Differential transformations of parabolic second-order operators in the plane

Differen tial T ransformations of P arab olic Second-Order Op erators in the Plane S.P . Tsarev ∗ Institut e of Mat hematics, Sib eria n F ederal Uni v ersity , Sv ob o d nyi a v en ue, 79 6600 4 1, Krasno y arsk . e-mail : spts arev@m ail.ru E. Shemy ak o v a † Researc h Instit ute for Symbol ic Computati on, J. Kepler Uni v ersity , Altenber g er Str. 69, Linz, Austri a. e-mail : kath @risc. uni-linz.ac.at No vem b er 2, 20 18 T o Ser gey Petr ovich Novikov, as a dev elopment of one of his ide as. 1 In t ro duct i o n The t heory o f transformations for h yp erb olic second-order equations in the pla ne, dev elop ed b y Darb oux, Laplace and Moutard, ha s man y applications in class ical differen t ia l geometry [12, 13], and b ey ond it in the theory of in tegrable sys tems [14, 19]. These results, whic h w ere obtained for the linear case, can b e applied t o non- linear D arb oux-in tegrable equations [2, 7, 15 , 16]. In the last decade, num erous generalizations of the classical theory hav e been deve lop ed. Among them there are generalizations to the case of systems of h yp erb olic equations in the plane [3 , 5, 6, 22], and generalizations to the case o f h yp erb olic equations with more than ∗ The author was suppor ted b y the Russian F o undation for Basic Res earch 06-01-008 14. † The a uthor was supp orted by the Austrian Science F und (FWF) under pro ject DIFFOP , Nr. P2033 6-N18. 1 t w o independen t v ariables [9, 2 3 ]. The non-hyperb olic case has b een mu ch less in v estigated [1 8, 20, 21]. Here, Darb oux’s classical results ab out transformations with differen tial substi- tutions for h yp erb olic equations are extended to the case of parab olic equations. Th us, consider for an arbitrary solution u of the e quation Lu = 0 , L = D 2 x + a ( x, y ) D x + b ( x, y ) D y + c ( x, y ) , b ( x, y ) 6 = 0 , (1) some Linear Partial Differential Op erat o r (LPDO) M and a new function v ( x, y ) = M u . One can easily compute that in the generic case v satisfies an o v erdetermined system of linear differen tial equ ations. Ho w ev er, there is some c hoice of M whic h leads to only one equation for v , namely , L 1 v = 0, where L 1 is an op erato r of the same fo rm (1) allb eit with possibly differen t co efficien ts a 1 ( x, y ), c 1 ( x, y ), b 1 ≡ b . In this case w e sa y that w e ha v e a differ ential tr ansformation of op erator L in to op erator L 1 with M , a nd denote this fact as L M − → L 1 . Also it is e asy to notice that in this case there m ust exist an op erator M 1 suc h that the follo wing equalit y holds: M 1 ◦ L = L 1 ◦ M , (2) that is the b oth parts of (2) define the left least common multiple lLC M ( L, M ) in the ring K [ D ] = K [ D x , D y ] of LPDOs in the plane. F or the case of h yp erb olic op erator s o f the form L H = D x D y + a ( x, y ) D x + b ( x, y ) D y + c ( x, y ) (3) there are quite complete results on the p ossible form of the op erators M that satisfy (2) (see [25, Ch. VII I]): in the generic case the op erator M can be determined (up to an arbitrar y multiplier) from M z i = 0, i = 1 , . . . , k , where z i ( x, y ) ar e indep enden t solutions of L H z i = 0. There are also some degenerate cases. As w as disco v ered b y Darb oux, one of those degenerate cases is the c lassical Laplace t r ansformation, whic h is defined b y the co efficien ts of op erato r (3) only . Relation (2) for the “intert wining op erator” M is widely used in the study of in tegrability problems in tw o- and one- dimensional cases [1 , 11]. In this pap er, w e prov e general Theorem 3.1 that provide