Classification of multidimensional Darboux transformations: first order ...

arXiv:1605.04362v1 [math.DG] 14 May 2016

CLASSIFICATION OF DARBOUX TRANSFORMATIONS: FIRST ORDER AND ITERATED TYPE DAVID HOBBY AND EKATERINA SHEMYAKOVA Abstract. We analyze Darboux transformations (DT) for general multi-dimensional linear partial differential operators. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first order Darboux transformations. We also introduce two new types of Darboux transformations of higher order: a class of invertible transformations generalizing previously introduced ”Type I” transformations (which are a generalization of classical Laplace transformations for 2D Schroedinger operators); and a class ”weak Wronskian type” transformations.

Contents 1. Introduction 2. Types of Darboux transformations 3. Operators of general form admitting Wronskian type Darboux transformations 4. Classification of Darboux transformations of first order 5. Iterated Darboux transformations 6. Darboux transformations of Weak Wronskian Type References

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1. Introduction Darboux transformations (DT), in the general sense of the word, are transformations between differential operators that simultaneously transform their kernels (solution spaces) or eigenspaces. In this paper, we analyze several particular types of Darboux transformations for (in general) multi-dimensional operators. We fully classify those operators that admit Wronskian type Darboux transformations of first order. Furthermore, we completely describe all possible first order Darboux transformations. We also discover new types of Darboux transformations of higher order: those that we call Iterated Type I transformations (such transformations are invertible and we give explicit formulas for the inverses) and those that we call Weak Wronskian type transformations. (The latter generalize Wronskian transformations in some particular cases.) We start by recalling the algebraic framework for Darboux transformations introduced in [24, 28]. Date: May 17, 2016. 1

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Darboux transformations originated in the work of Darboux and others on the theory of surfaces, as in [8], while particular examples were known to Euler and Laplace. They were rediscovered in Integrable Systems theory in the 1970s, where they were used for obtaining solutions of soliton-type equations with remarkable properties. They have also been studied by physicists for quantum-mechanical applications. The very name ‘Darboux transformations’ was introduced by V. B. Matveev [18, 19], who played a central role in creating the Darboux transformation method in soliton theory. As has been pointed out by experts in the field such as Novikov, the role of Darboux transformations is more important than as mere technical tools for constructing solutions. While a large amount of work has been devoted to Darboux transformations, it has been mainly concentrated on particular examples and specific operators, while a general theory was largely missing. Although starting from Darboux’s own work, there have been ideas, observations, and conjectures that can be interpreted as elements of this sought-after general theory of Darboux transformations. In view of the numerous applications of Darboux transformations, the importance of developing such a theory cannot be overestimated. There has been substantial progress in this direction in recent years. In [24, 28], a general algebraic (categorical) framework for Darboux transformations was put forward, which in particular allowed the proof of a long-standing conjecture of Darboux on factorization of the DTs for 2D second-order hyperbolic operators (the ‘2D Schr¨odinger operator’). In [25], a completely new class of transformations, Darboux transformations of Type I, for operators of very general form analogous to Laplace transformations was described. In [13], a complete classification of Darboux transformations of arbitrary nondegenerate operators on the superline was obtained. The model example of Darboux transformations is the following transformation of single variable Sturm–Liouville operators: L → L1 , where L = ∂x2 + u(x), L1 = ∂x2 + u1 (x), and u1 (x) is obtained from u(x) by the formula u1 (x) = u(x) + 2(ln ϕ0 (x))xx . Here ϕ0 (x) is a ‘seed’ solution of the Sturm–Liouville equation Lϕ0 = λ0 ϕ0 (with some fixed λ0 ). Then the transformation ϕ 7→ (∂x − (ln ϕ0 )x)ϕ sends solutions of Lϕ = λϕ to solutions of L1 ψ = λψ (with the same λ). The seed solution is mapped to zero. (According to [20], this transformation was already known to Euler.) This example is a model in two ways. Firstly, the formula for the transformation of solutions can be written in terms of Wronskian determinants: ϕ1 = W (ϕ0 , ϕ)/ϕ0 . This generalizes to a construction based on several linearly independent seed solutions and higher order Wronskian determinants (Crum [7] for Sturm–Liouville operators and Matveev [18] for general operators on the line). Secondly, if one denotes M = ∂x − (ln ϕ0 )x , then the following identity is satisfied: ML = L1 M .

(1)

This identity is equivalent to the relation between the old potential u(x) and the new potential u1 (x). In an abstract framework, if two operators with the same principal symbol, L and L1 , satisfy (1) for some M, then (1) is called the intertwining relation and M is (often) called the transformation operator. One can see that if (1) is satisfied, then the operator M

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defines a linear transformation of the eigenspaces of L to the eigenspaces of L1 (for all eigenvalues). The relation (1) can be taken as a definition of the DT. The intertwining relation (1) appeared, for Sturm–Liouville operators, in work of Shabat [22, eq. 19], Veselov– Shabat [32], and Bagrov–Samsonov [2]. It also appeared for higher dimensions, in Berest– Veselov [4, 5] for the Laplace type operators L = −∆ + u. The same relation was used in [13] for differential operators on the superline. The natural task that arises for such an algebraic definition of DT, is the classification of the DT satisfying it. In the case of differential operators on the line, it was established (in steps) that all DT defined this way arise from seed solutions and are given by Wronskian formulas. For the Sturm–Liouville operators, this was proved in [32, Theorem 5] when the new potential u1 (x) differs from the initial u(x) by a constant, u1 (x) = u(x) + c. It was proved in [23] for transformations of order two; and, finally in the general case, in [2] and the follow-up paper [3], see also [21, Sec.3]. For general operators on the line, it was proved in [1]. For the superline, the classification was obtained in [13]. THe intertwining relation (1), with a single transformation operator M, turns out to be already too restrictive for differential operators in 2D. We use the more flexible intertwining relation NL = L1 M . (2) which can be extracted from the work of Darboux himself [8] (see for example [30, eq. 2]). There are several important differences between the two kinds of intertwining relations. If (1) is satisfied, then, as already mentioned, the operator M transforms each eigenspace of L to the eigenspace of L1 with the same eigenvalue. In contrast with that, if (2) is satisfied, then we only have that M maps the kernel of L to the kernel of L1 . (It is in general false that M maps eigenspaces of L with nonzero eigenvalues to eigenspaces of L1 .) As we explain in greater detail in the paper, it is natural to introduce an equivalence relation between pairs (M, N) in conjunction with the intertwining relation (2). This gives extra flexibility and (as explained below, see Section 2) makes it possible to have invertible DT. We will see that DT defined by (1) are almost never invertible. Among DT defined by our intertwining relation (2) there are non-Wronskian type Darboux transformations such as the classical Laplace transformations for the hyperbolic second order operator in the plane, sometimes called the 2D Schr¨odinger operator. In [24, 28] it was proved that every Darboux transformation for the class of operators is a composition of atomic Darboux transformations of two types: Wronskian type and Laplace transformations (a conjecture that can be traced back to Darboux). Since (2) includes (1) as a special case, we will henceforth use intertwining relation to refer to (2). In this paper we recall the general algebraic framework for Darboux transformations based on intertwining relations and then analyze the various types of Darboux transformations that may arise. We partially answer the following questions in the general theory of Darboux transformations: • Which classes of operators admit Darboux transformations of Wronskian type? • Are there any new types of invertible Darboux transformations?

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• What is the qualitative difference between these two types? For more than a century, there had been known only two types of transformations of linear partial differential operators satisfying our intertwining relation (Darboux transformations): the transformations defined by Wronskian formulas, and Laplace transformations. Note that Wronskian formulas define DTs for several different kinds of operators, see [18, 19], while Laplace transformations are defined only for 2D Schr¨odinger operators. Laplace transformations have a number of good properties including invertibility. Recently, two different generalizations of Laplace transformations were proposed: “intertwining Laplace transformations” in [12] and “Type I transformations” (the latter are always invertible). In the present paper we clarify their relation and introduce a further generalization, which we call “iterated Type I transformations”. We obtain a full description of differential operators that admit a Wronskian type DT of first order. Moreover, by an analysis of general first order DT, we show how they fall into two main types: Wronskian type and Type I (plus the degenerate case arising from a factorization of the operator in question). These two different cases possess a unified algebraic presentation (see Theorem 4.3). We describe also large new classes of DT of higher order that can be perceived as “iterations” (not in literary sense) of Type I and Wronskian transformations: “iterated Type I transformations”, already mentioned, and “weak Wronskian transformations”. The structure of the paper is as follows. In Section 2, we recall key results on the category of Darboux transformations as introduced in [24, 28] and list known types of Darboux transformations. In Section 3, we give a criterion for a general operator to admit a first-order Wronskian Darboux transformation (Theorem 3.4). In Section 4 we classify first order DT. In Section 5, we introduce a generalization of Type I transformations, the iterated Type I transformations. We prove that they are also always invertible. In Section 6, we introduce the class of weak Wronskian transformations.

