Symmetry, Integrability and Geometry: Methods and Applications Invertible Darboux Transformations ⋆

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.


Introduction
Darboux transformations are shape preserving transformations of linear partial differential operators. These transformations were originally introduced in [2] first for operators corresponding to Sturm-Liouville equations, where u ∈ K, and where K is some differential field (see Section 2), and λ is a constant, and, secondly, for operators of the form where a, b, c ∈ K, which is a form equivalent up to a change of variable to stationary 2D Schrödinger operator. These transformations were suggested as a possible method for the solution of the corresponding linear partial differential equations 1 . By exploiting these ideas many new kinds of integrable 1D and 2D Schrödinger equations have been discovered. See, for example, this influential paper [8].
Decades later, Darboux transformations became a standard tool of the inverse scattering method, where they are applied to the linear partial differential operators forming the Lax pair and thus eventually serve for the solution of nonlinear partial differential equations [6,9].
A Darboux transformation for an arbitrary linear partial differential operator L can be defined as follows. An operator L is transformed into operator L 1 with the same principal symbol (see Section 2) by means of operator M if there is a linear partial differential operator N such that In this case we shall say that there is a Darboux transformation for pair (L, M); we also say that L admits a Darboux transformation generated by M. We define the order of a Darboux transformation as the order of the M corresponding to it. If a linear partial differential operator L ∈ K[D x , D y ] has a right factor D x + l, then it admits a Darboux transformation generated by M = D x + l. Indeed, suppose L = F (D x + l), then for arbitrary t ∈ K: (D x + t)L = (D x + t)F(D x + l), that is L transforms into (D x + t)F. However, a general Darboux transformation is not implied by a factorization.
For operator (1) the only possible Darboux transformations are those generated by M = D x − ψ 1,x ψ −1 1 , where ψ 1 is in the kernel of the operator (1). For operator (2) Darboux transformations can be generated by each of the following: where ψ 1 is in the kernel of operator (2). For operator (2), however, there are also Darboux transformations that are generated by M of a form D x + m or D y + m, m ∈ K, but cannot be represented as either (4) or (5). These transformations are known as Laplace transformations. Specifically, these transformations are transformations of operators L of the form (2) and are generated by . See more about Laplace transformations in [12]. It is also known that Laplace transformations are invertible [3]. Moreover, it has been proved [10] that Laplace transformations are the only two Darboux transformations of (2) generated by M of a form D x + m or D y + m, m ∈ K that cannot be represented as either (4) or (5). Operators (4) and (5) can be equivalently defined by their action on functions in terms of Wronskians: The consecutive application of a sequence of n Darboux transformations leads to one Darboux transformation of order n, which then can be defined in terms of Wronskians of order n. Indeed, if we define a (t, s) Wronkian, t + s = n − 1, of functions f 1 , . . . , f t+s as the n × n determinant then we can reformulate Darboux's original result [2] and join it to the result [13] for operators of the form as follows. In [1], which preceded [13], this theorem was proved for operators of the form (6) with constant coefficients. The general question whether a Darboux transformation of order n can be represented as a sequence of Darboux transformations of order one is open for operators of the form (2). In [11] the statement was proved for operators of the form (2) with arbitrary (i.e. not necessarily constant) coefficients, but for transformations of order two only. For operators of the form (1), the statement is an implication of the Crum theorem [6], which states that a Darboux transformation of order n is generated by M which can be defined by the same Wronskian formulae (with the only difference that this is a case of a single variable): Generalizations of Darboux transformations have been introduced for other types of operators and for systems of operators. Thus, in [5] one is defined in terms of a twisted derivation D satisfying D(AB) = D(A) + σ(A)B, where σ is a homomorphism. Such twisted derivations include regular derivations, difference and q-difference operators and superderivatives as special cases. The corresponding M can be expressed by the same formulae in terms of Wronskians.
However, a straightforward generalization of Theorem 1 would not be true for many other kinds of operators. This fact has not been much stressed in relevant papers. Below is possibly the first explicit example of such a situation.
x D y + yD 2 x + xD 2 y + 1 and an element of its kernel, Straightforward computations show that no M in the form (4), nor in the form (5), generates a Darboux transformation for operator L. There are, however, other ψ 1 ∈ ker L, which generate Darboux transformations with M in the form (4) or (5).
In the present paper we are interested in answering the following questions. Specifically, in the present paper we show that there are many more kinds of operators than just the one (2) which admit invertible (see the precise definition in Section 4) Darboux transformations. We give criteria which allow us to describe all possible invertible Darboux transformations of arbitrary linear partial differential operators L and generated by M in the form D x +m or D y +m. Given an arbitrary operator L ∈ K[D x , D y ], we give sufficient conditions that guarantee that Wronskian formulae (4) and (5) work. This paper is organized as follows. Section 2 outlines notation. Section 3 starts with the simple fact that a Darboux transformation for a pair (L, M = D x +m), m ∈ K exists if and only if a Darboux transformation for pair (L g , D x ) exists (analogously, for pairs with M = D y + m). Then Theorem 2 states necessary and sufficient conditions that guarantee that a Darboux transformation for pair (L g , D x ) exists. In Section 4 we define invertible transformations as transformations such that the corresponding mapping ker L → ker L 1 is invertible. Theorem 3 states necessary and sufficient conditions that guarantee that a Darboux transformation for pair (L g , D x ) is invertible (as well as for pairs with M = D y + m). It also describes all other possible cases for the dimension of the kernel of mapping ker L → ker L 1 . In Section 5 Theorem 4 states that if in the kernel of operator L there is a subspace of large enough dimension generated by elements that differ from each other by a multiplication of a function of variable y, then each of those elements ψ 1 gives the same M = D x − ψ 1,x ψ −1 1 , and this M generates a Darboux transformation. In other words, we state when Darboux Wronskian formulae work for arbitrary bivariate linear partial differential operator. There is also an analogous statement for the case M = D y − ψ 1,y ψ −1 1 .

