂ p ( L ) ∂ L ∂ L ∂ t n . (22) Using (1 ) , and (22) with n = 1, w e hav e ∂ p ( L ) ∂ t n     L fixed = − ∂ p ( L ) ∂ L  A n ( p ) ∂ L ∂ x − B n ( p ) ∂ L ∂ p  = A n ( p ( L )) ∂ p ( L ) ∂ x     L fixed + B n ( p ( L )) .  Prop osition 2.3 The function J 0 = ∂ p L ∂ x M − ∂ x L ∂ p M a n d its inverse G = J − 1 0 satisfy ∂ t n J 0 = ( A n J 0 ) x − ( B n J 0 ) p , (23) ∂ t n G = h A n , G i x − h B n , G i p , (24) wher e h U, V i i := U ( ∂ i V ) − ( ∂ i U ) V . Pr o of. Using the t n -flo ws of L , M in (1) and t he definition of J 0 , we ha v e ∂ t n J 0 = ( L p ) t n M x + L p ( M x ) t n − ( L x ) t n M p − L x ( M p ) t n , = − B np J 0 + A nx J 0 + A n J 0 x − B n J 0 p , = ( A n J 0 ) x − ( B n J 0 ) p . Moreo ver, substituting J 0 = G − 1 in to the ab ov e w e obt a in (24).  As w e will see, Prop osition 2.3 can provide a crucial w ay t o determine the hierarch y flo ws. 5 3 W aterbag-t yp e reduction Consider the waterbag-t yp e reduction of the generalized dKP hierarch y represen ted b y [8] L = p + N X i =1 ǫ i log( p − U i ) , (25) M = ∞ X n =1 nt n L n − 1 + M X i =1 δ i log( p − V i ) , (26) where ǫ i and δ i are assumed to satisfy N X i =1 ǫ i = M X i =1 δ i = 0 . (2 7 ) The ansatz (25,2 6) is consis ten t with the dynamics defined b y Manak o v-San t ini hierarc hy (1), i.e., the form of ansatz is preserv ed b y the dynamics. Condition (27) guaran tees that expansion of L , M at infinity is of the form ( 3 ,4). Reduced hierarc hy is represen ted a s infinite set of (1+1 )-dimensional systems of equations for the functions U i , V i , whic h are obta ined b y the substitution of ansatz ( 2 5,26) to equations of Manako v-San tini hierarc h y (1). Let us consider first flows o f reduced hierarch y . F or expansion of L , M at infinity from (25 ,26) we get L = p − ∞ X n =1 N X i =1 ǫ i U n i n ! p − n , (28) M = ∞ X n =1 nt n L n − 1 − ∞ X n =1 M X i =1 δ i V n i n ! p − n . (29) Comparing these expansions with formulae (3,4), w e come to the conclusion that u n = − P N i =1 ǫ i U n i n . T o calculate v n , w e should in v ert the series (28) to find p ( L ) that can b e done recursiv ely , and substitute p ( L ) to (29) . F or the first co efficien ts u n , v n w e get u 1 = − N X i =1 ǫ i U i , u 2 = − 1 2 N X i =1 ǫ i U 2 i , v 1 = − M X i =1 δ i V i , v 2 = − 1 2 M X i =1 δ i V 2 i . 