2. Types of Darboux transformations In this section we recall some general facts about Darboux transformations. Consider a differential field K of characteristic zero with commuting derivations ∂t , ∂x1 , . . . , ∂xn . We use one distinguished letter t to denote the variable which will play a special role. The letter t need not represent time, and was chosen for convenience. By K[∂t , ∂x1 , . . . , ∂xn ] denote the corresponding ring of linear partial differential operators over K. One can either assume the field K to be differentially closed, or simply assume that K contains the solutions of those partial differential equations that we encounter on the way. Darboux transformations viewed as mappings of linear partial differential operators can be defined algebraically as follows [24]. Definition 2.1. Consider a category, whose objects are all operators in K[∂t , ∂x1 , . . . , ∂xn ] and with morphisms are Darboux transformations. Such a morphism from objects L to L1 of the same principal symbol is an equivalence classes of a pairs (M, N) ∈ K[∂t , ∂x1 , . . . , ∂xn ] ×

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K[∂t , ∂x1 , . . . , ∂xn ] satisfying the intertwining relation NL = L1 M ,

(3)

where the equivalence relation is defined by (M, N) ∼ (M + AL , N + L1 A) ,

for any A ∈ K[∂t , ∂x1 , . . . , ∂xn ] .

(4)

Since there are only morphisms between objects with the same principal symbol, this category is partitioned into subcategories for each principal symbol. Composition of Darboux transformations is defined on representatives, as (M, N) ◦ (M1 , N1 ) = (M1 M, N1 N), where the left DT is applied first. The identity morphism 1L is given by the class of (1, 1). Lemma 2.2. The equivalence relation in the definition 2.1 is well-defined and is compatible with the composition. Proof. First of all, suppose a pair (M, N) satisfies (3). Consider an equivalent pair (M + AL, N + L1 A). We have (N + L1 A)L = NL + L1 AL = L1 M + L1 AL = L1 (M + AL), so it also satisfies the intertwining relation. Hence the equivalence relation is well defined. Consider now Darboux transformations (M, N) : L → L1 , and (M1 , N1 ) : L1 → L2 , where all operators are in K[∂t , ∂x1 , . . . , ∂xn ]. Taking different representatives in the same equivalence classes, (M + AL, N + L1 A) : L → L1 , and (M1 + BL1 , N1 + L2 B) : L1 → L2 for some A, B ∈ K[∂t , ∂x1 , . . . , ∂xn ]. That is we have: (N +L1 A)L  = L1 (M +AL), and (N1 +L2 B)L1 = c, N b : L → L2 with N b = (N1 + L2 B)(N + L2 (M1 + BL1 ). For the composition we have M c = (M1 + BL1 )(M + AL). Using NL = L1 M and N1 L1 = L2 M1 we have that L1 A), M b = N1 N + L2 (C + M1 A), and M c = M1 M + (E + BN) L, where C = BN + BL1 A, and N E = M1 A + BL1 A. This concludes the proof.  A Darboux transformation specified by the equivalence class of a pair (M, N) unambiguously defines a linear mapping from ker L to ker L1 where phi goes to Mφ. For if φ ∈ ker L then (M + AL)[φ] = M[φ] + A[L[φ]] = M[φ] + A[0] = M[φ], so M and M + AL give the same linear map. This motivates equivalence relation (4) in the definition. We define the order of a Darboux transformation (M, N) as the minimum possible order of a transformation in its equivalence class, that is the least possible order of an operator of the form M + AL. There are nontrivial automorphisms in this category, as shown in Example 4.7. Now, the above definition implies that the Darboux transformation (M, N) : L → L1 is invertible iff there exists (M ′ , N ′ ) : L1 → L such that their compositions in both orders are

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identity morphisms, which means that for some A, B ∈ K[∂t , ∂x1 , . . . , ∂xn ] we have: M ′ M = 1 + AL , ′

(5)

N N = 1 + LA ,

(6)

MM ′ = 1 + BL1 ,

(7)

′

NN = 1 + L1 B .

(8)

Lemma 2.3. Invertible Darboux transformations induce isomorphisms on the kernels of the operators. Proof. Consider ψ ∈ ker L, then from (5) we obtain that M ′ M[ψ] = ψ. Similarly for ψ1 ∈ ker L1 , 7 gives us MM ′ [ψ1 ] = ψ1 . Therefore, M and M1 induce mutually invertible maps between ker L and ker L1 .  In particular, this implies that ker L ∩ ker M = {0} is necessary for a Darboux transformation to be invertible. Also note that the order of a Darboux transformation is not related in an obvious way with the order of its inverse. The first steps in the general study of invertible Darboux transformations along with the analysis of one particular invertible class of Darboux transformations – Darboux transformations of Type I – can be found in [27, 25]. Lemma 2.4. (1) Given the intertwining relations NL = L1 M and N ′ L1 = LM ′ , the equalities for N, N ′ follow from the equalities for M, M ′ . Specifically, (6) follows from (5), and (8) follows from (7), (2) If (M, N) and (M ′ , N ′ ) are mutually inverse morphisms satisfying (5) through (8), then BN = MA. Proof. 1. Assume that (5) holds, so M ′ M = 1 + AL. Then N ′ NL = LM ′ M = L(1 + AL) = (1 + LA)L. Since K[∂t , ∂x1 , . . . , ∂xn ] is an integral domain, we get N ′ N = 1 + LA by cancellation. The other one is proved analogously. 2. We know that NN ′ = 1 + L1B. Then NN ′ N = N + L1 BN, which implies N(1 + LA) = N +L1 BN. Thus, NLA = L1 BN, which implies L1 MA = L1 BN using intertwining relation NL = L1 M. Therefore, L1 (MA − BN) = 0, and BN = MA.  Recall that a gauge transformation of an operator L is defined as Lg = g −1Lg, where g ∈ K. Gauge transformations commute with Darboux transformations in the sense that if there is a Darboux transformation (M, N) : L → L1 , then there are also Darboux transformations (M g , N g ) : Lg → Lg1 , (Mg, Ng) : Lg → L1 , (g −1M, g −1 N) : L → Lg1 . Definition 2.5. If a pair (M, N) defines a Darboux transformation from L to L1 , then the same pair defines a different Darboux transformation from L + AM to L1 + NA for every A ∈ K[∂t , ∂x1 , . . . , ∂xn ]. We shall say that Darboux transformations L → L1 and L + AM → L1 + NA are related by a shift. Note that this is a symmetric relation.

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The shift of the DT (M, N) : L → L1 is a DT. We have that N(L+AM) = (L1 +NA)M for the intertwining relation, and using ς(L) for the principal symbol of L, we have ς(L+AM) = ς(L1 + NA), since ς(L) = ς(L1 ) and ς(AM) = ς(NA). In general, shifts of Darboux transformations do not commute with compositions of Darboux transformations. However, shifts are useful when dealing with invertible Darboux transformations. Lemma 2.6. Shifts of Darboux transformations preserve invertibility. Proof. Suppose we have a Darboux transformation (M, N) : L → L1 with NL = L1 M, which is invertible, and there is a Darboux transformation (M ′ , N ′ ) : L′ → L with N ′ L1 = LM ′ , and (5) and (7) hold, giving M ′ M = 1 + AL and MM ′ = 1 + BL1 . Then for every e → L f1 , where C ∈ K[∂t , ∂x1 , . . . , ∂xn ] we have the Darboux transformation (M, N) : L e = L + CM, and L f1 = L1 + NC.. The inverse Darboux transformation for the shifted L f′ = M ′ + AC and N f′ = N ′ + CB. We first Darboux transformation is then given by M ′ ′ f1 = L eM f′ , or (N + CB)(L1 + NC) = (L + CM)(M + AC). This is equivalent to f′ L show N ′ ′ N L1 + N NC + CBL1 + CBNC = LM ′ + CMM ′ + CMAC + LAC. Taking into account that N ′ L1 = LM ′ and BN = MA, the last equality is equivalent to N ′ NC + CBL1 = CMM ′ + LAC, which is equivalent to (1 + LA)C + CBL1 = C(1 + BL1 ) + LAC, which is true. f′ M = 1 + AL. e This equality is equivalent to (M ′ + AC)M = Let us now prove that M ′ 1 + A(L + CM), which is true since M M = 1 + AL. f′ = 1+B L f1 , or M(M ′ +AC) = 1+B(L1 +NC). Since MA = BN It remains to show M M by Lemma 2.4, this is equivalent to MM ′ = 1 + BL1 , which is true. The equalities for N and N ′ follow from Lemma 2.4 as well.  Another useful idea is what we call the dual of a Darboux transformation. Definition 2.7. If (M, N) : L → L1 is a Darboux transformation, its dual is the Darboux transformation (L, L1 ) : M → N. This defines a Darboux transformation. If (M, N) : L → L1 is a DT, then NL = L1 M and ς(L) = ς(L1 ). Thus L1 M = NL, and ς(M) = ς(N), and (L, L1 ) : M → N is also a DT. Note that this is not a duality in the sense of category theory; our dual is quite unusual in the sense that it switches objects and morphisms. Lemma 2.8. A Darboux transformation is invertible iff its dual is. Proof. Since the dual of the dual of a DT is the original transformation, it suffices to show that the dual of an invertible DT is invertible. So let (M, N) : L → L1 be an invertible DT. This gives us a DT (M ′ , N ′ ) : L1 → L with M ′ M = 1 + AL and MM ′ = 1 + BL1 for some operators A and B. Lemma 2.4 gives us N ′ N = 1 + LA and BN = MA. Now we must show that (L, L1 ) : M → N is invertible. Using a new font for clarity, we have M = L, N = L1 , L = M, L1 = N and need to show this is invertible. That is, we need a DT (M′ , N ′) : L1 → L, with M′ M = 1 + AL and MM′ = 1 + BL1 .