s a w a y to determine transformations L M − → L 1 for parab olic equations (1). It turned out (Theorem 4.2) that transforming o p erat o rs M of some higher order can b e alwa ys r epresen ted as a composition of some first-order op erators that consecutiv ely define a series of transformations of t he op erators of the form (1 ) . Unlik e the classical case o f t he Laplace and Moutard transformatio ns, the trans- formations considered in this pap er are not inv ertible. In this resp ect the problem in question is analo gous to the generic case that w as considered in [25, Ch. VI I I]) for op erators (3). As follow s from Theorems 3.1, 4.2 for parab olic op erators (1) there are no degenerate cases lik e L a place transformations for arbitr ary op erators (3): an y differen t ia l transformation of the op erator (1) can b e determined b y an o p erator M of the form (11). It is of interes t to consider the problem of the existence of an in v erse transformation L 1 N − → L . The order of the in v erse ma y b e higher than the 2 order of the initia l transformat io n L M − → L 1 . Examples show that the existence of suc h an inv erse implies some differential constrains on the co efficien ts of the initial op erator L . In Sec. 5 w e show that these relat io ns can imply famous inte grable equations, in particular, the Boussinesq equation. This result is an analogue of re- sults [10, 14, 2 4] for p erio dic c hains of La pla ce t r a nsformations for the op erators (3), whic h also lead to in tegrable non-linear equations. Authors a r e thankful to M.V. P a vlov for useful discussions. 2 Basic Definit i ons and Auxiliary Results Consider a field K of characteristic zero with commuting deriv a tions ∂ x , ∂ y , and the ring of linear differen t ial op erators K [ D ] = K [ D x , D y ], where D x , D y corresp ond to the deriv a tions ∂ x , ∂ y , resp ectiv ely . In K [ D ] the v ariables D x , D y comm ute with eac h other, but not with elemen ts of K . F or a ∈ K w e hav e D i a = aD i + ∂ i ( a ). An y op erator L ∈ K [ D ] has the for m L = P d i + j =0 a ij D i x D j y , where a ij ∈ K . The p olynomial Sym L = P i + j = d a ij X i Y j in formal v aria bles X , Y is called the ( principal) symb ol of L . Belo w w e assume that the field K is differentially closed unless stated otherwise, that is it con tains solutions of (non- linear in t he generic case) differen tial equations with co efficien ts from K . Let K ∗ denote the set of in v ertible elemen ts in K . F or L ∈ K [ D ] and eve ry g ∈ K ∗ consider the g auge transformation L → L g = g − 1 ◦ L ◦ g . Then an algebraic differen t ia l expression I in the coefficien ts of L is (differen tial) invariant under the gauge transformations (w e consider only these in the presen t pap er) if it is unaltered b y these transformations. T rivial examples of inv a rian ts are the co efficien ts of the sym b ol of an operat o r. A generating set of in v ariants is a set using whic h all p ossible differen t ia l in v ariants can b e expressed. Theorem 2.1. [17, 26] The action of the ga uge g r oup on op er ators of the form (1) has the fol lowing gener ating system of inv a riants: I 1 = b , I 2 = c x − aa x / 2 − ba y / 2 − a xx / 2 +( b x a 2 / 4 − b x c + b x a x / 2) /b . Note that if an op erato r (1) has only constan t co efficien ts then I 1 is a constan t and I 2 = 0. If the field of co efficien ts K con tains quadratures (differentially closed), it is easy to prov e the in v erse statemen t: Prop osition 2.2. L et the field of c o effi c ients K b e differ ential ly close d. Th e e quiv- alenc e class of (1) with r esp e ct to gauge tr ansformations c on tains