Cartans structure of symmetry pseudo-group and coverings for the r-th modified dispersionless Kadomtsev--Petviashvili equation

Cartan’s structu r e of symmetry pseudo - g roup and co v erings for the r-th mo difie d d isp ersionl e ss Kadom tsev–P etviash vili equatio n Oleg I. Morozo v Department of Mathematics, Moscow State T echnical Universit y of Civ il Aviation, Kronshtadtskiy Blvd 20, Moscow 1 25993 , Russ ia oim@foxcub.org Abstract. W e derive tw o non-equiv alent cov erings for the r-th mdKP eq uation from Maurer–C artan fo rms of its symmetry pseudo-gro up. Also we find B¨ acklund tra nsfor- mations betw een the cov ering eq uations. AMS classification sc heme n umbers: 58H0 5, 58 J70, 35A30 Symmetries and c overings for the r-th mdKP e quation 2 1. In tro duction The role of co v erings in studying nonlinear differen tial equations ( de s) is w ell-know n, [20, 21, 22 , 23]. They lead to a n um b er of useful tec hniques suc h as in v erse scattering transformations, B¨ ac klund transformations, recursion op erators, nonlo cal symme tries and no nlo cal conserv ation la ws. F or a giv en de , a problem of constructing a co v ering is v ery difficult, see, e.g., [41, 8 , 6, 14, 29, 30, 38, 15, 35, 36, 42, 40, 28, 12, 13]. One of the p ossible approache s t o solution lies in the framew ork of ´ Elie Carta n’s structure theory of Lie pseudo-groups, [27, 3, 33, 34]. In the presen t pap er w e apply the metho d o f [33, 34] to the r-th mo dified disp ersi- onless Kadom tsev–P etviash vili equation (r- mdKP), [1]. W e use ´ Elie Cartan’s metho d of equiv alence, [4, 11 , 17, 3 7], t o compute Maurer–Cartan ( mc ) forms o f the pseudo-group of con tact symmetries of r-mdKP , and then find tw o linear com binations of these for ms, whose horizon talizations provide cov ering equations of r- mdKP . Previosly this approach w as applied in [34 ] to a particular case of r-mdKP – mo dified dispersionless Kadomtse v– P etviash vili equation (mdKP), or mo dified Khokhlo v–Zab olot sk a y a equation, [27, 2 5, 26]. Co v erings for particular cases of r-mdKP were found in [10, 7, 18, 5] via other metho ds. 2. Preliminaries 2.1. Covering s of DEs Let π ∞ : J ∞ ( π ) → R n b e the infinite jet bundle of lo cal sections of the bundle π : R n × R → R . The co ordinates on J ∞ ( π ) are ( x i , u I ), where I = ( i 1 , ..., i k ) are symmetric m ulti-indices, i 1 , ..., i k ∈ { 1 , ..., n } , u ∅ = u , and for any lo cal section f of π there exists a section j ∞ ( f ) : R n → J ∞ ( π ) suc h that u I ( j ∞ ( f )) = ∂ # I ( f ) /∂ x i 1 ...