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Now we take M′ = −A, N ′ = −B, A = −M ′ and B = −N ′ . We have N ′ L1 = −BN = M(−A) = LM′ , so (M′ , N ′) : L1 → L is a DT. Also M′ M = −AL = 1+(−M ′ )M = 1+AL, and MM′ = L(−A) = 1 + (−N ′ )N = 1 + BL1 , as required.  There are several known types of Darboux transformations. 1 Darboux transformations of Wronskian type These are the classical ones. The operator M is given by M(f ) =

W (f1 , f2 , . . . , fn , f ) W (f1 , f2 , . . . , fn )

where W (f1 , f2 , . . . , fn ) denotes a Wronski determinant with respect to one of the variables t, x1 , . . . , xn of m linearly independent fj ∈ K, which are either eigenfunctions of L (in the single-variable case), or elements of ker L (in the general case). Wronskian type Darboux transformations are proved to be admitted by several different types of operators [18, 19, 14, 13], and direct applications of these transformations to solve famous nonlinear equations are well known. Wronskian type transformations are never invertible, since ker L ∩ ker M 6= {0}. (Wronskian type DT formulas in abstract frameworks were introduced in [9] and [14]. An analog of Wronskian DT for super Sturm–Liouvlle operators was discovered in [15, 17, 16].) 2 Darboux transformations obtained from a factorization Suppose L = CM for some C, M ∈ K[∂t , ∂x1 , . . . , ∂xn ] − K. Then for any operator N with ς(N) = ς(M), there is a DT (M, N) : L → NC, since NL = NCM. Since ker M ⊆ ker L, these transformations are never invertible. (For the 1D case, this class overlaps that of Wronskian type transformations, namely, with those built on elements of the kernel.) 3 Laplace transformations These are another type of Darboux transformations, introduced in [8]. They are distinct from the Wronskian type. They are only defined for 2D Schr¨odinger type operators, which have the form L = ∂x ∂y + a∂x + b∂y + c, where a, b, c ∈ K. If the Laplace invariant k = by + ab − c is nonzero, then L admits a Darboux transformation with M = ∂x + b. (Explicit formulas for L1 and N are given below.) If the other Laplace invariant, h = ax + ab − c is nonzero, then L admits a Darboux transformation with M = ∂y + a. Laplace transformations are invertible, and are (almost) inverses of each other. This has been mentioned in the literature, e.g. [29]. Classically, the invertibility of Laplace transformations was understood to mean that they induced isomorphisms of the kernels of the operators in question. Interestingly, it is exactly the equivalence relation in the algebraic definition of Darboux transformations above that makes it possible to understand the invertibility of Laplace transformations in the precise algebraic sense. The following statement appeared in a brief form in [27]. Theorem 2.9. The composition of the Laplace transformation L → L1 given by the operator M = ∂x + b and the Laplace transformations L1 → L given by the operator M ′ = ∂y + a1 (where in fact a1 for L1 equals a for L) is equal to the gauge transformation L → kLk −1 .

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Similarly, the composition of the Laplace transformation L → L1 given by the operator M = ∂y + a and the Laplace transformations L1 → L given by the operator M ′ = ∂x + b1 (where b1 for L1 equals b for L) is equal to the gauge transformation L → hLh−1 . Both kinds of Laplace transformations are invertible. The inverse for the Laplace transformation L → L1 given by the operator M = ∂x + b is (M ′ , N ′ ) : L1 → L, where
M ′ = −k −1 (∂y + a) , N ′ = −k −1 ∂y + a − ky k −1 .

A similar statement holds for the Laplace transformation L → L1 given by the operator M = ∂y + a. Proof. Consider the Laplace transformation L → L1 given by the operator M = ∂x + b. The operators M and L in this case completely define operators L1 and N, and we have (∂x + b, N) : L → L1 , where N = ∂x + b − kx k −1 , and L1 = ∂x ∂y + a1 ∂x + b1 ∂y + c1 = ∂x ∂y + a∂x + (b − kx k −1 )∂y + ab − akx k −1 − k − ax . Also consider the Laplace transformations L1 → L given by the operator M1 = ∂y + a1 (where a1 for L1 equals a for L). The rest of the operators involved in the intertwining relation are completely defined and we can check that (∂y + a, N1 ) : L1 → L2 , where N1 = ∂y + a − ky k −1 , and L2 = L1/k . Now observe that for the resulting Darboux transformation (the composition) c, N) b : L → L1/k , (M

c = M1 M = (∂y + a)(∂x + b) = ∂x ∂y + a∂x + b∂y + ab + by = k + L, and we have M b = N1 N = k + L1/k = k + L2 . From here we can readily see how M1 and N1 can be N changed to make the composition an identity morphism from L to L. We conclude that the inverse Darboux transformation for the Laplace transformation with M = ∂x + b is   1 1 −1 (∂y + a1 ), (∂y − h1x h1 ) : L1 → L . k k  (Note that the formulas for the inverse for Laplace transformations can be generalized, as we do below, for Darboux transformations of Type I, which is a generalization of Laplace transformations to operators of a more general form.) 4 Intertwining Laplace transformations These were introduced in [12], and generalize Laplace transformations to linear partial differential operators L ∈ K[∂t , ∂x1 , . . . , ∂xn ] of very general form. One starts with any representation L = X1 X2 − H, where L, X1 , X2 , H ∈ K[∂t , ∂x1 , . . . , ∂xn ]. Then it was proved that

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there is a Darboux transformation for such operator, (X2 , X2 + ω) : L → L1 , where L1 = X2 X1 + ωX1 − H, and ω = −[X2 , H]H −1 . The latter is a pseudodifferential operator in the general case, an element of the skew Ore field over K that extends K[∂t , ∂x1 , . . . , ∂xn ]. In [12], E. Ganzha then adds the requirement that ω ∈ K[∂t , ∂x1 , . . . , ∂xn ]. This is a very general class of transformations and can be used for theoretical investigations. The class contains both invertible and non-invertible Darboux transformations. 5 Darboux transformations of Type I These were introduced by the second author in [25], and are admitted by operators in K[∂t , ∂x1 , . . . , ∂xn ] that can be written in the form L = CM + f , where C, M ∈ K[∂t , ∂x1 , . . . , ∂xn ], and f ∈ K. We have (M, M 1/f ) : L → L1 , where L1 = M 1/f C + f . The inverse Darboux transformation always exists and is as follows:   1 1 − C, − C : L1 → L . f f The original theory of Laplace transformations is very naturally formulated in terms of differential invariants. In [25], analogous ideas were developed for Darboux tranformations of type I for operators of third order and in two independent variables. This can also be done for operators of a general form using regularized moving frames [10, 11] and ideas from [26]. The classical Laplace transformations are special case of transformations of Type I. We will see that Darboux transformations of Type I can be identified with a subclass of the Intertwining Laplace transformations defined above. Note that the composition of two Darboux transformations of Type I is (in general) not of Type I. 3. Operators of general form admitting Wronskian type Darboux transformations Several different types of operators have been proved to admit Wronskian type Darboux transformations. From classical results, it is known that arbitrary 1D operators and 2D Schr¨odinger operator admit Wronskian type Darboux transformations. Also operators in two independent variables of the form L = ∂t − ai ∂xi , i = 1, . . . , n admit Wronskian type Darboux transformations with M of the form M = ∂x + m [18]. Analogues of Wronskian type Darboux transformations are admitted by super Sturm-Liouville operators [15, 16, 17], and, as was recently proved, non-degenerate operators on the superline [13]. However, there is an abundance of examples where a Wronskian type operator based on a element of ker L does not define a Darboux transformation. Example 3.1. For the following operator L = ∂t + ∂x2 + t∂x −