6 Substituting these expressions to relations (5), (6 ) and using equations (1 ), we obtain equations of reduced hierarc h y . Equations of the flow corr esp o nding to y = t 2 read ∂ y U k =  2 U k + ∂ x M X i =1 δ i V i  ∂ x U k − 2 ∂ x  N X i =1 ǫ i U i  , ∂ y V k =  2 V k + ∂ x M X i =1 δ i V i  ∂ x V k − 2 ∂ x  N X i =1 ǫ i U i  . (30) F or the flo w corresp onding t o t = t 3 w e get ∂ t U k = 3 U 2 k + 3 U k ∂ x M X i =1 δ i V i − 6 N X i =1 ǫ i U i + 3( ∂ x M X i =1 δ i V i ) 2 + 3 ∂ x M X i =1 δ i V 2 i 2 ! ∂ x U k − 3 U k ∂ x N X i =1 ǫ i U i + 3( ∂ x N X i =1 ǫ i U i )( ∂ x M X i =1 δ i V i ) + 3 ∂ x N X i =1 ǫ i U 2 i 2 ! , ∂ t V k = 3 V 2 k + 3 V k ∂ x M X i =1 δ i V i − 6 N X i =1 ǫ i U i + 3( ∂ x M X i =1 δ i V i ) 2 + 3 ∂ x M X i =1 δ i V 2 i 2 ! ∂ x V k − 3 V k ∂ x N X i =1 ǫ i U i + 3( ∂ x N X i =1 ǫ i U i )( ∂ x M X i =1 δ i V i ) + 3 ∂ x N X i =1 ǫ i U 2 i 2 ! . (31) A common solution to the systems (30 ) , (31) g iv es a solution to Manako v-San tini equation (1 8) defined as u = − P N i =1 ǫ i U i , v = − P M i =1 δ i V i . 4 Diagonal form of reduced hierarc h y F or the w aterbag r eduction (25, 26) one can sho w that the G f unction can b e ex- pressed in the follo wing f orm G = J − 1 0 = Q N i =1 ( p − U i ) Q M j =1 ( p − V j ) F ( U n , U nx , V m , V mx ; p ) , n = 1 , . . . , N ; m = 1 , . . . , M , (32) where the function F in denominator is a p olynomial of p with degree N + M . In gen- eral, F can also b e factorized in t o Q N + M k =1 ( p − W k ), fo r whic h W k = W k ( U n , U nx , V m , V mx ) are ro ots of J 0 . W e lik e to men tion here that the deriv ativ es U nx , V mx can b e in v ersely expresse d as function of the form U nx = f n ( U i , V j , W k ) , V mx = g m ( U i , V j , W k ) . (33) Therefore, w e hav e J 0 = Q N + M k =1 ( p − W k ) Q N i =1 ( p − U i ) Q M j =1 ( p − V j ) , n = 1 , . . . , N ; m = 1 , . . . , M . (34) 7 As the result, the ev alua t io n of G at U i or V i , i.e., G ( p = U i ) = 0 or G ( p = V i ) = 0 sho ws that Eq.(24) can b e written in to the follo wing ev olution equations of U i , V i : ∂ U i ∂ t n = A n ( p = U i ) ∂ U i ∂ x + B n ( p = U i ) , (35) ∂ V i ∂ t n = A n ( p = V i ) ∂ V i ∂ x + B n ( p = V i ) . (36) Similarly , Eq.(23) with J 0 ( p = W i ) = 0 giv es rise ∂ W i ∂ t n = A n ( p = W i ) ∂ W i ∂ x + B n ( p = W i ) . (37) In summary , combinin g (35), (36), (37) and replacing those U nx ’s and V mx ’s in A n , B n with the transformations (33), w e obtain the non-homo ge n e ous Riemann inva ri a n t form as ∂ t n R i = A n ( p = R i ) R ix + B n ( p = R i ) , i = 1 , . . . , 2 N + 2 M , (3 8 ) for whic h ( R 1 , . . . , R 2 N +2 M ) = ( U 1 , . . . , U N ; V 1 , . . . , V