Preliminaries
Let K be a differential field of characteristic zero with commuting derivations ∂ x , ∂ y . Let K[D x , D y ] be the corresponding ring of linear partial differential operators over The formal polynomial One can either assume the field K to be differentially closed, in other words containing all the solutions of, in general nonlinear, partial differential equations with coefficients in K, or simply assume that K contains the solutions of those partial differential equations that we encounter on the way.
Let f ∈ K, and L ∈ K[D x , D y ]; by Lf we denote the composition of operator L with the operator of multiplication by a function f , while L(f ) mean the application of operator L to f . For example, The second lower index attached to a symbol denoting a function means the derivative of that function with respect to the variables listed there. For example, Definition 1. Given some operator R ∈ K[D x , D y ] and an invertible function g ∈ K, the corresponding gauge transformation is defined as Then one exists also for (M g , L g ), where g is an arbitrary invertible element of K.  Theorem 2. Let L be an arbitrary linear partial differential operator in K[D x , D y ], that is, an operator of the form (7). Operator for all non-zero a i0 and a k0 , i = 0, . . . , d, k = 0, . . . , d. Here F ik (x) are some functions of the variable x.
Proof . Let there be a Darboux transformation for the pair (L, D x ). This means that for some The right hand side of this operator equality does not contain terms of the form cD j y , c ∈ K. On the other hand for j = 1, . . . , d the coefficient of D j y in the left hand side is na 0j + a 0j,x = 0.
The 'free' coefficient on the right hand side of (10) is zero, while on the left hand side it is na 00 + a 00,x = 0.
Equating the expressions for n obtained from each of the equalities (11) and (12), we see that condition (8) is satisfied. The analogous statement for the case M = D y can be proved similarly.
To prove the theorem in the other direction, suppose that the conditions (8) are satisfied. Consider then the operator equality (10) defining a Darboux transformation for the pair (L, D x ). It implies equalities (11), (12), from which n can be determined uniquely and without contradiction. In general, equality (10), which is an equality of operators of orders d + 1, implies (d + 3)(d + 2)/2 equalities of the corresponding coefficients, of which (d + 1) are equalities (11) and (12). Thus, we have a linear algebraic system of a maximum (d + 3)(d + 2)/2 − (d + 1) equations to solve. The unknowns in this system are the coefficients of L 1 , an operator of order d and their number is (d + 2)(d + 1)/2. Thus the number of equations in this system is less than or equal to the number of unknowns, so there is at least one non-zero solution.
Analogously we can prove that conditions (9) guarantee that there is a Darboux transformation for pair (L, D y ). This mapping is invertible if it is an isomorphism, that is if its kernel is zero

Invertible Darboux transformations
In this case we shall say that the Darboux transformation is invertible. Proof . In [10] it has been proved that a Darboux transformation generated by M = D x + m or by M = D y + m is either a Laplace transformation, that is generated by M = D x + b or M = D y + a, or generated by M in the form D x − ψ 1,x ψ −1 1 , or D y − ψ 1,y ψ −1 1 , where ψ 1 ∈ ker L. In the latter case, ψ 1 ∈ (ker L ∩ ker M), and, therefore, the mapping ker L → ker L 1 is not invertible.
Consider a Darboux transformation for pair (L, M = D x + b). To use the criteria obtained in Theorem 3, we consider a gauge transformation of both operators of this pair. We use g ∈ K such that M becomes D x , that is M g = D x . Such g can be easily found as any solution of g x g −1 = −b. Consider now L g , where k is one of two Laplace invariants of L (see Section 1). Thus, k = 0 is the necessary and sufficient condition for the Darboux transformation for the pair (L, M = D x +b) to be invertible. Analogously, h = 0 is the necessary and sufficient condition for Darboux transformation for pair (L, M = D y + a) to be invertible.