∂ x i k , # I = #( i 1 , ..., i k ) = k . The total derivatives on J ∞ ( π ) are defined in the lo cal coordintes as D i = ∂ ∂ x i + X # I ≥ 0 u I i ∂ ∂ u I . W e ha v e [ D i , D j ] = 0 fo r i, j ∈ { 1 , ..., n } . A de F ( x i , u K ) = 0 defines a submanifold E ∞ = { D I ( F ) = 0 | # I ≥ 0 } ⊂ J ∞ ( π ), where D I = D i 1 ◦ ... ◦ D i k for I = ( i 1 , ..., i k ). W e denote restrictions of D i on E ∞ as D i . In lo cal co ordinates, a c overing o v er E ∞ is a bundle e E ∞ = E ∞ × Q → E ∞ with fibre co ordinates q α , α ∈ { 1 , ..., N } o r α ∈ N , equipp ed with extende d total de ri v atives e D i = D i + X α T α i ( x j , u I , q β ) ∂ ∂ q α suc h that [ e D i , e D j ] = 0 whenev er ( x i , u I ) ∈ E ∞ . Dually , the co v ering is defined b y the followin g differential 1-forms, [41], ω α = dq α − T α i ( x j , u I , q β ) dx i Symmetries and c overings for the r-th m dKP e quation 3 suc h that dω α ≡ 0 ( mo d ω β , ¯ ϑ I ) iff ( x i , u I ) ∈ E ∞ , where ¯ ϑ I are restrictions of the con tact forms ϑ I = du I − u I ,k dx k on E ∞ . 2.2. Cartan ’s structur e the ory of c ontact symmetry pseudo- g r oups o f DEs A pse udo-gr oup on a manifold M is a collection of lo cal diffeomorphisms of M , whic h is closed under comp osition when define d , con tains an iden tity and is closed under in v erse. A Lie pseudo-g r oup is a pseudo-group whose diff eomor phisms are lo cal analytic solutions of an inv o lutiv e system of partial differen tial equations. ´ Elie Cartan’s a ppro ac h to L ie pseudo-groups is based on a p ossibilit y to c haracterize transforma t io ns from a pseudo- group in terms of a set of inv arian t differential 1-f orms called Maur er–Cartan forms . The mc forms for a Lie pseudo-group can be computed by means o f algebraic o p erations and differentiation. Expres sions of differen tials of the mc fo rms in terms of themselv es giv e structur e e quations of the pseudo-group. The structure equations con tain the full information ab out their pseudo-group. EXAMPLE 1. Consider the bundle J 2 ( π ) of jets of the second order of the bundle π . A differential 1-form ϑ on J 2 ( π ) is called a c ontact form if it is annihilated by all 2 - jets of lo cal sections: j 2 ( f ) ∗ ϑ = 0. In the lo cal co ordinat es ev ery contact 1- f orm is a linear com bination of the forms ϑ 0 = d u − u i dx i , ϑ i = du i − u ij dx j , i, j ∈ { 1 , ..., n } , u j i = u ij . A lo cal diffeomorphism ∆ : J 2 ( π ) → J 2 ( π ), ∆ : ( x i , u, u i , u ij ) 7→ ( x i , u, u i , u ij ), is called a c ontact tr a n sformation if for ev ery contact 1-form ϑ the fo rm ∆ ∗ ϑ is also con tact. W e denote b y Con t( J 2 ( π )) the pseudo-group of con tact transformations on J 2 ( π ). Consider the follow ing 1- forms Θ 0 = a ϑ 0 , Θ i = g i Θ 0 + a B k i ϑ k , Ξ i = c i Θ 0 + f ik Θ k + b i k dx k , Θ ij = a B i k B j l ( du k l − u k lm dx m ) + s ij Θ 0 + w k ij Θ k + z ij k Ξ k , (1) defined o n J 2 ( π ) × H , where H is an op en subset of R (2 n +1)( n +3)( n +1) / 3 with lo cal co ordinates ( a , b i k , c i , f ik , g i , s ij , w k ij , u ij k ), i, j, k ∈ { 1 , ..., n } , i ≤ j , suc h that a 6 = 0, det( b i k ) 6 = 0 , f ik = f k i , u ij k = u ik j = u j i k , while ( B i k ) is the in v erse mat r ix for the matrix ( b k l ). As it is sho wn in [32], the forms (1) are mc forms for Con t( J 2 ( π )), that is, a lo cal diffeomorphism b ∆ : J 2 ( π ) × H → J 2 ( π ) × H satisfies the conditions b ∆ ∗ Θ 0 = Θ 0 , b ∆ ∗ Θ i = Θ i , b ∆ ∗ Ξ i = Ξ i , and b ∆ ∗ Θ ij = Θ ij if and only if it is pro jectable on J 2 ( π ), a nd its pro jection ∆ : J 2 ( π ) → J 2 ( π ) is a con tact transformatio n. The structure equations for Con t( J 2 ( π )) hav e the form d Θ 0 = Φ 0 0 ∧ Θ 0 + Ξ i ∧ Θ i , d Θ i = Φ 0 i ∧ Θ 0 + Φ k i ∧ Θ k + Ξ k ∧ Θ ik , d Ξ i = Φ 0 0 ∧ Ξ i − Φ i k ∧ Ξ k + Ψ i 0 ∧ Θ 0 + Ψ ik ∧ Θ k , d Θ ij = Φ k i ∧ Θ k j + Φ k j ∧ Θ k i − Φ 0 0 ∧ Θ ij + Υ 0 ij ∧ Θ 0 + Υ k ij ∧ Θ k + Λ ij k ∧ Ξ k , where the additiona l forms Φ 0 0 , Φ 0 i , Φ k i , Ψ i 0 , Ψ ij , Υ 0 ij , Υ k ij , and Λ ij k dep end on differen tials of the co ordinates of H . Symmetries and c overings for the r-th m dKP e quation 4 EXAMPLE 2 . Supp ose E is a second-order diff erential equation in one dep enden t and n indep enden t v ariables. W e consider E as a submanifold in J 2 ( π ). Let Con t( E ) be the group of con tact symmetries for E . It consists of all t he con tact transformations on J 2 ( π ) mapping E to itself. Let ι 0 : E → J 2 ( π ) b e an embedding, and ι = ι 0 × id : E × H → J 2 ( π ) × H . The mc forms of Con t( E ) can b e deriv ed from t he for ms θ 0 = ι ∗ Θ 0 , θ i = ι ∗ Θ i , ξ i = ι ∗ Ξ i , and θ ij = ι ∗ Θ ij b y means of Cartan’s metho d of equiv alence, see details and examples in [9, 31 , 32]. 3. Cartan’s structure of the con tact symmetry pseudo-group for r-mdKP The r-th mdKP u tx = − (3 − r ) (1 − r ) 2 u 2 x u xx + r ( 3 − r ) 2 − r u x u xy + 3 − r (2 − r ) 2 u y y + (3 − r ) (1 − r ) 2 − r u y u xx , r ∈ Z \{ 2 } , w as deriv ed in [1]. F or a con v enience of computations we use t he f o llo wing c hange of v ariables: ˜ t = (3 − r ) t, ˜ x = x, ˜ y = (2 − r ) y , ˜ u = − (1 − r ) u , where r 6∈ { 1 , 2 , 3 } . Then we hav e ˜ u ˜ y ˜ y = ˜ u ˜ t ˜ x +  1 2 (1 − r ) ˜ u 2 ˜ x + ˜ u ˜ y  ˜ u ˜ x ˜ x + r 1 − r ˜ u ˜ x ˜ u ˜ x ˜ y . W e drop t ildes and denote κ = r 1 − r ; this yields u y y = u tx +  κ + 1 2 u 2 x + u y  u xx + κ u x u xy . (2) The exceptional cases r = 2 and r = 3 corresp ond to κ = − 2 and κ = − 3 2 , resp ectiv ely . W e will not consider the case of r = 1. The case of κ = − 1 is exceptional, t o o, since r → ∞ when κ → − 1. In the cases of κ = 0, κ = 1, and κ = − 1 Eq. (2) g ets the forms of the mdKP equation, [27, 25, 26], u y y = u tx +  1 2 u 2 x + u y  u xx , (3) the dBKP equation, [39, 18], u y y = u tx +  u 2 x + u