an op er ator with c onstant c o efficie n ts i f and only if I 1 is a c o nstant and I 2 = 0 . Pr o of . Let I 1 = b hav e a constant v alue and I 2 = 0. Consider an op erator L = D 2 x + aD x + bD y + c from the equiv alence class. Using t he gauge tra nsformation with 3 g = exp  − 1 2 R a dx  one can make a = 0. Then I 2 = 0 implies 0 = c x − b x c/b . Since I 1 = b is a constan t, w e hav e c = c ( y ). Applying the ga uge transformation with g = e R − c/bdy to L w e obta in L g = D 2 x + bD y , whic h has constan t co efficien ts. So ev ery op erator (1) with constan t I 1 = b and I 2 = 0 can be transformed in to op erator D 2 x + D y using substitution y 7→ const · y and gauge transformat ions. Lemma 2.3. Without loss o f gener ality on e c an divide the s ymb ols Sym ( M ) = Sym( M 1 ) by any non- z er o g ∈ K . The op er ator L an d the symb ol o f L 1 ar e left unchange d. Pr o of . Indeed, m ultiply the b oth sides of (2 ) b y 1 /g on the left: 1 g M ◦ L = 1 g L 1 g ◦ 1 g M 1 = L g 1 ◦ 1 g M 1 . Then “ new” M and M 1 ha v e the co efficien ts of t he “ o ld” ones divided b y g , while L 1 is sub jected to the ga uge transformation with g , and, therefore, its sym b ol is unch anged, while the other co efficien ts can b e c hanged. Lemma 2.4 (Simplification b y gauge transformations) . In (2) one c an assume without los s of gene r ality that a = 0 , that is ther e e x i sts a gauge tr ansformation that tr a nsforms L , M , L 1 and M 1 into op er a tors of the same form such that the c o effici e nt of L at D x is 0 , and the e quality M ◦ L = L 1 ◦ M 1 (2) is pr eserve d. Pr o of . It is enough to apply the g auge transformation with g = exp( − 1 2 R a dx ) to all op erato r s in (2). This gauge transformation do not alter the sym b ols o f the op erators, and, therefore, do es not in terfere with the simplifications from Lemma 2.3. 3 First-Order T ransformations Consider L of the form (1) and an opera t o r L 1 of the same form: L 1 = D 2 x + a 1 ( x, y ) D x + b 1 ( x, y ) D y + c 1 ( x, y ). Then a differen tial transformat io n of the first- order that tra nsfor ms L in to L 1 exists if there exist M = p ( x, y ) D x + q ( x, y ) D y + r ( x, y ) , M 1 = p 1 ( x, y ) D x + q 1 ( x, y ) D y + r 1 ( x, y ) suc h that ( 2 ) holds. The comparison of the sym b ols implies p 1 = p , q 1 = q . First consider the c ase p 6 = 0 , q 6 = 0 . By lemma 2.3 without loss of generalit y one can assume p = 1, and a = 0 b y lemma 2.4. Equating the co efficien ts in (2) w e ha v e a 1 = − 2 q x q , b 1 = b , c 1 = ( − 2 bq x + b x q + q 2 c + q 2 b y + 2 q 2 x − bq y q − q xx q ) /q 2 , r 1 = r − 2(ln q ) x , and tw o constrains on the co efficien ts of t he op erato rs L and M : C 1 = 0, C 0 = 0, where C 0 = − 2 q cq x + c x q 2 + q 3 c y + 2 r bq x − r b x q − r q 2 b y − 2 r q 2 x + + r bq y q + r q xx q + 2 q x r x q − br y q 2 − r xx q 2 , (4) C 1 = − 2 bq x + b x q + q 2 b y + 2 q 2 x − bq y q − q xx q − 2 q x r q + 2 r x q 2 . (5) 4 W e see f rom (4), (5) that giv en the co efficien ts o f the op erator L , one can alw a ys find solutions r , q of these equations in the differen tially closed field K , that is ev- ery o p era t or (1) a dmits infinitely many transformations with differen t op erators M . The equations (4), (5) f or r , q can b e solve d explicitly with the help of tw o a rbi- trary (indep enden t) generic solutions o f the equation (1). Indeed, g iv en a first-order op erator M that satisfies the constrain (2), the following system of equations  Lu = 0 , M u = 0 , (6) is consisten t and has a tw o-dimensional space of solutions, whic h is para meterized, for example, by the v alues u ( x 0 , y 0 ), u y ( x 0 , y 0 ). In fact, we can express the deriv ativ es of u of a ny order with resp ect to x in terms o f its deriv atives with resp ect to y from the second eq uation M u = 0. Substituting those in to the first equation Lu = 0, w e hav e an expression for the second deriv ative u y y , pr ovided q 6 = 0. On the other ha nd the consistency of (6) is guaranteed b y (2), whic h can b e rewritten as q D y Lu − D 2 x M u = 0 mo d ( L, M ) . Conv ersely , a basis z 1 ( x, y ), z 2 ( x, y ) in the space of solutions of (6) allo ws us t o reconstruct M : the conditio ns M z 1 = 0, M z 2 = 0 giv e a system of tw o linear algebraic equations for the co efficien ts r , q , and w e can easily determine t he op erato r M : M u =       u u y u x z 1 ( z 1 ) y ( z 1 ) x z 2 ( z 2 ) y ( z 2 ) x       ·     z 1 ( z 1 ) y z 2 ( z 2 ) y     − 1 . (7) Since the v alues z i ( x 0 , y 0 ), ( z i ) y ( x 0 , y 0 ) are lineally indep enden t, t he denominator of this expression is non-zero. Vice v ersa, the c hoice of tw o ar bit r a ry lineally indep enden t solutions z 1 , z 2 of the equation (1) defines the op erator M by the f o rm ula (7). The op erator M in its turn implies a differential transformat ion of L , that is the equality (2). Indeed, compute the deriv ativ es v x , v y , v xx of the function v = M u for an arbitrary solution u of t he equation (1), then using (1) we can remo v e all the terms that contain u xx , u xxx , u xxy . Using an appropriate combination ˜ Lv = v xx + a 1 ( x, y ) v x + b 1 ( x, y ) v y w e can also remo v e the terms with u xy , u y y , leaving u x , u y , u only . Since the expression ˜ Lv v anishes after the substitution u = z i it mus t b e prop ortiona l to M u : ˜ Lv = ˜ L ( M u ) = c 1 ( x, y ) M u , whic h implies (2) with L 1 = ˜ L − c 1 for an arbit r a ry function u ( x, y ). Note that in the considered case the co efficien ts a t D y , D x in M are non- zero. F rom now on w e refer to suc h transformatio ns as X + q Y -tra nsfor ma t ions. Below w e consider the cases when one or another of the co efficien ts is zero separately . Therefore, w e will pro v e the following statemen t: Theorem 3.1. F or every op er a tor L = D 2 x + aD y + bD y + c ther e e xist infinitely many differ ential tr ansformation s w ith op er ators M = D x + q ( x, y ) D y + r ( x, y ) . If q 6 = 0 then the op er ator M is define d by the c ondition s M z 1 = 0 , M z 2 = 0 , wher e z i ar e two arbitr ary chosen ind e p endent solutions of the e quation (1 ) . The o p er ators 5 of the form M = D x + r ( x, y ) ar e define d by the choic e o f one solution z 1 of the e quation (1 ) a nd by the c on dition M z 1 = 0 . The intertwining op er ator o f the form M = D y + r ( x, y ) do es not ex i s t fo r generic L . The degenerate cases of o p era t o rs M of forms M = D x + r and M = D y + r are considered b elo w. Case p 6 = 0 , q = 0 ( M = D x + r ) Without lo ss of generality one can assume p = 1 a nd a = 0. If w e equate the corresp onding co efficien ts in (2), w e hav e a 1 = − ln( b ) x , b 1 = b , c 1 = c + r ln( b ) x − 2 r x , r 1 = r − ln( b ) x and an equation 0 = − c ln( b ) x + c x − r 2 ln( b ) x + 2 r r x + r x ln( b ) x − br y − r xx , (8) for r . W e apply the same tric k as in the non- degenerate case in order to determine the o p erat or M in terms of solutions of the initial equation ( 1). Now w e c ho ose one solution z 1 and require M to satisfy the condition M z 1 = 0. W e get M ( u ) =     u u x z 1 ( z 1 ) x     · z − 1 1 . (9) Indeed, given an op erator M suc h that the in tert wining equality (2) holds, a n ap- propriate z 1 is f o und as a solution of the consisten t system (6), whic h no w has a one-dimensional solution space. Con v ersely , g iv en a solution z 1 of the equation (1), M can b e fo und f r o m ( 9 ), then fo r v = M u the deriv ativ es v x , v y , v xx are simplified using (1). Then an appropriate combination ˜ Lv = v xx + a 1 ( x, y ) v x + b 1 ( x, y ) v y con tains only u x and u (there are no terms with u y y !). The obtained expression ˜ Lv v anishes if we substitute u = z 1 and therefore it m ust b e prop ortional to M u , whic h implies (2). Later on we refer to suc h transforma t ions as X -transformations. Case p = 0 , q 6 = 0 ( M = D y + r ) Without lo ss of generalit y w e can assume q = 1, a = 0. If w e equate the corresp onding co efficien ts in (2), w e obtain in part icular r x = 0, c y − r b y − br y = 0. Th us, r = r ( y ) can b e found only for some particular functions b, c and for an arbitrarily c hosen L = D 2 x + aD x + bD y + c t here is no differen tia l transformatio ns with M = D y + r . Notice also that an attempt to construct M by the formula M ( u ) =     u u y z 1 ( z 1 ) y     · z − 1 1 . w ould not lead to a ny success either: for suc h an op erator M and v = M u the deriv ativ es v x , v y , v xx simplified with (1) would con tain u xy , u y y , u x , u y , u , and we cannot not find a n appropriate com bination ˜ Lv = v xx + a 1 ( x, y ) v x + b 1 ( x, y ) v y ha ving only u y , u . Therefore, Theorem 3.1 is pro v ed. 6 Note that when differential transformations with M = D x + q D y + r ar e applied to the op erator (1), the new v alues of the basic inv arian t s (that is the v alues of in v ariants I 1 and I 2 for L 1 ) are I 1 1 = I 1 = b , I 1 2 = I 2 − 2 bq xx /q 2 − b 2 x / ( q b ) − b x b y /b + b xx /q + b xy − b x q x /q 2 + 4 q 2 x b/q 3 . When differen tial transforma t io ns with M = D x + r are applied the new v alues of the basic inv a rian ts ar e I 1 1 = I 1 = b , I 1 2 = I 2 − 1 / 4(8 b 3 r xx − 12 b x r x b 2 + 8 r bb 2 x − 4 r b 2 b xx + 2 b 2 b y b x − 9 b 3 x − 2 b 2 b xxx − 10 b x bb xx − 2 b 3 b xy ) /b 3 . Example 3.2. Consider an op erator L = D xx + 2 x + 2 y x 2 D y − 2 x 2 . The equation L ( z ) = 0 has the following solutio ns z 1 = x 2 , z 2 = x + y . Using the determinan t a l form ula (7) compute M = D x + x + 2 y x D y − 2 x , and M 1 = D x + x +2 y x D y − 2 x +2 y , L 1 = D 2 x − 4 y x ( x + 2 y ) D x + ( 2 x + 2 y x 2 ) D y − 6 ( x + 2 y ) x − 4 y ( x + 2 y ) x 2 . Note t hat L 1 cannot be obtained fro m L by an y gauge transformation. Indeed, the v alue of t he inv aria n t I 2 for L is I 2 = 2 x 2 ( x + y ) , while the v alue of I 2 for L 1 is I 1 2 = 2( x 2 − 2 xy − 4 y 2 ) x ( x + y )( x +2 y ) 3 . Example 3.3. Applying the differen tial transformation with M = D x + q ( x, y ) D y + r ( x, y ) to L = D 2 x + D y (pro vided conditions (4) and (5) are satisfied or equiv alen tly , pro vided M is in the for m (7)) w e ha v e M 1 = D x + q ( x, y ) D y + r − 2(ln q ) x and L = D 2 x + D y − → L 1 = D 2 x − 2 q x /q D x + D y + ( q y q + q xx q − 2 q 2 x + 2 q x ) /q 2 , I 2 = 0 − → I 1 2 = − 2 q xx /q 2 + 4 q 2 x /q 3 . Example 3.4. Applying the differen tial tra nsformation with M = D x + r ( x, y ) to L = D 2 x + D y (pro vided t he condition (8) is satisfied or equiv alen tly , pro vided M is in the fo r m(9 )) w e hav e M 1 = M and L = D 2 x + D y − → L 1 = D 2 x + D y − 2 r x , I 2 = 0 − → I 1 2 = − 2 r xx . 