M ; W 1 , . . . , W N + M ). Some linearly degenerate no n-homogeneous R iemann inv ariants forms, asso ci- ated with comm uting quadratic Hamiltonians and the Killing ve ctor fields of the giv en metric, w ere in v estigated in [9, 10]. Ho w ev er, in our case equation (38) is ob viously not linearly degenerate. Remark. F or the type of non-homogeneous Riemann inv ariant fo rm ∂ t n R i = Λ i n ( R ) R i x + Q i n ( R ) , (39) the requireme n ts o f the comm utativity are equiv a len t to the follo wing restrictions on their characteristic sp eeds and non-homogeneous terms (see app endix A) ∂ j Λ i n Λ j n − Λ i n = ∂ j Λ i m Λ j m − Λ i m , ∂ j Q i n Q j n = ∂ j Q i m Q j m , Q j n Λ j n − Λ i n = Q j m Λ j m − Λ i m , i 6 = j, n 6 = m. where ∂ i ≡ ∂ /∂ R i . Example 1. ( N , M ) = (1 , 1) reduction. In this case, L = p + log (1 − U /p ) , M = ∞ X n =1 nt n L n − 1 + log(1 − V /p ) . Comparing to the expansion of (3,4) we ha ve u n = − U n /n for n ≥ 1 and v 1 = − V , v 2 = − V 2 / 2 , v 3 = U V − V 3 / 3, etc. These t ransformations allo w us to g et A n , B n (b y Eqs.(5), (6)) whic h correspo nd to the reduced system . T he G function is given by G = p ( p − U )( p − V ) Q 3 i =1 ( p − W i ) , (40) 8 where W i satisfy 3 X i =1 W i = U + V + V x , 3 X i,j =1 ( i>j ) W i W j = U + U V + U V x , 3 Y i =1 W i = U V + U V x − U x V . (41) Notice that (40) is not coinciden t with that in (32), there is one more ro ot of p = 0 to b e considered. By (24), it turns o ut that the ev aluation o f p = 0 giv es an additio nal condition, namely U V B n ( p = 0) = 0 , ∀ n ≥ 1 . (42) There are tw o simple cases: (i) V = 0 , U 6 = 0, (ii) V 6 = 0 , U = 0. One can easily deduce considering t 2 -flo w of (38) that case (i) is a trivial reduction. F or the case (ii), w e hav e the fact that B n ( U = 0 ) = 0 for n ≥ 1, and Eq.(41) will r eveal us the only one relation: V x = W − V . T o this end, system (38) reduces t o the t yp e of homogeneous one in (39) with Q i n = 0, namely ∂ t n R i = Λ i n ( R ) ∂ x R i , (43) where R = ( R 1 , R 2 ) = ( V , W ) and the c har acteristic sp eeds Λ i n = A n ( p = R i , U = 0). F or instance, for t 2 = y flo w, w e hav e A 2 ( U = 0 ) = 2 p + 2 V x = 2 p + 2( W − V ), then Eq.