1 , t

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and the element of its kernel ψ = t, M = ∂t − ψt ψ −1 does not generate a Darboux transformation. This mean that it is not possible to find L1 = ∂t + a1 ∂x2 + a2 ∂x + a3 and N = ∂t + n such that NL = L1 M. Below we investigate if an arbitrary operator in K[∂t , ∂x1 , . . . , ∂xn ] admits a Wronskian type Darboux transformation with M = ∂t − ψt ψ −1 . Here t is a fixed but arbitrary variable. We do not a priori assume any special dependence on ∂t . The idea is that we shall treat operators L in K[∂t , ∂x1 , . . . , ∂xn ] as ordinary differential operators with respect to t with coefficients in the ring K[∂x1 , . . . , ∂xn ]. Now K[∂x1 , . . . , ∂xn ] is embedded in K[∂t , ∂x1 , . . . , ∂xn ], so we may say that elements of K[∂x1 , . . . , ∂xn ] are the t-free operators in K[∂t , ∂x1 , . . . , ∂xn ]. Alternatively, we can define this as follows. Definition 3.2. For a given variable t, call a differential operator t-free if it does not contain ∂t and none of its coefficients depend on t. Theorem 3.3. Given an operator L ∈ K[∂t , ∂x1 , . . . , ∂xn ], let M = ∂t . Then we have the DT (M, N) : L → L1 for some operator L1 and a first-order operator N if and only if there exist some A, B ∈ K[∂t , ∂x1 , . . . , ∂xn ] and c ∈ K, c 6= 0, such that L = A∂t + cB , where B is t-free. If the Darboux transformation exists, then L1 and N are given by the formulas: L1 = L − A∂t + NA = NA + cB , N = M 1/c . Here we may take any non-zero coefficient of our choice in B to be 1 by a suitable choice of c. Proof. Let L be given, and assume NL = L1 M for some N, L1 ∈ K[∂t , ∂x1 , . . . , ∂xn ], and M = ∂t . Since ς(L) = ς(L1 ), we have ς(M) = ς(N) so N = ∂t + n, for some n ∈ K. We shall now construct a certain shift of this Darboux transformation. By division with remainder in K[∂x1 , . . . , ∂xn ][∂t ], we can choose A ∈ K[∂t , ∂x1 , . . . , ∂xn ] such that L′ = L−A∂t does not contain ∂t . We write: L′ = cp1 p2 ...pn ∂xp11 ∂xp22 . . . ∂xpnn ,

p1 , p2 , . . . , pn ≥ 0 ,

cp1 p2 ...pn ∈ K ,

where we assume summation in upper and lower indices. We also write L′1 = L1 − NA. We then get the Darboux transformation: (M, N) : L′ → L′1 , since NL′ = NL − NA∂t = L1 M − NAM = (L1 − NA)M. Now, in NL′ = L′1 ∂t every term of NL′ must be a left multiple of ∂t , and all the terms that do not contain ∂t must have their coefficients zero. So, (∂t + n) [L′ ] = 0 ,

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where brackets means application of ∂t + n to all the coefficients of L′ . So, for each term cp1 p2 ...pn ∂xp11 ∂xp22 . . . ∂xpnn in L′ we must have (∂t + n)[cp1 p2 ...pn ] = 0. Writing c for cp1 p2 ...pn , we have ct + nc = 0 or −n = ct /c. Thus ct /c = ∂t (ln(c)) is the same for all of the cp1 p2 ...pn , so all the ln(c) differ by functions that do not depend on t, and all the c are equal to within multiplication by functions which do not depend on t. We may also take some non-zero coefficient in B of our choice to be 1, and write all the cp1 p2 ...pn in the form cb with b t-free, using a common function c that satisfies ct /c = −n, hence N = ∂t − ct /c , and L′ is of the form L′ = cB , where B is t-free. We have NL′ = L′1 ∂t , and calculate NL′ = (∂t −ct /c)cB = ∂t cB−ct B = c∂t B+ct B−ct B = c∂t B = cB∂t , since B is t-free. Thus, L′1 ∂t = cB∂t , and cancelling ∂t gives us L′1 = cB. So L′1 = cB = L′ . Thus, L = L′ + A∂t = A∂t + cB. Also L1 = L′1 + NA = cB + NA. The other direction is an easy calculation.



This theorem also gives us a similar result for general transformations of Wronskian type. Theorem 3.4. Given an operator L ∈ K[∂t , ∂x1 , . . . , ∂xn ]. Then ψ ∈ ker L is such that M = ∂t − ψt ψ −1 generates a Darboux transformation with NL = L1 M for some N, L1 ∈ K[∂t , ∂x1 , . . . , ∂xn ] if and only if there exists some A ∈ K[∂t , ∂x1 , . . . , ∂xn ], and a t-free B ∈ K[∂t , ∂x1 , . . . , ∂xn ] such that Lψ = A∂t + cB . Here any one non-zero coefficient in B may be taken to be 1. If the Darboux transformation exists, then L1 and N are given by the formulas: Lψ1 = Lψ − A∂t + N ψ A , 1/c

N ψ = ∂t

.

Proof. Let L be given, together with ψ ∈ ker(L), and let M be ∂t − ψt ψ −1 . Note that M ψ = ψ −1 (∂t − ψt ψ −1 )ψ = ψ −1 (ψ∂t + ψt − ψt ) = ∂t , so we conjugate NL = L1 M by ψ. This gives N ψ Lψ = Lψ1 M ψ = Lψ1 ∂t . Now apply the previous theorem.  Corollary 3.5. Given an operator L ∈ K[∂t , ∂x1 , . . . , ∂xn ], and ψ ∈ ker L is such that Mψ = ∂t − ψt ψ −1 generates a Darboux transformation. Then there exists some A ∈ K[∂t , ∂x1 , . . . , ∂xn ], and a t-free B ∈ K[∂t , ∂x1 , . . . , ∂xn ] such that L = AMψ + cB , Here B is not necessarily t-free.

[Mψ , B] = 0 .

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Proof. From the previous theorem, Lψ = A∂t + cB and [∂t , B]. Conjugating by ψ −1 , L = −1 −1 −1 −1 −1 −1 Aψ ∂tψ + cψ Bψ −1 , with [∂tψ , B ψ ] = 0, Note that ∂tψ = ψ∂t ψ −1 = ψ(ψ −1 ∂t − −1 −1 ψt ψ −2 ) = ∂t − ψt ψ −1 = Mψ , so L = Aψ Mψ + cBψ −1 , and [Mψ , B ψ ] = 0. Now rename −1 Aψ as A, and Bψ −1 as B.  4. Classification of Darboux transformations of first order In this section by a first-order transformation we understand a Darboux transformation with M = ∂t + m, where as above t is arbitrary but fixed variable, and m ∈ K. A key factor in our classification of Darboux transformations (M, N) : L → L1 is the intersection of the kernels, K = ker L ∩ ker M. For example, if (M, N) : L → L1 is invertible, it must have K = {0}, and we conjecture that the converse is true. So in honor of Wronski, we make the following definition. Definition 4.1. We say that the Darboux transformation (M, N) : L → L1 is of Wronskian type iff it has K = ker L ∩ ker M 6= {0}. We further split this type into two subtypes. If K is large enough so that ker A ⊆ K for some operator A ∈ / K, we say that (M, N) : L → L1 is of strong Wronskian type. Otherwise, we say that (M, N) : L → L1 is of weak Wronskian type. Summarizing what we know so far, Type I Darboux transformations are never of Wronskian type, while DTs obtained from factorizations are of strong Wronskian type. Theorem 4.2. Suppose there is a first-order Darboux transformation (M, N) : L → L1 with N, M, L1 ∈ K[∂t , ∂x1 , . . . , ∂xn ], and M = ∂t + m. Then this transformation is either of Type I, weak Wronskian type, or obtained from a factorization of L. Proof. While M = ∂t + a for some a ∈ K, it is more convenient to write M as ∂tu = u∂t u−1 = ∂t − ut u−1 , which is easily done by solving −au = ut . −1 Then we have NL = L1 ∂tu , and conjugate by u to get N u Lu = Lu1 ∂t . By Theorem 3.3, we have Lu = A∂t + cB, where B is t-free. We have three cases. −1 If cB = 0 then L = Au M and we have a Darboux transformation obtained from a factorization. Suppose now that cB is the non-zero function c, which we have since we are free to pick one coefficient of B to be 1. Then Lu = A∂t +c, and conjugation gives L = uAu−1 M +c. We let C be uAu−1 and f be c, giving L = CM + f . Theorem 3.3 also gives N u = ∂t − ct c−1 = c∂t c−1 , so N = cu∂t u−1 c−1 = f Mf −1 , as expected. Similarly, Lu1 = Lu − A∂t + (∂t − ct c−1 ) A = Lu − A∂t + N u A, yielding L1 = L − uAu−1 M + NuAu−1 = CM + f − CM + NC = NC + f . This shows the transformation is of Type I. We now suppose that cB is zero, or that B is an t-free differential operator that is not a function. In either case, we have a non-zero t-free function φ in ker(cB). We let ψ be uφ, and claim ψ is in both ker(L) and ker(M). We have L = uA∂t u−1 + ucBu−1 , so L[ψ] is (uA∂t u−1 + ucBu−1)[uφ] = uA∂t [φ] + ucB[φ] = 0, because φ is t-free. Since M is u∂t u−1 , M[ψ] is u∂t u−1 uφ = u∂t φ = 0 also. Thus the transformation is of Wronskian type in this case.  −1