y  u xx + u x u xy . (4) and the equation describing Lorentzian h yp er-CR Einstein–W eil structures, [7, 10], u y y = u tx + u y u xx − u x u xy . (5) W e use the metho d outlined in the previous section to compute mc forms a nd structure equations of the pseudo-group of contact symmetries for Eq. (2). The results dep end on κ . Symmetries and c overings for the r-th m dKP e quation 5 When κ 6∈ {− 2 , − 1 } , the structure equations hav e the f orm dθ 0 =  η 1 + ξ 2 − κ − 4 8 ξ 3 − κ κ +2 θ 22  ∧ θ 0 + ξ 1 ∧ θ 1 + ξ 2 ∧ θ 2 + ξ 3 ∧ θ 3 , dθ 1 =  3 2 η 1 − κ − 4 8 ξ 3 − 3( κ +1) κ +2 θ 22  ∧ θ 1 +  ( κ + 1) θ 2 + ( κ + 2) ξ 2  ∧ θ 3 + ξ 1 ∧ θ 11 +  2 κ 2 +15 κ +4 8 θ 2 − κ 2 − 10 κ +8 8 ξ 2 + κ θ 23  ∧ θ 0 + ξ 2 ∧ θ 12 + ξ 3 ∧ θ 13 , dθ 2 =  1 2 η 1 − κ +1 κ +2 θ 22 + 3 κ +4 8 ξ 3  ∧ θ 2 + ξ 1 ∧ θ 12 + ξ 2 ∧ θ 22 + ξ 3 ∧ θ 23 , dθ 3 =  η 1 − 2( κ +1) κ +2 θ 22 + κ +4 8 ξ 3  ∧ θ 3 +  κ − 4 8 θ 22 − κ 2 − 16 64 ξ 3  ∧ θ 0 + κ +2 2 ξ 2 ∧ θ 2 + ξ 1 ∧ θ 13 + ξ 2 ∧ θ 23 + ξ 3 ∧ θ 12 , dξ 1 = −  1 2 η 1 − 2 κ +3 κ +2 θ 22  ∧ ξ 1 , dξ 2 =  1 2 η 1 + 1 κ +2 θ 22 + ξ 3  ∧ ξ 2 +  κ − 4 8 θ 0 − θ 3  ∧ ξ 1 − θ 2 ∧ ξ 3 , dξ 3 = − ( κ + 2 ) ( θ 2 + ξ 2 ) ∧ ξ 1 + θ 22 ∧ ξ 3 , dθ 11 = 2 η 1 ∧ θ 11 − κ η 2 ∧ θ 0 + η 3 ∧ ξ 2 + η 4 ∧ ξ 3 + η 5 ∧ ξ 1 + κ  θ 3 + κ +8 4 θ 12  ∧ θ 0 +  (4 κ + 3) θ 23 − κ 2 − 18 κ +20 4 ξ 2  ∧ θ 1 −  (2 κ + 3 ) θ 13 + 5 κ 2 +31 κ +24 4 θ 1  ∧ θ 2 − κ θ 3 ∧ θ 12 −  5 κ +6 κ +2 θ 22 + κ − 4 8 ξ 3  ∧ θ 11 + (2 κ + 4) ξ 2 ∧ θ 13 , dθ 12 = η 1 ∧ θ 12 + η 2 ∧ ξ 3 + η 3 ∧ ξ 1 + θ 2 ∧  θ 23 − ξ 2  +  θ 3 − κ − 4 8 θ 0 + 3 κ +4 κ +2 θ 12  ∧ θ 22 + 3 κ +4 8 ξ 3 ∧ θ 12 , dθ 13 = 3 2 η 1 ∧ θ 13 + η 2 ∧ ξ 2 + η 3 ∧ ξ 3 + η 4 ∧ ξ 1 + κ 3 +10 κ 2 − 32 κ − 96 64 θ 0 ∧ ξ 2 −  κ 3 + κ 2 − 14 κ − 24 16 θ 2 + κ 2 − 3 κ − 4 8 θ 23  ∧ θ 0 +  κ − 4 8 θ 22 − κ 2 − 16 64 ξ 3  ∧ θ 1 −  7 κ 2 +32 κ +24 8 θ 3 + ( κ + 2) θ 12  ∧ θ 2 +  (2 κ + 1) θ 23 + κ 2 +22 κ +24 8 ξ 2  ∧ θ 3 + 3 ( κ +2) 2 ξ 2 ∧ θ 12 −  4 κ +5 κ +2 θ 22 − κ +4 8 ξ 3  ∧ θ 13 , dθ 22 =  3 κ 2 +18 κ +24 8 θ 2 + ( κ + 2) ( θ 23 + ξ 2 )  ∧ ξ 1 dθ 23 = 1 2 η 1 ∧ θ 23 + η 2 ∧ ξ 1 −  3( κ +4) 8 θ 22 + 9 κ 2 +48 κ +112 64 ξ 3  ∧ θ 2 +  3 ( κ +4) 8 θ 23 − ξ 2  ∧ ξ 3 +  2 κ +3 κ +2 θ 23 − 3 κ − 4 8 ξ 2  ∧ θ 22 , dη 1 = 0 , dη 2 = η 6 ∧ ξ 1 +  η 1 + 2 (2 κ +3) κ +2 θ 22 − 3 ( κ +4) 8 ξ 3  ∧ η 2 + θ 3 ∧  3 ( κ − 4) 8 θ 22 − ξ 3  +  3 κ 2 − 16 κ +16 64 θ 22 − κ − 4 8 ξ 3  ∧ θ 0 +  3 ( κ +4) 8 θ 2 + ξ 2  ∧ θ 23 +  3 ( κ +4) 8 θ 22 + 9 κ 2 +48 κ +112 64 ξ 3  ∧ θ 12 , dη 3 = η 6 ∧ ξ 3 + η 7 ∧ ξ 1 +  ( κ + 3) θ 2 + ( κ + 2) ξ 2  ∧ η 2 +  κ − 4 8 θ 1 − θ 13  ∧ θ 22 +  3 2 η 1 + 5 κ +7 κ +2 θ 22 + 3 κ +4 8 ξ 3  ∧ η 3 + ( κ + 2) θ 3 ∧ ( θ 23 + ξ 2 ) + κ − 4 64  (3 κ 2 + 18 κ + 32) θ 2 + 8 ( κ + 2) ( θ 23 + ξ 2 )  ∧ θ 0 +  3 κ 2 +18 κ +32 8 θ 3 + 6 κ 2 +29 κ +28 4 θ 12  ∧ θ 2 −  3 ( κ + 1) θ 23 − 3 κ 2 +34 κ +32 8 ξ 2  ∧ θ 12 , dη 4 = η 6 ∧ ξ 2 + η 7 ∧ ξ 3 + η 8 ∧ ξ 1 +  2 η 1 − 2 (3 κ + 4) κ +2 θ 22 + κ +4 8 ξ 3  ∧ η 4 Symmetries and c overings for the r-th m dKP e quation 6 + κ  κ − 4 8 θ 0 − 2 θ 3  ∧ η 2 +  (2 κ + 4 ) θ 2 + 5 ( κ +2) 2 ξ 2  ∧ η 3 − κ 16  2 ( κ − 4) θ 3 − ( κ 2 − 2 κ − 4) θ 12  ∧ θ 0 + κ 8 (7 κ + 5 ) θ 3 ∧ θ 12 + 1 16  (2 κ 3 + 3 κ 2 − 30 κ − 112) θ 2 − 2 (2 κ 2 − 5 κ + 4) θ 23  ∧ θ 1 − 1 32 ( κ 3 + 10 κ 2 − 32 κ − 96) θ 1 ∧ ξ 2 −  6 ( κ + 1) θ 23 + 1 4 ( κ 2 + 30 κ + 36 ) ξ 2  ∧ θ 13 − 1 8  20 κ 2 + 101 κ + 92  θ 2 ∧ θ 13 − 1 64  8 ( κ − 4) θ 22 − ( κ 2 − 16 ) ξ 3  ∧ θ 11 , dη 5 = − κ η 6 ∧ θ 0 + η 7 ∧ ξ 2 + η 8 ∧ ξ 3 + η 9 ∧ ξ 1 +  5 2 η 1 − 7 κ + 9 κ +2 θ 22 − κ − 4 8 ξ 3  ∧ η 5 + (5 κ + 3) η 2 ∧ θ 1 + η 3 ∧  2 κ 2 +17 κ − 4 8 θ 0 + ( κ − 1) θ 3  − 3 ( κ + 1) θ 12 ∧ θ 13 . +  (3 κ + 5 ) θ 2 + 3 ( κ + 2) ξ 2  ∧ η 4 −  9 ( κ + 1) θ 23 − 3 κ 2 − 78 κ − 96 8 ξ 2  ∧ θ 11 +  κ 3 − 22 κ 2 +52 κ +90 32 θ 1 − κ 2 +2 κ − 8 4 θ 13  ∧ θ 0 − 2 ( κ + 2) θ 3 ∧ θ 13 +  κ 2 − 22 κ − 20 4 θ 3 − 6 κ 2 +39 κ +24 4 θ 12  ∧ θ 1 − 24 κ 2 +71 κ +64) 4 θ 2 ∧ θ 11 , where ξ 1 = q − 1 dt, ξ 2 = q u 2 xx  κ +1 2 u 2 x − u y  dt + d x − u x dy  , ξ 3 = u xx ( dy − ( κ + 2) u x dt ) , (6) η 1 = 2 dq q + 2 (2 κ + 3) κ + 2 du xx u xx , and q = B 1 1 6 = 0. W e need not explicit expressions f or the other mc fo rms in the sequel. In the case of κ = − 1 the con tact symmetry pseudo-group of Eq. (5) has the follo wing structure equations dθ 0 =  η 1 + 5 8 ξ 3 + θ 22  ∧ θ 0 + ξ 1 ∧ θ 1 + ξ 2 ∧ θ 2 + ξ 3 ∧ θ 3 , dθ 1 = 3 2 η 1 ∧ θ 1 + η 2 ∧  1 8 θ 0 + θ 3  −  9 8 θ 2 + θ 23 − 1 2 ξ 2  ∧ θ 0 − 5 8 θ 1 ∧ ξ 3 + ξ 1 ∧ θ 11 + ξ 2 ∧ θ 12 + ξ 3 ∧ θ 13 , dθ 2 = 1 8  4 η 1 + ξ 3  ∧ θ 2 + ξ 1 ∧ θ 12 + ξ 2 ∧ θ 22 + ξ 3 ∧ θ 23 , dθ 3 = η 1 ∧ θ 3 + 1 2 η 2 ∧ θ 2 − 5 64  8 θ 22 + 3 ξ 3  ∧ θ 0 + 3 8 ξ 3 ∧ θ 3 + ξ 1 ∧ θ 13 + ξ 2 ∧ θ 23 + ξ 3 ∧ θ 12 , dξ 1 = − 1 2 ( η 1 − 2 θ 22 ) ∧ ξ 1 , dξ 2 = − 1 8 (5 θ 0 + 8 θ 3 ) ∧ ξ 1 + 1 2  η 1 + 2 θ 22 + ξ 3  ∧ ξ 2 − 1 2 ( η 2 + θ 2 ) ∧ ξ 3 , dξ 3 = − ( η 2 + θ 2 ) ∧ ξ 1 + θ 22 ∧ ξ 3 , dθ 11 = 2 η 1 ∧ θ 11 + η 2 ∧  3 4 θ 1 + 2 θ 13  + η 4 ∧ ξ 2 + η 5 ∧ ξ 3 + η 6 ∧ ξ 1 −  θ 22 − 5 8 ξ 3  ∧ θ 11 +  η 3 − θ 3 − 7 4 θ 1 2  ∧ θ 0 −  1 2 θ 2 − θ 23 − 1 2 ξ 2  ∧ θ 1 + θ 2 ∧ θ 13 + θ 3 ∧ θ 12 , dθ 12 = η 1 ∧ θ 12 + η 2 ∧  1 8 θ 2 + θ 23 + ξ 2  + η 3 ∧ ξ 3 + η 4 ∧ ξ 1 + 5 8 θ 0 ∧ θ 22 − ξ 2 ∧ θ 23 −  θ 23 − 9 8 ξ 2  ∧ θ 2 + ( θ 3 + θ 12 ) ∧ θ 22 + 1 8 ξ 3 ∧ θ 12 , dθ 13 = 3 2 η 1 ∧ θ 13 + 1 64 η 2 ∧ (35 θ 0 + 24 θ 3 + 96 θ 12 ) + η 3 ∧ ξ 2 + η 4 ∧ ξ 3 + η 5 ∧ ξ 1 Symmetries and c overings for the r-th m dKP e quation 7 + 5 16  2 θ 2 + ξ 2  ∧ θ 0 − 5 64  8 θ 22 − 3 ξ 3  ∧ θ 1 +  1 8 θ 3 + θ 12  ∧ θ 2 + θ 3 ∧ θ 23 −  θ 22 − 3 8 ξ 3  ∧ θ 13 , dθ 22 = 1 8  4 η 2 + 9 θ 2 + 4 ξ 2 + 8 θ 23  ∧ ξ 1 , dθ 23 = 1 2 η 1 ∧ θ 23 + 1 2 η 2 ∧  θ 22 + ξ 3  + η 3 ∧ ξ 1 + 1 64 θ 2 ∧  72 θ 22 − 73 