(43 ) b ecomes  V W  y =  2 W 0 0 4 W − 2 V   V W  x . (44) F or t 3 = t flo w, w e derive A 3 ( U = 0 ) = 3 p 2 + 3( W − V ) p + 3( W − V ) 2 + 3 V ( W − V ), th us  V W  t =  3 W 2 0 0 9 W 2 − 6 V W   V W  x . (45) F ro m the tw o non trivial flo ws ( 4 4), (45), w e readily obtain the following set of ho dograph equation x + 2 W y + 3 W 2 t = ˆ F ( V , W ) , x + (4 W − 2 V ) y + (9 W 2 − 6 V W ) t = ˆ G ( V , W ) , ( 4 6) where ˆ F and ˆ G satisfy the linear equations ( W − V ) ˆ G V = ˆ F − ˆ G, ( W − V ) ˆ F W = ˆ G − ˆ F . Dividing these tw o equations for V 6 = W w e g et ˆ G V = − ˆ F W . It follow s that t here exists a f unction φ suc h that ˆ F = φ V , ˆ G = − φ W , whence φ satisfies the defining equation ( V − W ) φ V W = φ V + φ W . (47) 9 Eq. (47) has a g eneral solution of the form φ = ( V − W )  f ( W ) + Z g ( V ) ( V − W ) 2 dV  , where f ( W ) and g ( V ) are ar bitr ary functions of W and V , respectiv ely . Choo sing, for example f ( W ) = W 3 , g ( V ) = Const., then w e hav e ˆ F = W 3 and ˆ G = − 3 V W 2 + 4 W 3 . Substituting back in to the ho dograph equation (46) w e solv e V = W = 1 6 h − 1 / 3 (24 y + 36 t 2 + 6 th 1 / 3 + h 2 / 3 ) , h = 216 y t + 108 x + 216 t 3 + 12 p − 96 y 3 − 108 y 2 t 2 + 324 y tx + 81 x 2 + 324 xt 3 , whic h satisfies the t 2 - a nd t 3 -flo ws (4 4), (45). Ho wev er, V = W con tradicts to the relation W = V + V x and V do es not satisfy equation (20). Actually , from equation (47) we can see that when V = W one can get ˆ F = ˆ G . Then w e obta in all the solutions will satisfy V = W . Consequen tly , there is no (1,1)- reduction. Similar considerations can show that there are no (1,2)- and (2 ,1 )- reductions, either. Example 2. ( N , M ) = (2 , 2) reduction. In this case, L = p + ǫ 1 log p − U 1 p − U 2 , M = ∞ X n =1 nt n L n − 1 + δ 1 log p − V 1 p − V 2 . F or simplicit y , w e set ǫ 1 = δ 1 = 1. Comparing to the expansion o f (3,4), w e ha v e u n = ( U n 2 − U n 1 ) /n for n ≥ 1 and v 1 = V 2 − V 1 , v 2 = ( V 2 2 − V 2 1 ) / 2 , v 3 = u 1 v 1 +( V 3 2 − V 3 1 ) / 3 , . . . . No w w e expand the hierarch y flow of U i , V i and W i up to t 2 = y , t 3 = t . F rom (32) with ( N , M ) = (2 , 2) we ha v e G = Q 2 i =1 ( p − U i ) Q 2 j =1 ( p − V j ) Q 4 k =1 ( p − W k ) . where W i satisfy 4 X i =1 W i = U 1 + U 2 + V 1 + V 2 + V 1 x − V 2 x , (48) 4 X i,j =1 ( i>j ) W i W j = U 1 − U 2 + U 1 U 2 + V 1 V 2 + V 1 x V 2 − V 1 V 2 x +( U 1 + U 2 )( V 1 + V 2 + V 1 x − V 2 x ) , (49) 4 X i =1 W − 1 i 4 Y j =1 W j = ( U 1 + U 2 )( V 1 V 2 + V 1 x V 2 − V 1 V 2 x ) + ( U 2 x − U 1 x )( V 1 − V 2 ) +( U 1 − U 2 + U 1 U 2 )( V 1 + V 2 + V 1 x − V 2 x ) , (50) 4 Y i =1 W i = ( U 1 − U 2 + U 1 U 2 )( V 1 V 2 + V 1 x V 2 − V 1 V 2 x ) − ( V 1 − V 2 )( U 1 x U 2 − U 1 U 2 x ) , (51) 10 from whic h, one can substitute