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We can put first-order Darboux transformations of both Type I and Wronskian Type on a common footing, and show that both of them depend on the existence of a partial factorization of L of a special form. Theorem 4.3. A Darboux tranformation (M, N) : L → L1 where M, N are first order exists if and only if L can be written in the form L = CM + cB ,

[M, B] = 0 ,

c∈K .

(9)

If c = 0 then it is a Darboux transformation obtained from a factorization. If cB ∈ K, cB 6= 0 then it is a transformation of Type I. If cB 6∈ K (an operator of order larger than zero) then it is a transformation of Wronskian Type. The corresponding operators N and L1 for the Darboux transformation are N = cMc−1 ,

L1 = NC + cB ,

(10)

if c 6= 0; and N = M, L1 = NC in the case of c = 0. Proof. By Theorem 4.2, any transformation of first order is either of Type I or Wronkian type. If this is a Darboux transformation of Type I, it means that L can be written in the form L = CM + f , where f ∈ K. Then NL = L1 M, where N = f Mf −1 and L1 is NC + f . The equality depends on the fact that Nf = f Mf −1 f = f M. In transformations of Wronskian type, we have L = CM + F , where as before we may change variables and apply gauge transformations to make M = ∂u . By Theorem 3.3, this gives L = C∂u + F , where F is an operator of the form cB, where B is u-free. Then NL = L1 M, where N = cMc−1 and L1 is NC + F . The equality NL = L1 M then relies on the fact that NF = F M, or that cMc−1 cB = cMB = cBM = F M, where the key step is that MB = BM because M = ∂u and B is u-free. Now let us prove the statement in the opposite direction and suppose that L = CM + cB, where MB = BM. Then we let N = cMc−1 , and L1 = NC + cB. This gives us NL = cMc−1 (CM + cB) = cMc−1 CM + cMB = NCM + cBM = (NC + cB)M = L1 M.  Remark 4.4. As Type I Darboux transformations do, the transformations of Wronskian type of order one fit the framework developed by E.I. Ganzha in [12]. Given L = CM + cB with [M, B] = 0, we write L = X1 X2 − H by taking X1 = C, X2 = M and H = −cB. We need ω = −[X2 , H]H −1 to be a strictly differential operator, and calculate ω = −[M, −cB](−cB)−1 = (cBM − McB)B −1 c−1 = (cMB − McB)B −1 c−1 = (cM − Mc)c−1 = cMc−1 − M = N − M. This also gives us that N is M + (N − M) = X2 + ω, as required. Remark 4.5. A natural question is then which L have partial factorizations of the form L = CM + cB, with [M, B] = 0, where c is an arbitrary element in K, and if they do, how many do they have? We do not have uniqueness of factorization for partial differential operators, as, e.g., shown by Landau’s Example (mentioned in [6]). Thus the general question of how many partial factorizations an operator L has may be difficult. We may however note the following for Darboux transformations of Type I, where cB = c. Call a differential

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operator nontrivial if it is not a function. Then for the partial factorization L = CM + c where C and M are nontrivial, we have that the principal symbols of C and M are factors of the principal symbol of L. If M is first order, it is almost determined by its principal symbol. Given a first order M, we may change variables if need be, and write M as a∂t + b. Grouping a with C, we may assume that the principal symbol of M is pt , which corresponds to ∂t . A partial factorization of L is determined by M, so it is natural to ask if there can be more than one first order factor M with the same principal symbol pt . For Darboux transformations of Type I, the answer is no, unless the principal symbol of L has a repeated factor of pt . For suppose we have two first order factors M1 and M2 of L, that both have principal symbol pt . Applying an appropriate gauge transformation, we may assume M = ∂t and N = ∂t + n. Then L = C∂t + f = E(∂t + n) + g, where f and g are in K. Rearranging, En = (C − E)∂t + f − g. Thus pt divides the principal symbol of En, and hence of E. Thus p2t divides the principal symbol of L. This gives us that when the principal symbol of L has no repeated factors, that there is at most one first order Type I transformation per factor, and thus at most deg(L) many first order Type I Darboux transformations of L. In the case of Darboux transformations of Wronskian type, the situation is not as simple, since the principal symbol of M does not necessarily divide that of L. See the following example. Example 4.6. We let L be ∂xxy + ∂xyy + (1 − x/2)∂xx + (3 − x)/2∂xy + (−1/x + 1/2x2 )∂yy + 1/2∂x + (−1/x + 1/x2 )∂y . Letting M = x∂x + ∂y we have L = CM + cB, where BM = MB. Here c = x(x − 1)/(8e3y ), and C = (1/8)((1 − x)∂xx + (4 + 4/x)∂xy + (1/x − 1/x2 )∂yy + (1 + 3/x)∂x − (2/x + 2/x2 )∂y ), B = e3y x−3 (x3 ∂xxx − 3x2 ∂xxy + 3x∂xyy − ∂yyy − 3x2 ∂xx + 9x∂xy − 6∂yy + 3x∂x − 8∂y ). The following example is derived from the famous Landau example [31] of non-unique factorization of linear partial differential operators. The resulting Darboux transformation of higher order exhibits different patters than Darboux transformations of order one. Example 4.7. Landau example [31] of non-unique factorization of operators is RQ = QQP for P = ∂x + x∂y , Q = ∂x + 1 and operator R = ∂xx + x∂xy + ∂x + (2 + x)∂y is irreducible over any extension of Q(x, y). This implies Darboux transformation of higher order, (QP, R) : Q → Q , which does not comply to formula (10) for N. Indeed, assuming R = cQP c−1 , we obtain R = ∂xx + x∂xy + (1 −2cx c−1 −xcy c−1 )∂x + (x−xcx c−1 + 1), which implies 2cx c−1 + xcy c−1 = 0 and xcx c−1 = −1. The latter gives c = f (y)/x, but plugging it in the former gives c = 0.