ξ 3  +  θ 23 + 3 8 ξ 2  ∧ θ 22 + 1 8  9 θ 23 + 4 ξ 2  ∧ ξ 3 , dη 1 = 0 , dη 2 = 1 2  η 1 + ξ 3  ∧ η 2 − 5 8 θ 0 ∧ ξ 1 −  θ 2 + 1 2 ξ 2  ∧ ξ 3 − θ 3 ∧ ξ 1 − θ 22 ∧ ξ 2 , dη 3 = η 7 ∧ ξ 1 + η 1 ∧ η 3 + 1 64 η 2 ∧  41 θ 2 + 72 θ 23 + 36 ξ 3  + 1 8 η 3 ∧  16 θ 22 + 9 ξ 3  + 5 64  7 θ 22 + 8 ξ 3  ∧ θ 0 − 1 64  72 θ 23 + 41 ξ 2  ∧ θ 2 −  1 8 θ 22 + ξ 3  ∧ θ 3 + 1 64  72 θ 22 + 73 ξ 3  ∧ θ 12 + 1 8 θ 23 ∧ ξ 2 , dη 4 = η 7 ∧ ξ 3 + η 8 ∧ ξ 1 + 1 8 (5 θ 0 + 8 θ 3 + 6 θ 12 ) ∧ η 2 +  2 η 2 + 2 θ 2 − ξ 2  ∧ η 3 + 1 8  12 η 1 − 16 θ 22 + ξ 3  ∧ η 4 + 5 64 θ 0 ∧ (17 θ 2 + 8 θ 23 ) − 5 8 θ 1 ∧ θ 22 + 1 8 (17 θ 3 + 10 θ 12 ) ∧ θ 2 + θ 3 ∧ θ 23 − 5 8 θ 12 ∧ ξ 2 − θ 13 ∧ θ 22 , dη 5 = η 7 ∧ ξ 2 + η 8 ∧ ξ 3 + η 9 ∧ ξ 1 − 5 32 η 2 ∧ (7 θ 1 + 8 θ 13 ) +  5 8 θ 0 + 2 θ 3  ∧ η 3 +  5 2 η 2 + 2 θ 2  ∧ η 4 +  2 η 1 − 2 θ 22 + 3 8 ξ 3  ∧ η 5 + 5 16 ( θ 12 − 2 θ 3 ) ∧ θ 0 − 5 16  5 θ 2 + 2 θ 23 + 2 ξ 2  ∧ θ 1 + 13 8 θ 12 ∧ θ 3 − 5 64  8 θ 22 + 3 ξ 3  ∧ θ 11 − 1 8  11 θ 2 + 4 ξ 2  ∧ θ 13 , dη 6 = η 7 ∧ θ 0 + η 8 ∧ ξ 2 + η 9 ∧ ξ 3 + η 10 ∧ ξ 1 + η 2 ∧  η 5 − 15 8 θ 11  − 2 η 3 ∧ θ 1 − η 4 ∧  19 8 θ 0 + 2 θ 3  − 2 η 5 ∧ θ 2 − 1 8 η 6 ∧  20 η 1 + 16 θ 22 − 5 ξ 3  + 1 32 (5 θ 1 + 72 θ 13 ) ∧ θ 0 + 3 4 ( θ 3 + 3 θ 12 ) ∧ θ 1 + 5 4 θ 11 ∧ θ 2 + 2 θ 13 ∧ θ 3 with ξ 1 = q − 1 dt, ξ 2 = q u 2 xx  ( s 2 u 2 xx − s u x u xx − u y ) dt + d x − s u xx dy  , ξ 3 = u xx (( u x − 2 s u xx ) dt + d y ) , (7) η 1 = 2 dq q + 2 du xx u xx , where s = B 2 3 ∈ R . 4. Cov erings of r-mdKP F ollo wing [33, 34], w e find linear combinations o f the mc forms (6) and (7), whic h pro - vide co v erings of Eq. (2) in the cases of κ 6∈ { − 2 , − 3 2 , − 1 } , κ = − 3 2 , and κ = − 1, resp ectiv ely . 4.1. Gener al c ase When κ 6∈ {− 2 , − 3 2 , − 1 } , we ta ke the following linear combination of the mc forms (6) ω = η 1 − λ 1 ξ 1 − λ 2 ξ 2 − λ 3 ξ 3 Symmetries and c overings for the r-th m dKP e quation 8 = 2 dq q + 2 (2 κ + 3) κ + 2 du xx u xx − λ 2 q u xx dx + ( λ 2 q u x u xx − λ 3 u xx ) dy +  λ 2 q  u y − κ + 1 2 u 2 x  u 2 xx + λ 3 ( κ + 2) u x u xx − λ 1 q − 1  dt, with λ 1 , λ 2 , λ 3 ∈ R , and put q = − ( κ + 1) v ( κ + 2) v 2 κ +3 1 , u xx = v κ +2 1 v , where v and v 1 are new indep enden t v ar iables. This giv es ω = − 2 ( κ + 1 ) ( κ + 2) v ( dv − A v 1 dt − v 1 dx − B v 1 dy ) (8) with A = − ( κ + 2) 2 4 ( κ + 1)  λ 1 λ 2 v 2( κ +1) 1 + 2 λ 3 ( κ + 1) u x v κ +1 1  + κ + 1 2 u 2 x − 2 u y , B = − u x − λ 3 ( κ + 2) 2 ( κ + 1) v κ +1 1 . The form (8) is equal to zero whenev er v 1 = v x and v t =  λ 1 λ 2 ( κ + 2) 2 4 ( κ + 1 ) 2 v 2( κ +1) x + λ 3 ( κ + 2) 2 2 ( κ + 1 ) v κ +1 x + κ + 1 2 u 2 x − u y  v x , v y = −  λ 3 ( κ + 2) κ + 1 v κ +1 x + u x  v x . This system is compatible, i.e., ( v t ) y = ( v y ) t , whenev er u y y − u tx −  κ + 1 2 u 2 x + u y  u xx − κ u x u xy + ( κ + 2) 2 4 ( κ + 1 ) 2 v 2( κ +1) x  λ 2 3 ( κ + 2) 2 − λ 1 λ 2 (2 κ + 3 )  = 0 . This equation coincides with Eq. (2) iff λ 2 3 ( κ + 2) 2 − λ 1 λ 2 (2 κ + 3 ) = 0. So we put λ 1 = λ 2 3 ( κ + 2) 2 λ 2 (2 κ + 3 ) . This yields v t =  λ 2 3 ( κ + 2) 4 (2 κ + 3) ( κ + 1) 2 v 2( κ +1) x + λ 3 ( κ + 2) 2 2 ( κ + 1) u x v κ +1 x + κ + 1 2 u 2 x − u y  v x , v y = −  λ 3 ( κ + 2) κ + 1 v κ +1 x + u x  v x . When λ 3 = 0, we hav e v t =  κ + 1 2 u 2 x − u y  v x , v y = − u x v x . (9) In t he case of κ = 0 this co v ering for Eq. (3) was obtained in [3 4]. When λ 3 6 = 0, w e put v =  λ 3 ( κ +2) 2 ( κ +1)  1 / ( κ +1) w . Then w t =  ( κ + 2) 2 2 κ + 3 w 2( κ +1) x + ( κ + 2) u x w κ +1 x + κ + 1 2 u 2 x − u y  w x , w y = −  w κ +1 x + u x  w x . (10) Symmetries and c overings for the r-th m dKP e quation 9 F o r κ = 0 this co v ering of (3) w as found in [5] by means of another tec hnique and in [34] via the method described ab ov e. F or κ = 1 the cov ering (10) of Eq. (4) was obtained in [18]. F ro m (9) we hav e u x = − v y v x , u y = κ + 1 2  v y v x  2 − v t v x . (11) The in tegrability condition ( u x ) y = ( u y ) x of this system giv es v y y = v tx +  ( κ + 1) v 2 y v 2 x − v t v x  v xx − κ v y v x v xy . (12) F o r κ = 0 this equation was obtained in [2]. Also, fro m (10 ) we get u x = − w y w x − w κ +1 x , u y = − w t w x + ( κ + 1) w 2 y 2 w 2 x − w κ x w y − ( κ + 1) w 2( κ +1) x 2(2 κ + 3) (13) This system yields w y y = w tx +  ( κ + 1) w 2 y w 2 x − w t w x + κ w κ x w y + ( κ +1) 2 2 κ +3 w 2( κ +1) x  w xx − κ  w y w x + w κ x  w xy . (14) Substitution for (11) in ( 10) give s a B¨ ac klund tr na sfo r ma t ion w t = ( κ + 2) 2 2 κ + 3 w 2 κ +3 x − ( κ + 2) v y v x w κ +2 x + v t v x w x , w y = − w κ +2 x + v y v x w x from Eq. (12) to Eq. (14). The inv erse B¨ ac klund transformation app ears f rom substi- tution for (13) in (9). 4.2. Case of κ = − 3 2 In the case of κ = − 3 2 w e take the followin g combination o f the mc forms (6) ω = η 1 − λ 1 ξ 1 − 4 ξ 2 = 2 dq q +  q u 2 xx ( u 2 x + 4 u y ) − λ 1 q − 1  dt − 4 q u xx ( dx − u x dy ) and the following c hange of v ariables: q = − v − 2 , u xx = ( v v 1 ) 1 / 2 . Then w e ha v e ω = − 4 dv v −  ( u 2 x + 4 u y ) v 1 v − λ 1 v 2  dt + 4 v 1 v dx − 4 u x v 1 v dy . This form is equal to zero whenev er v 1 = v x and v t = 1 4 λ 1 v 3 −  1 4 u 2 x + u y  v x , v y = − u x v x . This syste m is compatible for ev ery v alue o f λ 1 whenev er Eq. (2) with κ = − 3 2 is satisfied. When λ 1 = 0, we hav e Eqs. (9) with κ = − 3 2 : v t = −  1 4 u 2 x + u y  v x , v y = − u x v x . (15) When λ 1 6 = 0, we put v = 2 λ − 1 / 2 1 w . Then we g et w t = w 3 −  1 4 u 2 x + u y  w x , w y = − u x w x . (16) Symmetries and c overings for the r-th m dKP e quation 10 Exclusion of u x and u y from Eqs. (15) and (16) giv es equations v y y = v tx −  v 2 y 2 v 2 x + v t v x  v xx + 3 v y 2 v x v xy , (17) w y y = w tx −  w 2 y 2 w 2 x + w t w x − w 3 w x  w xx + 3 w y 2 w x w xy − 3 w 2 w x , (18) and a B¨ ac klund transforma t ion from (17) to (18): w t = w 3 + v t v x w x , w y = v y v x w x . 