into A n , B n to eliminate U ix , V ix , etc. F or n = 2, using (4 8 ), (50 ) w e ha ve A 2 ( p ) = 2 p + 2( V 1 x − V 2 x ) = 2 p + 2  − U 1 − U 2 − V 1 − V 2 + 4 X i =1 W i  , = 2( p − R 1 − R 2 − R 3 − R 4 + R 5 + R 6 + R 7 + R 8 ) , and the non-ho mo g eneous term B 2 ( p ) = 2( U 2 x − U 1 x ) , = 2 V 1 − V 2  4 X i =1 W − 1 i 4 Y j =1 W j + ( U 1 + U 2 )  U 1 − U 2 + U 1 U 2 − 4 X i>j W i W j  +  U 1 + U 2 − 4 X i =1 W i  U 1 − U 2 + U 1 U 2 − ( U 1 + U 2 ) 2   , = 2 R 3 − R 4 h ( R 1 + R 2 − R 5 − R 6 − R 7 − R 8 )( R 1 − R 2 − R 1 R 2 − R 2 1 − R 2 2 ) +( R 1 + R 2 )( R 1 − R 2 + R 1 R 2 − R 5 R 6 − R 5 R 7 − R 5 R 8 − R 6 R 7 − R 6 R 8 − R 7 R 8 ) + R 5 R 6 R 7 + R 6 R 7 R 8 + R 7 R 8 R 5 + R 8 R 5 R 6 i . Then the t 2 = y flo w in (38) is now read ∂ y R i = 2( R i − R 1 − R 2 − R 3 − R 4 + R 5 + R 6 + R 7 + R 8 ) R ix + B 2 . (52) F or n = 3, Eq.(38) b ecomes ∂ R i ∂ t = A 3 ( p = R i ) R ix + B 3 ( p = R i ) , =  3 p 2 + 3( V 1 x − V 2 x ) p + 6 ( U 2 − U 1 ) + 3 ( V 1 x − V 2 x ) 2 + 3 2 ( V 2 1 − V 2 2 ) x  p = R i R ix +  3( U 2 x − U 1 x ) p − 3( U 2 x − U 1 x )( V 2 x − V 1 x ) + 3 2 ( U 2 2 − U 2 1 ) x  p = R i =  3 R 2 i + 3 R i ( V 1 x − V 2 x ) + 6 ( U 2 − U 1 ) + 3 ( V 1 x − V 2 x ) 2 + 3 2 ( V 2 1 − V 2 2 ) x  R ix +3 R i ( U 2 x − U 1 x ) − 3( U 2 x − U 1 x )( V 2 x − V 1 x ) + 3 ( U 2 U 2 x − U 1 U 1 x ) . Using Eqs.(48 ) –(51) w e arriv e ∂ R i ∂ t = 3 R ix  U 2 − U 1 + R i  R i − U 1 − U 2 − V 1 − V 2 + 4 X i =1 W i  + U 1 U 2 − V 1 V 2 − V 2 1 − V 2 2 − 4 X i>j W i W j +  U 1 + U 2 + V 1 + V 2 − 4 X i =1 W i  2 11 +( U 1 + U 2 + V 1 + V 2 )  − U 1 − U 2 + 4 X i =1 W i   + 3 R i V 1 − V 2  ( U 1 + U 2 )  U 1 − U 2 + U 1 U 2 − 4 X i>j W i W j  + 4 X i =1 W − 1 i 4 Y j =1 W j +  U 1 + U 2 − 4 X i =1 W i  U 1 − U 2 + U 1 U 2 − ( U 1 + U 2 ) 2   + +3  − U 1 − U 2 − V 1 − V 2 + 4 X i =1 W i  × × 1 V 1 − V 2   U 1 + U 2 − 4 X i =1 W i  U 1 − U 2 + U 1 U 2 − ( U 1 + U 2 ) 2  +( U 1 + U 2 )  U 1 − U 2 + U 1 U 2 − X i>j W i W j  + 4 X i − 1 W − 1 i 4 Y j =1 W j  + 3 V 1 − V 2  ( U 1 + U 2 ) 4 X i − 1 W − 1 i 4 Y j =1 W j + ( U 1 + U 2 ) 2  U 1 − U 2 + U 1 U 2 − X i>j W i W j  − 4 Y i =1 W i + ( U 1 + U 2 )  U 1 + U 2 − 4 X i =1 W i  U 1 − U 2 + U 1 U 2 − ( U 1 + U 2 ) 2  +( U 1 − U 2 + U 1 U 2 )  4 X i>j W i W j − ( U 1 − U 2 ) − U 1 U 2 + ( U 1 + U 2 ) 2 − ( U 1 + U 2 ) 4 X i =1 W i   . Expressing in terms of R i , i = 1 , . . . , 8, w e get ∂ R i ∂ t = 3 R ix  R 2 − R 1 + R i ( R i − R 1 − R 2 − R 3 − R 4 + R 5 + R 6 + R 7 + R 8 ) + R 1 R 2 + R 3 R 4 + R 5 R 6 + R 5 