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For any operator C, we may shift (the definition of a shift is given in Sec. 2) this Darboux transformation into another one: (QP, R) : CM + Q → NC + Q For C = ∂x we have Darboux transformation that is not of the form L = CM + cB where MB = BM. For we would have cB = Q, or B = c−1 Q for some function c. Then BM = MB gives us c−1 QQP = QP c−1 Q or c−1 RQ = QP c−1 Q. This implies c−1 R = QP c−1 and R = cQP c−1 , contradicting the fact that R is not a gauge transformation of QP . Note that transformation with C = ∂x is not of Type I as e−x is in both ker(M) and ker(L). 5. Iterated Darboux transformations The following is an example of a second order Darboux transformation that is of neither Type I or Wronskian type. Example 5.1. Let M = ∂xx + 1, F = ∂x , C = ∂y + x, consider Darboux transformation (M, M) : CM + F → MC + F . Indeed, NL = M(CM + F ) = MCM + MF = MCM + F M = (MC + F )M = L1 M. If ψ is in the kernels of both M and F , it is in the kernel of M −∂x F = 1, so the intersection of the kernels of M and F is {0}. Now, if ψ were in the kernels of both L and M, it would be in the kernel of L − CM = F , so this transformation is not of Wronskian type. F is not in K, so the transformation is also not of Type I. Our long term goal is to classify Darboux transformations for an operator of a general form. The example above is of neither known to us types. It can be understood in terms of the following general phenomenon. Suppose that we have operators L and M so that L = CM + F for some operators C and F 6= f ∈ K. So the corresponding Darboux transformation cannot be of Type I. We can rewrite this as F = L − CM, where we have written F as a particular kind of linear combination of L and M. If F is a function f , we can get a Type I transformation where NL = L1 M. Now suppose F is not a function, but that we have M = AF + f for some operator A and nonzero function f . We can write this as f = M − AF , and view this as reaching f in two linear combination steps. The first gives F , and the second gives f . Note that if φ ∈ ker(L) ∩ ker(M), then φ is in ker(F ), since we have F [φ] = L[φ] − C[M[φ]] = 0 − 0 = 0. Thus φ ∈ ker(M) ∩ ker(F ), and since f = M − AF , we similarly get φ ∈ ker(f ) = {0}. Thus ker(L) ∩ ker(M) = {0}, and the corresponding Darboux transformation, if exists, cannot be of Wronskian type. However, below we prove that there is a Darboux transformation in this situation. These Darboux transformations are “reaching” for Darboux transformations of Type I in two steps. Below we also show that this can be also extended to “reaching” for Darboux transformations of Type I in several steps. Let us first show that such a construction lead to a Darboux transformation.

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Theorem 5.2. Let L ∈ K[∂t , ∂x1 , . . . , ∂xn ], and suppose there are operators A1 , A2 . . . Ak , M = M1 , M2 , . . . Mk ∈ K[∂t , ∂x1 , . . . , ∂xn ] for some k ≥ 1 and a nonzero f = Mk+1 ∈ K so that Mi−1 = Ai Mi + Mi+1 , 1 ≤ i ≤ k, where M0 = L, i.e. L = A1 M1 + M2 M1 = A2 M2 + M3 ... Mi−1 = Ai Mi + Mi+1 , ... Mk−1 = Ak Mk + f

1≤i≤k

Then there exists a Darboux transformation for operator L (M1 , N) : L → L1 defined as follows. Define Nk+1 = Mk+1 = f and Nk = f Mk f −1 , and define Ni for 0 ≤ i ≤ k − 1 by downward recursion using Ni = Ni+1 Ai+1 + Ni+2 . Finally let N = N1 , and let L1 = N0 . Then M and N have the same principal symbol, and the intertwining relation NL = L1 M holds. The corresponding Darboux transformation is not of Wronskian type, and if k > 1 it is not of Type I. Proof. Fix k. If k = 1, we have M2 = f , so L = M0 = A1 M1 + M2 = A1 M1 + f . Then N = N1 = f M1 f −1 , and L1 = N0 = N1 A1 + N2 = f M1 f −1 A1 + f . We have NL = L1 M, since this is a Darboux transformation of Type I. Henceforth assume k > 1. We have that ker(L)∩ker(M) = ker(M0 )∩ker(M1 ) ⊆ ker(M1 )∩ ker(M2 ), the last step because M2 is a linear combination of M0 and M1 . Now Mi−1 = Ai Mi +Mi+1 for all i, so each Mi+1 is a linear combination of Mi−1 and Mi , giving us similarly that ker(Mi−1 ) ∩ ker(Mi ) ⊆ ker(Mi ) ∩ ker(Mi+1 ) for all 1 ≤ i ≤ k. Thus ker(M0 ) ∩ ker(M1 ) ⊆ ker(M1 ) ∩ ker(M2 ) ⊆ ker(M2 ) ∩ ker(M3 ) ⊆ . . . ker(Mk ) ∩ ker(Mk+1 ). But ker(Mk+1 ) = ker(f ) = {0}, so ker(L) ∩ ker(M) = {0}, and our transformation will not be Wronskian. We will now do two inductive proofs of statements Si , starting with i = k as the basis and showing Si ⇒ Si−1 for all i ≥ 1. First, we have that Mi and Ni have the same principal symbols for 0 ≤ i ≤ k, which will give ς(L) = ς(L1 ) since M0 = L and N0 = L1 . To start with, we have Nk+1 = f = Mk+1 and Nk = f Mk f −1 , so ς(Nk+1 ) = ς(Mk+1 ) and ς(Nk ) = ς(Mk ). Now we let 0 ≤ i < k, and assume ς(Ni+2 ) = ς(Mi+2 ) and ς(Ni+1 ) = ς(Mi+1 ). Then Ni = Ni+1 Ai+1 + Ni+2 and Mi = Ai+1 Mi+1 + Mi+2 , and so ς(Ni ) = ς(Mi ). It remains to show Ni Mi−1 = Ni−1 Mi for 1 ≤ i ≤ k + 1. When i = 1, we have N = N1 , L = M0 , L1 = N0 and M = M1 , so this will be NL = L1 M. Our basis is when i = k + 1. Then we have Ni Mi−1 = Nk+1 Mk = f Mk = f Mk f −1 f = Nk Mk+1 = Ni−1 Mi , as desired. Now let 1 < i ≤ k + 1, and assume Ni Mi−1 = Ni−1 Mi . Adding Ni−1 Ai−1 Mi−1 to both sides, we get Ni Mi−1 + Ni−1 Ai−1 Mi−1 = Ni−1 Mi + Ni−1 Ai−1 Mi−1 . But Ni + Ni−1 Ai−1 = Ni−2 and Mi + Ai−1 Mi−1 = Mi−2 , so we have Ni−2 Mi−1 = Ni Mi−1 + Ni−1 Ai−1 Mi−1 = Ni−1 Mi +

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Ni−1 Ai−1 Mi−1 = Ni−1 Mi−2 . Reversing the order of equality, Ni−1 Mi−2 = Ni−2 Mi−1 , which is the required statement with i replaced by i − 1 throughout. Thus Ni Mi−1 = Ni−1 Mi for 1 ≤ i ≤ k + 1 by induction.  This theorem then gives us a new type of Darboux transformation, which generalizes Type I transformations. Definition 5.3. Darboux transformations defined in Theorem 5.2 we shall call Darboux transformation of Iterated type. A natural question is how Darboux transformations of Iterated type relate to the theory developed by E.I. Ganzha in [12]. Let us consider transformations of iterated type for k = 2, so we have L = A1 M1 + M2 and M1 = A2 M2 + M3 where M3 = f . Intertwining Laplace transformations depend on writing L in a particular form, where L = X1 X2 − H. In our notation, the definition from [12] then gives M = X2 , ω = −[X2 , H]H −1 and N = X2 + ω, where we would like ω to be a differential operator and not merely pseudodifferential. Substituting M = M1 for X2 , we have L = A1 M1 + M2 = X1 X2 − H. So it would be natural to try setting X1 = A1 , giving H = −M2 . In this case, we get ω = −[M1 , H]H −1 = −(M1 (−M2 ) − (−M2 )M1 )(−M2 )−1 = (M2 M1 − M1 M2 )M2−1 . Expanding this using M1 = A2 M2 + f , it becomes (M2 (A2 M2 + f ) − (A2 M2 + f )M2 )M2−1 = ((M2 A2 − A2 M2 − f )M2 + M2 f )M2−1 , which is not a differential operator unless M2 and f commute, which is seldom true. We can continue this by setting X1 = A1 + Y for an unknown operator Y , and trying to find Y so that ω is differential. Calculating as before, we get H = X1 X2 − L = X1 M1 − (A1 M1 + M2 ) = Y M1 − M2 . So ω = −(X2 H − HX2 )H −1 = HX2 H −1 − X2 . For this to be differential, we need HX2 = ZH for some operator Z. Substituting in, we want (Y M1 −M2 )M1 = Z(Y M1 −M2 ) or (Y (A2 M2 +f )−M2)(A2 M2 +f ) = Z(Y (A2 M2 +f )−M2 ). No value of Y seems to satisfy this equation. It is hard to be sure since there is no bound on the orders of Y and Z, but we conjecture that almost all Darboux Transformations of iterated type do not fit the form of [12]. We have that transformations of iterated type are not Wronskian, and note that when M is first order, that the iterated transformation NL = L1 M must be of Type I. It turns out that all DTs of Iterated type are invertible. Some insight into why this is can be obtained by looking at the construction of the iterated DT (M1 , N1 ) : L → L1 by alternately forming shifts and duals of a simple starting DT. With notation as in Theorem 5.2, we have Mk+1 = Nk+1 = f , and Nk = f Mk f −1 . Then (Mk , Nk ) : Mk+1 → Nk+1 is an invertible DT, since it is Type 1. Now (Mk , Nk ) : Mk+1 + Ak Mk → Nk+1 + Nk Ak is a shift of (Mk , Nk ) : Mk+1 → Nk+1 , and also an invertible DT. But Mk+1 + Ak Mk = Mk−1 and Nk+1 + Nk Ak = Nk−1 , so this DT is actually (Mk , Nk ) : Mk−1 → Nk−1 . Then its dual (Mk−1 , Nk−1 ) : Mk → Nk is also an invertible DT. Now we shift again, getting that (Mk−1 , Nk−1) : Ak−1 Mk−1 + Mk → Nk−1 Ak−1 + Nk or (Mk−1 , Nk−1) : Mk−2 → Nk−2 is an invertible DT. Then we take the dual of this, and continue the process, eventually obtaining that (M1 , N1 ) : M0 → N0 or (M, N) : L → L1 is an invertible DT.

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The above process is constructive, in the sense that the inverse of (M, N) : L → L1 can be recovered using the descriptions of the inverses of a shift and a dual in Lemma 2.6 and Lemma 2.8. Doing this gives recursive definitions, as in the inductive proof of the following result. Theorem 5.4. All Darboux transformations of Iterated type are invertible. Proof. Let NL = L1 M be of iterated type, and let k, L = M0 , A1 , A2 . . . Ak , M = M1 , M2 , . . . Mk and a nonzero function f = Mk+1 be as in Theorem 5.2. As there, we have Mi−1 = Ai Mi + Mi+1 for 1 ≤ i ≤ k, and also define Ni by Nk+1 = Mk+1 = f , Nk = f Mk f −1 , and Ni = Ni+1 Ai+1 + Ni+2 . for 0 ≤ i ≤ k − 1. Using Mi−1 = Ai Mi + Mi+1 repeatedly, we can write L = M0 = Pk Mk + Qk Mk+1 . The quantity Pk is of interest to us, so we define it inductively as follows. We let P0 = 1, P1 = A1 , and Pi+1 = Pi Ai+1 + Pi−1 for all i ≥ 1. For example, P2 = P1 A2 + P0 = A1 A2 + 1. We also need related quantities Pi′, created by reversing the order of all the products in Pk . For example, P2′ is A2 A1 + 1. We define these inductively by letting P0′ = 1, P1′ = A1 , ′ ′ and Pi+1 = Ai+1 Pi′ + Pi−1 for all i ≥ 1. ′ We need the fact that Pi Pi+1 = Pi+1 Pi′ for all i ≥ 0, and prove this by induction. For the basis, we have i = 0 and get P0 P1′ = 1A1 = A1 1 = P1 P0′ . Now assume that i ≥ 1 ′ ′ ′ ′ and that Pi−1 Pi′ = Pi Pi−1 . We get Pi Pi+1 = Pi (Ai+1 Pi′ + Pi−1 ) = Pi Ai+1 Pi′ + Pi Pi−1 = ′ ′ ′ ′ Pi Ai+1 Pi + Pi−1 Pi = (Pi Ai+1 + Pi−1 )Pi = Pi+1 Pi , which proves the claim. We have L = M0 = Pk Mk + Pk−1 Mk+1 , and show this by proving that M0 = Pi Mi + Pi−1 Mi+1 for 1 ≤ i ≤ k by induction on i. Our basis is where i = 1. Here we have M0 = A1 M1 + 1M2 = P1 M1 + P0 M2 . Now assume that M0 = Pi Mi + Pi−1 Mi+1 , we will show M0 = Pi+1 Mi+1 + Pi Mi+2 . We have M0 = Pi Mi + Pi−1 Mi+1 = Pi (Ai+1 Mi+1 + Mi+2 ) + Pi−1 Mi+1 = (Pi Ai+1 + Pi−1 )Mi+1 + Pi Mi+2 = Pi+1 Mi+1 + Pi Mi+2 , as required. ′ A similar inductive proof shows that L1 = N0 = Nk Pk′ + Nk+1 Pk−1 . We leave it to the reader. To construct the inverse transformation N ′ L1 = LM ′ , we let N ′ = (−1)k Pk f −1 and M ′ = ′ (−1)k f −1 Pk′ . Then N ′ L1 = (−1)k Pk f −1 (Nk Pk′ + Nk+1 Pk−1 ) = (−1)k Pk f −1 (f Mk f −1 Pk′ + ′ ′ ′ ). Similarly, we ) = (−1)k (Pk Mk f −1 Pk′ + Pk Pk−1 f Pk−1 ) = (−1)k Pk (Mk f −1 Pk′ + Pk−1 ′ k −1 ′ k −1 ′ have LM = (Pk Mk + Pk−1 Mk+1 )((−1) f Pk ) = (−1) (Pk Mk f Pk + Pk−1 f f −1 Pk′ ) = ′ ), where the last step follows (−1)k (Pk Mk f −1 Pk′ + Pk−1 Pk′ ) = (−1)k (Pk Mk f −1 Pk′ + Pk Pk−1 ′ ′ ′ ′ from Pk−1 Pk = Pk Pk−1 . Thus N L1 = LM as required. To establish invertibility, we also need to show that there are operators A and B with M ′ M = 1 + AL and MM ′ = 1 + BL1 as in equations 5 and 7 in Section 2. Similarly to the definitions of Pi and Pi′ , we define the sequences Ri and Ri′ recursively. We ′ ′ take R1 = R1′ = 1, R2 = R2′ = A2 and set Ri+1 = Ri Ai+1 + Ri−1 and Ri+1 = Ai+1 Ri′ + Ri−1 for i ≥ 2. We can now prove M1 = Ri Mi + Ri−1 Mi+1 for 2 ≤ i ≤ k, by induction. We have M1 = A2 M2 + M3 = R2 M2 + R1 M3 for our basis. And assuming M1 = Ri Mi + Ri−1 Mi+1 , we get M1 = Ri Mi + Ri−1 Mi+1 = Ri (Ai+1 Mi+1 + Mi+2 ) + Ri−1 Mi+1 = (Ri Ai+1 + Ri−1 )Mi+1 + Ri Mi+2 = Ri+1 Mi+1 + Ri Mi+2 . This gives us M = M1 = Rk Mk + Rk−1 Mk+1 .