Example 2 (an invertible Darboux transformation). Consider the operator
The coefficients of L satisfy condition (8), and, therefore, there is a Darboux transformation for L generated by M = D x . Since there is no term in the form D i y and L(1) = 1 = 0, then by Theorem 3 the Darboux transformation for pair (L, M) is invertible. This Darboux transformation takes L into The corresponding operator N is D x . In this case there is an inverse transformation of operators,  such that for some non-constant functions T i (y) ∈ K, i = 2, . . . , k, and then there is a Darboux transformation for L generated by Analogously, if instead of (13) and (14) conditions ψ i ψ −1 1 = F i (x) and W k−1,0 (1, T 2 , . . . , T k ) = 0 hold, then there is a Darboux transformation for L generated by M = D y − ψ 1,y ψ −1 1 .
The former means that a 00 = 0, while the latter implies that d j=1 a 0j D j y (T i (y)) = 0, i = 2, . . . , k which is a linear system of k − 1 equations. The number of unknowns a 0j is the number of non-zero a 0j , j = 1, . . . , d. If k = d, then the number of the unknowns is less or equal to k. If k = d − 1, then a 0d = 0, and since the principal symbol of an operator is invariant with respect to the gauge transformations, a 0d = a 0d = 0. Which means that in this case the number of unknowns is also less than or equal to k. The Wronkian in (14) has the first column (1, 0, . . . , 0) t , and the first row (1, T 2 , . . . T k ). Thus, if we remove the first column and the first row, the rank of the corresponding matrix is k − 1. Then the rank of the transpose of the latter matrix is also k − 1, This is a (k − 1) × (k − 1) matrix and adding extra column (T k 2 , . . . , T k k ) t on the right does not change the rank. Thus linear system (15) has matrix with rank k − 1 and less or equal to k unknowns, and, therefore, condition a 0j = G jk (y)a 0k holds for all non-zero a 0j and a 0k , j = 0, . . . , d, k = 0, . . . , d for some G jk (y) ∈ K. Thus, by Theorem 2 there exists a Darboux transformation for the pair (L , D x ). Therefore, there is a Darboux transformation for pair (L, ψ 1 D x ψ −1 1 ), that is for pair (L, D x − ψ 1,x ψ −1 1 ). The second statement, giving sufficient conditions for the existence of a Darboux transformation for L generated by M = D y − ψ 1,y ψ −1 1 , can be proved analogously.
Example 3. For operators of the form D x D y + aD x + bD y + c, a, b, c ∈ K k = 1, Theorem 4 then means that it is enough to have one ψ ∈ ker L to guarantee that a Darboux transformation for pair (L, M = D x − ψ 1,x ψ −1 1 ) exists. This agrees with result of Theorem 1. If d y is the largest j such that a 0j = 0, and since a 00 = 0 Theorem 3 implies that there are either d y (or infinitely many) linearly independent u i ∈ ker L ∩ ker M , i = 1, . . . , d y (or i = 1, . . . , ∞).
Example 4 (the largest j such that a 0j = 0 is not invariant under gauge transformation). Consider the operator and ψ 1 = x ∈ ker L. Then Thus, d y = 2 and d y = 1, and there exists (a non-invertible) Darboux transformation generated by M = D x − ψ 1,x ψ −1 and the corresponding N is N = M.
Theorem 4 states some conditions under which an operator has non-invertible Darboux transformations generated by M in the form D x − ψ 1,x ψ −1 1 or D y − ψ 1,y ψ −1 1 , where ψ 1 ∈ ker L.

Conclusions
The paper provides the first ideas for the general (algebraic) theory of invertible first-order Darboux transformations for bivariate linear partial differential operators of arbitrary order d and of arbitrary form. Although the order of the auxiliary operator has been restricted to the first order, the major operator is taken in general form. Future work may include strengthening of Theorem 4 to describe a criteria. It is not known yet how to extend these ideas to the important three dimensional case, the generalized Laplace transformations for which have been recently developed in [3]. It is also extremely important to extend these ideas for discrete analogies that are under active development now [4,7].