4.3. Case of κ = − 1 When κ = − 1, we take the fo llowing combination of the mc forms (7): ω = η 1 + 1 2 λ 2 3 ξ 1 + 2 ξ 2 − λ 3 ξ 3 = 2 q u 2 xx dx − u xx  2 q s u 2 xx + λ 3  dy + 2 dq q + 2 du xx u xx + 1 2 q  ( λ 3 + q u xx (2 s u xx − u x ) 2 − q 2 u 2 xx  u 2 x + 4 u y  dt. Then w e substitute f o r q = ( v v 1 ) − 1 , s = u x v − 1 1 , u xx = v 1 and o bta in ω = − 1 2 v  4 dv −  λ 2 3 v 2 + 2 λ 3 u x v − 4 u y  v 1 dt − 4 v 1 dx + 2 ( λ 3 v + 2 u x ) v 1 dy  . This form is equal to zero whenev er v 1 = v x and v t =  λ 2 3 4 v 2 + 1 2 u x v − u y  v x , v y = −  1 2 λ 3 v + u x  v x . This system is compatible for ev ery v alue of λ 3 whenev er Eq. (5) is satisfied. F or λ 3 = 0 w e hav e v t = − u y v x , v y = − u x v x . (19) When λ 3 6 = 0, we put v = 2 λ − 1 3 w . Then w e hav e w t =  w 2 + u x w − u y  w x , w y = − ( w + u x ) w x . (20) Excluding u x and u y from systems (19) and (20), we get equations v y y = v tx − v t v x v xx + v y v x v xy (21) and w y y = w tx − w t + w w x w x w xx + w y + w w x w x w xy , (22) corresp ondingly . The B¨ ac klund t r a nsformation b et w een Eqs. (21) and ( 2 2) has the form w t =  w 2 + v t − w v y v x  w x , w y =  w − v y v x  w x . REMARK 1. A one-parametric family of cov erings with a nonremo v able parameter v t = − ( u y − λ u x − λ 2 ) v x , v y = − ( u x + λ ) v x , (23) Symmetries and c overings for the r-th m dKP e quation 11 for Eq. (5) is presen ted in [7]. This family can b e obtained from (19) by means the follo wing tec hnique, [22, § 3.6] [19, ? , 30, 16]. Eq. ( 5) ha s t he infinitesimal symmetry X = y ∂ ∂ x + 2 x ∂ ∂ u , whic h can’t b e lifted into a symmert y of the cov ering (1 9). Then the deformat io n e λX transforms the co v ering (19) in to (2 3 ). Indeed, we hav e ˜ t = e λ X ( t ) = t, ˜ x = e λ X ( x ) = x + λ y , ˜ y = e λ X ( y ) = y , ˜ u = e λ X ( u ) = u + 2 λ x + λ 2 y , and therefore ˜ u ˜ t = e λ X ( u t ) = u t , ˜ u ˜ x = e λ X ( u x ) = u x + 2 λ, ˜ u ˜ y = e λ X ( u y ) = u y − λ u x − λ 2 . Since ˜ v = e λ X ( v ) = v , for the fo rm ˜ ω 1 = d ˜ v + ˜ u ˜ y ˜ v ˜ x d ˜ t − ˜ v ˜ x d ˜ x + ˜ u ˜ x ˜ v ˜ x d ˜ y , whic h defines the cov ering (19) in the t ilded v ariables, w e ha v e  e λX  ∗ ˜ ω 1 = d v + ( u y − λ u x − λ 2 ) v x dt − v x dx + ( u x + λ ) v x dy . This form defines the family of cov erings (23). Similarly , w e deriv e a new one-parametric family of co v erings with a nonremov able parameter from the co v ering (20). 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