R 7 + R 5 R 8 + R 6 R 7 + R 6 R 8 + R 7 R 8 + R 1 R 3 + R 1 R 4 − R 1 R 5 − R 1 R 6 − R 1 R 7 − R 1 R 8 + R 2 R 3 + R 2 R 4 − R 2 R 5 − R 2 R 6 − R 2 R 7 − R 2 R 8 − R 3 R 5 − R 3 R 6 − R 3 R 7 − R 3 R 8 − R 4 R 5 − R 4 R 6 − R 4 R 7 − R 4 R 8 + R 2 5 + R 2 6 + R 2 7 + R 2 8  + 3 R i R 3 − R 4  2 R 2 1 − 2 R 2 2 − R 1 R 5 − R 1 R 6 − R 1 R 7 − R 1 R 8 + R 2 R 5 + R 2 R 6 + R 2 R 7 + R 2 R 8 − R 3 1 − R 3 2 − R 2 1 R 2 + R 2 1 R 5 + R 2 1 R 6 + R 2 1 R 7 + R 2 1 R 8 − R 1 R 2 2 + R 2 2 R 5 + R 2 2 R 6 + R 2 2 R 7 + R 2 2 R 8 + R 1 R 2 R 5 + R 1 R 2 R 6 + R 1 R 2 R 7 + R 1 R 2 R 8 − R 1 R 5 R 6 − R 1 R 5 R 7 − R 1 R 5 R 8 − R 1 R 6 R 7 − R 1 R 6 R 8 − R 1 R 7 R 8 − R 2 R 5 R 6 − R 2 R 5 R 7 − R 2 R 5 R 8 − R 2 R 6 R 7 − R 2 R 6 R 8 − R 2 R 7 R 8 + R 5 R 6 R 7 + R 6 R 7 R 8 + R 7 R 8 R 5 + R 8 R 5 R 6  + 12 + 3 R 3 − R 4  − R 2 1 − R 2 2 + 2 R 1 R 2 + R 3 1 − R 3 2 − R 2 1 R 2 + R 1 R 2 2 − 2 R 2 1 R 3 + 2 R 2 2 R 3 − 2 R 2 1 R 4 + 2 R 2 2 R 4 + R 2 1 R 5 − R 2 2 R 5 + R 2 1 R 6 − R 2 2 R 6 + R 2 1 R 7 − R 2 2 R 7 + R 2 1 R 8 − R 2 2 R 8 − R 1 R 2 5 + R 2 R 2 5 + R 1 R 2 6 − R 2 R 2 6 + R 1 R 2 7 − R 2 R 2 7 + R 1 R 2 8 − R 2 R 2 8 + R 3 1 R 2 + R 3 1 R 3 + R 3 1 R 4 − R 3 1 R 5 − R 3 1 R 6 − R 3 1 R 7 − R 3 1 R 8 + R 1 R 3 2 + R 3 2 R 3 + R 3 2 R 4 − R 3 2 R 5 − R 3 2 R 6 − R 3 2 R 7 − R 3 2 R 8 + R 2 1 R 2 2 + R 2 1 R 2 5 + R 2 2 R 2 5 − R 2 1 R 2 6 − R 2 2 R 2 6 − R 2 1 R 2 7 − R 2 2 R 2 7 − R 2 1 R 2 8 − R 2 2 R 2 8 + R 1 R 5 R 6 + R 1 R 5 R 7 + R 1 R 5 R 8 + 3 R 1 R 6 R 7 + 3 R 1 R 6 R 8 + 3 R 1 R 7 R 8 − R 2 R 5 R 6 − R 2 R 5 R 7 − R 2 R 5 R 8 − 3 R 2 R 6 R 7 − 3 R 2 R 6 R 8 − 3 R 2 R 7 R 8 + R 3 R 5 R 1 − R 3 R 5 R 2 − R 3 R 6 R 1 + R 3 R 6 R 2 − R 3 R 7 R 1 + R 3 R 7 R 2 − R 3 R 8 R 1 + R 3 R 8 R 2 + R 4 R 5 R 1 − R 4 R 5 R 2 − R 4 R 6 R 1 + R 4 R 6 R 2 − R 4 R 7 R 1 + R 4 R 7 R 2 − R 4 R 8 R 1 + R 4 R 8 R 2 − 2 R 5 R 2 1 R 2 − 2 R 6 R 2 1 R 2 − 2 R 7 R 2 1 R 2 − 2 R 1 R 6 R 2 2 − 2 R 1 R 7 R 2 2 − 2 R 1 R 8 R 2 2 − 2 R 8 R 2 1 R 2 − 2 R 2 1 R 6 R 7 − 2 R 2 1 R 6 R 8 − 2 R 2 1 R 7 R 8 − 2 R 2 2 R 6 R 7 − 2 R 2 2 R 6 R 8 − 2 R 2 2 R 7 R 8 − 2 R 1 R 5 R 2 2 + R 2 5 R 6 R 7 + R 7 R 8 R 2 5 + R 8 R 2 5 R 6 + R 2 5 R 1 R 2 − R 1 R 2 5 R 6 − R 1 R 2 5 R 7 − R 1 R 2 5 R 8 − R 2 R 2 5 R 6 − R 2 R 2 5 R 7 − R 2 R 2 5 R 8 + R 5 R 2 6 R 7 + R 2 6 R 7 R 8 + R 8 R 5 R 2 6 − R 2 6 R 1 R 2 − R 1 R 5 R 2 6 − R 1 R 2 6 R 7 − R 1 R 2 6 R 8 − R 2 R 5 R 2 6 − R 2 R 2 6 R 7 − R 2 R 2 6 R 8 + R 5 R 6 R 2 7 + R 6 R 2 7 R 8 + R 2 7 R 8 R 5 − R 2 7 R 1 R 2 − R 1 R 5 R 2 7 − R 1 R 6 R 2 7 − R 1 R 2 7 R 