7 4 T ransformations of Arb i trary Order W e show that differen tial transformatio ns of arbitra r y order o f a generic op erator (1) can b e expressed in terms of some n um b er of partial solutions of ( 1 ). In [25, Ch.VI I I] analogous formulae w ere in tro duced for hyperb olic o p erators (3). First of all, giv en some transforming op erator M of higher order satisfying (2), w e can use the op erator L t o remov e all terms having deriv ativ es with resp ect to y (generally sp eaking, this manipulation increases the order of M ). The resulting op erator has the form M = m X i =0 q i ( x, y ) D i x , q m 6 = 0 . (10) Belo w w e call the cor r esponding transformation an ( m ) -tr a n sformation . Theorem 4.1. Given an op er ator (1) and m line al ly indep endent gene ric p artial solutions z 1 , . . . , z m of the c o rr esp ondi n g e quation L ( z ) = 0 , then ther e exis ts a differ ential tr ansformation w ith M u = ϑ ( x, y )          u ∂ u ∂ x . . . ∂ m u ∂ x m z 1 ∂ z 1 ∂ x . . . ∂ m z 1 ∂ x m . . . . . . . . . . . . z m ∂ z m ∂ x . . . ∂ m z m ∂ x m          (11) wher e ϑ ( x, y ) is arbitr ary. Con v ersely, every ( m ) -tr ansformation of an op er ator of the fo rm (1) c orr esp on d s to so m e op er ator M of the form (11) . Pr o of . Ha ving computed the deriv ativ es v x , v y , v xx of v = M u for an a rbitrary solution u of equation (1), we use (1) as ab o v e to remov e all terms that contain deriv ativ es with resp ect to y . The remaining terms will con tain only some linear com binations o f the deriv ativ es D s x u , s = 0 , . . . , m + 2. Cho osing some appropriate com bination ˜ Lv = v xx + a 1 ( x, y ) v x + b 1 ( x, y ) v y w e can remo v e terms with D m +2 x u , D m +1 x u , and leav e terms with D s x u , s = 0 , . . . , m o nly . Since the resulting express ion ˜ Lv v anishes when w e substitute a n y u = z i , we conclude that it m ust b e prop ortiona l to M u : ˜ Lv = ˜ L ( M u ) = c 1 ( x, y ) M u , whic h implies (2) with L 1 = ˜ L − c 1 for an arbitrary function u ( x, y ). The o nly requiremen t is the non-v a nishing o f the W ronskian det( D j x z i ), i = 1 , . . . , m , j = 0 , . . . , m − 1. Con v ersely , giv en the in tertw ining op erato r M of the form (10) satisfying ( 2 ), consider the system (6). The consistency o f the system is equiv alen t to ( 2 ), whic h al- lo ws us to c ho ose a basis of its m solutions with non- v anishing W ronskian det( D j x z i ), i = 1 , . . . , m , j = 0 , . . . , m − 1, and o btain the required form (11) o f the op erator M . Theorem 4.2. An arbi tr ary ( m ) -tr ansfo rm ation of an op er ator (1 ) with m > 1 c an b e r epr esente d as a c omp osition of first-or d e r differ ential tr an s f o rmations. 8 Pr o of . Consider a n op erator M in the form (11) and the corresp onding solutions z i . Then z 1 generates a first-order transformation with ˆ M of the form (9), whic h trans- forms L in t o some ˆ L of the same f orm (1). Others z i , i = 2 , . . . , m are transfor med in to solutions ˆ z i = ˆ M z i of the equation ˆ L ˆ z = 0. Since M z 1 = 0, ˆ M z 1 = 0, then if w e divide the ordinary differen tial op erat o r M b y ˆ M , the r emainder is zero: M = P ˆ M , P ∈ K [ D x ]. (2) implies that the op erator l LC M ( L, M ) = M 1 L = L 1 M is divisible b y l LC M ( L, ˆ M ) = ˆ M 1 L = ˆ L ˆ M , that is M 1 L = N 1 ˆ M 1 L = N 1 ˆ L ˆ M = L 1 M = L 1 P ˆ M , whic h implies N 1 ˆ L = L 1 P . Th us w e hav e o bt a ined a n in tert wining op erator P , whose o r der is less by one, suc h that ˆ L P − → L 1 . The induction by the order m of the intert wining op erator completes the pro of. 5 Generalize d Mo utard T ransformatio n s and Dif- feren tial T ransfor matio ns. P erio dical Differen- tial T rans formations An imp ortant sub class of the considered class of the pa r ab olic op erators are o p era - tors L = D 2 x − D y + c ( x, y ) . (12) In [8], a mo dification o f Moutar d transformations for suc h o p era t o rs w as suggested and applications to the construction of solutions in the Kadom tsev— P etviash vili (KP) hierar ch y of equations w ere giv en. As w e sho w b elow, some of the examples considered in [8] can also b e obtained b y our metho d. D irect application of the ab ov e results prov es the following lemma. Lemma 5.1. X -tr ansformations pr ese rve the class of the op er ators (12) . F or M = D x + r ( x, y ) the c ondition (8) for the existenc e of such tr an sformations has the fol lowing fo rm : c x + 2 r r x + r y − r xx = 0 , (13) and M 1 = M , L 1 = D 2 x − D y + c − 2 r x . (14) The b asic invariant I 2 tr ansforms as fol lows: I 2 = c x − → I 1 2 = c x − 2 r xx . If the op er ator M is given in the f orm (9) for some p artial solution z 1 = z 1 ( x, y ) of the e quation L ( z ) = 0 , we h ave L 1 = D 2 x − D y − 2 z 2 1 x − z 1 z 1 y − z 1 xx z 1 z 2 1 . Note that X + q Y -transformatio ns do no t preserv e the class of op erators (12): Example 5.2. The equation ( D 2 x − D y ) z = 0 has partial solutio ns z 1 = x , z 2 = e x + y . The form ula (7) implies M = D x +  1 x − 1  D y − 1 x and L 1 = D 2 x − 2 x ( x − 1) D x − D y − 2 x ( x − 1) . How ev er, the gauge transformatio n with g − ( x − 1) /x reduces L 1 to the form (12): L g 1 = D 2 x − D y − 2 ( x − 1) 2 . 9 This example and the o ne b elow show that classical examples of f unctions c ( x, y ) obtained in [8] can also b e obta ined by the application of one o r sev eral differen tial transformations. Actually , b oth approaches can b e considered as t w o-dimensional generalizations of Darb o ux transformations f o r the one-dimensional Sc hr¨ odinger op- erator D 2 x − c ( x, y ). Example 5.3. Consider a differen t ia l transformation of L = D 2 x + D y with M = D x + r ( x, y ). Cho osing r = 1 / 2 − tanh( x + y ) satisfying the condition (13) o f the existence o f the transformation, we hav e L = D 2 x + D y − → L 1 = D 2 x + D y + 2 cosh( x + y ) 2 , I 2 = 0 − → I 1 2 = − 4 sinh( x + y ) cosh( x + y ) 3 . No w w e study the inv ertibilit y of a g iven transformatio n L M − → L 1 , that is, the p ossibilit y of finding a transformatio n L 1 N − → L , p ossibly of higher order. Example 5.4. X -transformation of the op erator L = D 2 x − D y − x 4 + 2 x with M = D x + x 2 results in the follo wing op erator: L 1 = D 2 x − D y − x 4 − 2 x . This transformation has the in v erse X -transformation with N = D x − x 2 . As the simplest examples sho w, an inv erse transformat io n do es not exist for a generic op erator L . In fact the existence of a n in v erse transformation implies a system of constrains on the co efficien t s of L . In some cases, it pro duces known in tegrable equations. First, Theorem 4.2 implies that the existence of an in v erse transformation, that is t he existence of a comp osition L N · M − → L , is equiv alen t to the existence of a transformation P = N · M o f hig her order that transforms the op erator L into itself: P 1 · L = L · P . F or op erato r s (12) the existence of such an op erator implies a particular case of the standard problem of classific ation of Lax pa irs: for P of order one or tw o of the form (10) this leads to p oten tials c ( x, y ) of simple form; the existence of an o p erator P = p 3 ( x, y ) D 3 x + p 2 ( x, y ) D 2 x + p 1 ( x, y ) D x + p 0 ( x, y ) of the third or der implies P 1 = P and P = 4 D 3 x + 6 cD x + p 0 ( x, y ) (up to some simple transformations) and the system  ( p 0 ) x = 3( c y + c xx ) , ( p 0 ) y = 3 c xy − 6 cc x − c xxx , (15) that is t he famo us Boussinesq equation for c : c y y = − ( c 2 + c xx / 3) xx . 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