8 − R 2 R 5 R 2 7 − R 2 R 6 R 2 7 − R 2 R 2 7 R 8 + R 6 R 7 R 2 8 + R 7 R 2 8 R 5 + R 2 8 R 5 R 6 − R 2 8 R 1 R 2 − R 1 R 5 R 2 8 − R 1 R 6 R 2 8 − R 1 R 7 R 2 8 − R 2 R 5 R 2 8 − R 2 R 6 R 2 8 − R 2 R 7 R 2 8 + R 3 R 2 1 R 2 + R 3 R 1 R 2 2 − R 3 R 5 R 2 1 − R 3 R 5 R 2 2 + R 3 R 6 R 2 1 + R 3 R 6 R 2 2 + R 3 R 7 R 2 1 + R 3 R 7 R 2 2 + R 3 R 8 R 2 1 + R 3 R 8 R 2 2 + R 4 R 2 1 R 2 + R 4 R 1 R 2 2 − R 4 R 5 R 2 1 − R 4 R 5 R 2 2 + R 4 R 6 R 2 1 + R 4 R 6 R 2 2 + R 4 R 7 R 2 1 + R 4 R 7 R 2 2 + R 4 R 8 R 2 1 + R 4 R 8 R 2 2 − 3 R 1 R 5 R 6 R 7 − 3 R 1 R 6 R 7 R 8 − 3 R 1 R 5 R 7 R 8 − 3 R 1 R 5 R 6 R 8 − 3 R 2 R 5 R 6 R 7 − 3 R 2 R 6 R 7 R 8 + 3 R 5 R 6 R 7 R 8 − 3 R 2 R 7 R 8 R 5 − 3 R 2 R 8 R 5 R 6 + R 1 R 2 R 5 R 6 + R 1 R 2 R 5 R 7 + R 1 R 2 R 5 R 8 − R 1 R 2 R 6 R 7 − R 1 R 2 R 6 R 8 − R 1 R 2 R 7 R 8 − R 3 R 5 R 6 R 7 − R 3 R 6 R 7 R 8 − R 3 R 7 R 8 R 5 − R 3 R 8 R 5 R 6 − R 3 R 5 R 1 R 2 + R 3 R 6 R 1 R 2 + R 3 R 7 R 1 R 2 + R 3 R 8 R 1 R 2 + R 3 R 1 R 5 R 6 + R 3 R 1 R 5 R 7 + R 3 R 1 R 5 R 8 + R 3 R 1 R 6 R 7 + R 3 R 1 R 6 R 8 + R 3 R 1 R 7 R 8 + R 3 R 2 R 5 R 6 + R 3 R 2 R 5 R 7 + R 3 R 2 R 5 R 8 + R 3 R 2 R 6 R 7 + R 3 R 2 R 6 R 8 + R 3 R 2 R 7 R 8 − R 4 R 5 R 6 R 7 − R 4 R 6 R 7 R 8 − R 4 R 7 R 8 R 5 − R 4 R 8 R 5 R 6 − R 4 R 5 R 1 R 2 + R 4 R 6 R 1 R 2 + R 4 R 7 R 1 R 2 + R 4 R 8 R 1 R 2 + R 4 R 1 R 5 R 6 + R 4 R 1 R 5 R 7 + R 4 R 1 R 5 R 8 + R 4 R 1 R 6 R 7 + R 4 R 1 R 6 R 8 + R 4 R 1 R 7 R 8 + R 4 R 2 R 5 R 6 + R 4 R 2 R 5 R 7 + R 4 R 2 R 5 R 8 + R 4 R 2 R 6 R 7 + R 4 R 2 R 6 R 8 + R 4 R 2 R 7 R 8  . 13 5 Conclud ing Remarks In this art icle, w e inv estigate the Manak o v-San tini equation starting from Lax-Sato form ulation of asso ciated hierarc h y and obtain equations (23), (24), whic h generalize the results of [6 ]. F ro m these, one can in tro duce new co ordinates (32 ) suc h that the non-h ydro dynamic evolution (30), (31) of w aterbag reduction t ransforms to non- homogeneous Riemann inv ariants form of hy dro dynamic t yp e (38). The equation (38) is no t linearly degenerate. Hence the g eneralization o f [9, 10] fro m linearly degenerate case to the general one could b e in teresting. Also, the solution structures of (38) havin g infinite symmetries should b e in ve stigated. These issues will b e published elsewhere. App endix A Comm utabili t y prop ertie s of the non-homogen eous diagonal syst em W e start from the comm utabilit y of (39) by ∂ m ∂ n R i = ∂ n ∂ m R i : ∂ m ∂ n R i = ∂ m (Λ i n R i x ) + ∂ m Q i n , = X j ( ∂ j Λ i n )( ∂ m R j ) R i x + Λ i n ∂ x ( ∂ m R i ) + X j ( ∂ j Q i n )( ∂ m R j ) , = X j ( ∂ j Λ i n )(Λ j m R j x + Q j m ) R i x + Λ i n ∂ x (Λ i m R i x + Q i m ) + X j ( ∂ j Q i n )(Λ j m R j x + Q j m ) , = X j ( ∂ j Λ i n )(Λ j m R j x + Q j m ) R i x + Λ i n  X j ( ∂ j Λ i m ) R j x R i x + Λ i m R i xx + X j ( ∂ j Q i m ) R j x  + X j ( ∂ j Q i n )(Λ j m R j x + Q j m ) , = X j h ( ∂ j Λ i n )Λ j m + ( ∂ j Λ i m )Λ i n i R j x R i x + X j ( ∂ j Λ i n ) Q j m R i x + Λ i n Λ i m R i xx + X j h ( ∂ j Q i m )Λ i n + ( ∂ j Q i n )Λ j m i R j x + X j ( ∂ j Q i n ) Q j m . Similarly , ∂ n ∂ m R i = ∂ n (Λ i m R i x ) + ∂ n Q i m , = X j h ( ∂ j Λ i m )Λ j n + ( ∂ j Λ i n )Λ i m i R j x R i x + X j ( ∂ j Λ i m ) Q j n R i x + Λ i m Λ i n R i xx 14 + X j h ( ∂ j Q i n )Λ i m + ( ∂ j Q i m )Λ j n i R j x + X j ( ∂ j Q i m ) Q j n . Then, ∂ m ∂ n R i = ∂ n ∂ m R i pro vide t he fo llo wing compatibility conditions: (i) T aking the co efficien ts of R j x R i x w e ha v e ( ∂ j Λ i n )Λ j m + ( ∂ j Λ i m )Λ i n = ( ∂ j Λ i m )Λ j n + ( ∂ j Λ i n )Λ i m , whic h implies ∂ j Λ i n Λ j n − Λ i n = ∂ j Λ i m Λ j m − Λ i m . (A.1) (ii) T aking the co efficien ts of R i x w e ha v e ( ∂ j Λ i n ) Q j m = ( ∂ j Λ i m ) Q j n . Com bining (A.1), the ab ov e equation can b e written a s Q j n Λ j n − Λ i n = Q j m Λ j m − Λ i m . (A.2) (iii) T aking the co efficien ts of R j x w e get ∂ j Q i n Λ j n − Λ i n = ∂ j Q i m Λ j m − Λ i m . (A.3) (iv) The zero-th term o f ∂ m ∂ n R i = ∂ n ∂ m R i giv e us ∂ j Q i n Q j n = ∂ j Q i m Q j m . (A.4) Notice that according to (A.2), equation (A.4) is equiv alen t to (A.3). T o summarize, w e hav e three compatibilit y conditions (A.1 ) , (A.2) and (A.4). Ac knowledgmen ts The first a uthor (L VB) is grateful to National Defense Univ ersit y (T a oyuan, T ai- w an), where a part o f this work w as done, for hospitality . This researc h was par- ticularly supp orted by the R ussian-T aiw anese gra nt 95 WF E03 00007 (RFBR grant 06-01- 8 9507) a nd NSC-95-2923- M- 606-001 -MY3. L VB w as also suppor t ed in part b y R FBR grant 07-01 -00446 . 15 References [1] S. V. Manak ov and P . M. San tini, The Cauc hy pro blem o n the plane for the disp ersionless Kadomtse v-P etviash vili equation, JETP Lett. 83 ( 2 006) 462-4 66. [2] S. V. Manako v and P . M. San tini, A hierar ch y of in tegrable PDEs in 2 + 1 dimensions associated with 2-dimensional vec tor fields, Theor. Math. Phy s. 152 (200 7 ) 100 4–1011. [3] S. V. Manak o v and P . M. 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