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Next we need several identities, and will prove by induction that Pi′ Ri = Ri′ Pi , Pi′ Ri−1 = ′ ′ + (−1)i and Ri−1 Pi = Pi−1 Ri + (−1)i for 2 ≤ i ≤ k. When i = 2, these become (A2 A1 +1)A2 = A2 (A1 A2 +1), (A2 A1 +1)1 = A2 A1 +(−1)2 and 1(A1 A2 +1) = A1 A2 +(−1)2 , ′ all of which are true. Now we assume Pi′ Ri = Ri′ Pi , Pi′Ri−1 = Ri′ Pi−1 + (−1)i and Ri−1 Pi = ′ ′ ′ Pi−1 Ri + (−1)i , and calculate as follows. First, Pi+1 Ri+1 = (Ai+1 Pi′ + Pi−1 )(Ri Ai+1 + Ri−1 ) = ′ ′ Ai+1 Pi′ Ri Ai+1 +Ai+1 Pi′ Ri−1 +Pi−1 Ri Ai+1 +Pi−1 Ri−1 ) = Ai+1 Ri′ Pi Ai+1 +Ai+1 (Ri′ Pi−1 +(−1)i )+ ′ i ′ ′ ′ ′ (Ri−1 Pi − (−1) )Ai+1 + Ri−1 Pi−1 ) = Ai+1 Ri Pi Ai+1 + Ai+1 Ri′ Pi−1 + Ri−1 Pi Ai+1 + Ri−1 Pi−1 ) = ′ ′ ′ ′ ′ ′ (Ai+1 Ri + Ri−1 )(Pi Ai+1 + Pi−1 ) = Ri+1 Pi+1 , as desired. Next, Pi+1 Ri = (Ai+1 Pi + Pi−1 )Ri = ′ ′ ′ ′ Ai+1 Pi′ Ri + Pi−1 Ri = Ai+1 Ri′ Pi + (Ri−1 Pi − (−1)i ) = (Ai+1 Ri′ + Ri−1 )Pi + (−1)i+1 = Ri+1 Pi + i+1 ′ ′ ′ (−1) , showing the second equality. Finally, Ri Pi+1 = Ri (Pi Ai+1 + Pi−1 ) = Ri Pi Ai+1 + Ri′ Pi−1 = Pi′Ri Ai+1 + (Pi′Ri−1 − (−i)i ) = Pi′(Ri Ai+1 + Ri−1 ) + (−i)i+1 ) = Pi′ Ri+1 + (−i)i+1 , giving the third equality. Thus we have Pk′ Rk = Rk′ Pk , Pk′ Rk−1 = Rk′ Pk−1 + (−1)k and ′ ′ Rk−1 Pk = Pk−1 Rk + (−1)k . ′ We also need that Rk−1 Pk′ = Rk Pk−1 + (−1)k . We leave the easier inductive proof of this to the reader. Now we have M ′ M = (−1)k f −1 Pk′ (Rk Mk + Rk−1 Mk+1 ) = (−1)k f −1 Pk′ Rk Mk + (−1)k f −1 Pk′ Rk−1 Mk+1 = (−1)k f −1 Rk′ Pk Mk + (−1)k f −1 (Rk′ Pk−1 + (−1)k )Mk+1 = (−1)k f −1 Rk′ (Pk Mk +Pk−1)+(−1)k f −1 (−1)k Mk+1 = f −1 Mk+1 +(−1)k f −1 Rk′ (Pk Mk +Pk−1 ) = 1 + ((−1)k f −1 Rk′ )L. This is 1 + AL, where A is (−1)k f −1 Rk′ . Similarly, we get MM ′ = (Rk Mk + Rk−1 Mk+1 )(−1)k f −1 Pk′ = (−1)k (Rk f −1 f Mk + Rk−1 f −1 f Mk+1 )f −1 Pk′ = (−1)k (Rk f −1 Nk + Rk−1 f −1 Nk+1 )Pk′ = (−1)k (Rk f −1 Nk + Rk−1 )Pk′ , since Nk = f Mk f −1 and Nk+1 = f . Continuing, (−1)k (Rk f −1 Nk + Rk−1 )Pk′ = ′ +(−1)k ) = (−1)k (Rk f −1 Nk Pk′ + (−1)k (Rk f −1 Nk Pk′ +Rk−1 Pk′ ) = (−1)k (Rk f −1 Nk Pk′ +Rk Pk−1 −1 ′ 2 k −1 ′ ′ Rk f Nk+1 Pk−1 ) + (−1) k = (−1) Rk f (Nk Pk + Nk+1 Pk−1 ) + 1 = (−1)k Rk f −1 (Nk Pk′ + ′ Nk+1 Pk−1 ) + 1 = 1 + (−1)k Rk f −1 L1 . This is 1 + BL1 , where B = (−1)k Rk f −1 .  Ri′ Pi−1

6. Darboux transformations of Weak Wronskian Type We may generalize the Iterated type slightly, by removing the requirement that f be a function. Much of the theory still works, provided Mk+1 and Mk commute. Here is the analogue of Theorem 5.2. Theorem 6.1. Let k ≥ 1 be given, and suppose there are operators L M0 , A1 , A2 . . . Ak , M = M1 , M2 , . . . Mk+1 so that

=

Mi−1 = Ai Mi + Mi+1 for 1 ≤ i ≤ k, and there is a function c with Mk+1 = cB and Mk B = BMk . Then we can define a Darboux transformation NL = L1 M as follows. Define Nk+1 = Mk+1 and Nk = cMk c−1 , and define Ni for 0 ≤ i ≤ k − 1 by downward recursion using Ni =

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Ni+1 Ai+1 + Ni+2 . Finally let N = N1 , and let L1 = N0 . Then M and N have the same principal symbol, and NL = L1 M is a Darboux transformation. Proof. The proof is essentially the same as that of Theorem 5.2, so we will only indicate the parts that change. When k = 1 we will not have a Type I Darboux transformation. The transformation may be Wronskian, since we only get ker(L) ∩ ker(M) ⊆ ker(Mk+1 ), and Mk+1 need not be a function. In fact, Mi−1 = Ai Mi + Mi+1 gives us ker(Mi−1 ) ∩ ker(Mi ) = ker(Mi ) ∩ ker(Mi+1 ) for all i, so ker(L) ∩ ker(M) = ker(M0 ) ∩ ker(M1 ) = · · · = ker(Mk ) ∩ ker(Mk+1 ), and the transformation is Wronskian iff ker(Mk ) ∩ ker(Mk+1 ) 6= {0}. The proof for principal symbols works essentially unchanged. Since Mk+1 = Nk+1 and cMk c−1 = Nk , we start with ς(Mk+1 ) = ς(Nk+1 ) and ς(Mk ) = ς(Nk ). After that, the induction proceeds unchanged. The proof of NL = L1 M is also by induction. We need the hypothesis that Mk and B commute for the basis, and have Nk+1 Mk = Mk+1 Mk = cBMk = cMk B = cMk c−1 cB = Nk Mk+1 . After that, the proof works as before.  Transformations of this generalized Iterated type are seldom invertible, though. As in our remarks preceding Theorem 5.4, we have that the Darboux transformation (M, N) : L → L1 is obtained by repeatedly forming shifts and duals, starting with (Mk , Nk ) : Mk+1 → Nk+1 which is (Mk , cMk c−1 ) : cB → cB. So (M, N) : L → L1 is invertible iff (Mk , cMk c−1 ) : cB → cB is, by Lemma 2.6 and Lemma 2.8. We have not been able to construct a nontrivial example where (Mk , cMk c−1 ) : cB → cB is invertible. For instance, we can let Mk = ∂xx + 1. Then for the DT to be invertible, we need ker Mk ∩ ker cB = {0}. This is hard to achieve, but we can do it by taking c 6= 0 and B = ∂x . The reader may verify that the DT is then invertible, we can for instance use (1, 1) : cB → cB as the inverse. But this is not a satisfying example, since this is essentially an iterated Type I DT in disguise. Instead of stopping at Mk+1 = c∂x , one should have divided Mk by Mk+1 , giving Mk = ∂xx + 1 = (∂x c−1 )(c∂x ) + 1 = Ak+1 Mk+1 + Mk+2 , letting Ak+1 = ∂x c−1 and Mk+2 = 1. References ` Adler, V. G. Marikhin, and A. B. Shabat. Lagrangian lattices and canonical B¨acklund trans[1] V. E. formations. Theoret. and Math. Phys., 129(2):1448–1465, 2001. Theoretical and Mathematical Physics, Vol. 129, No. 2, pp. 1448–1465, 2001. [2] V. G. Bagrov and B. F. Samsonov. Darboux transformation, factorization and supersymmetry in onedimensional quantum mechanics. Theoret. and Math. Phys., 104(2):1051–1060, 1995. [3] V. G. Bagrov and B. F. Samsonov. Darboux transformation of the Schr¨odinger equation. Phys. Part. Nucl., 28(4):374–397, 1997. [4] Yu. Yu. Berest and A. P. Veselov. Singularities of the potentials of exactly solvable Schr¨odinger equations, and the Hadamard problem. Russian Math. Surveys, 53(1(319)):208–209, 1998. [5] Yuri Berest and Alexander Veselov. On the structure of singularities of integrable Schr¨odinger operators. Lett. Math. Phys., 52(2):103–111, 2000. ¨ [6] H. Blumberg. Uber algebraische Eigenschaften von linearen homogenen Differentialausdr¨ ucken. PhD thesis, G¨ ottingen, 1912.

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[7] M.M. Crum. Associated Sturm-Liouville systems. Quart. J. Math. Oxford, pages 121–127, August 1955. [8] G. Darboux. Le¸cons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriques du calcul infinit´esimal, volume 2. Gauthier-Villars, 1889. [9] Pavel Etingof, Israel Gelfand, and Vladimir Retakh. Factorization of differential operators, quasideterminants, and nonabelian Toda field equations. Math. Res. Lett., 4(2-3):413–425, 1997. [10] M. Fels and P.J. Olver. Moving coframes. I. A practical algorithm. Acta Appl. Math., 51(2):161–213, 1998. [11] M. Fels and P.J. Olver. Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math, 55:127–208, 1999. [12] Elena I. Ganzha. Algebraic and Algorithmic Aspects of Differential and Integral Operators: 5th International Meeting, AADIOS 2012, Held at the Applications of Computer Algebra Conference, ACA 2012, Sofia, Bulgaria, June 25-28, 2012, Selected and Invited Papers. pages 96–115, 2014. [13] Sean Hill, Ekaterina Shemyakova, and Theodore Voronov. Darboux transformations for differential operators on the superline. Russian Mathematical Surveys, 70(6):207–208, 2015. arXiv:1505.05194 [math.MP]. [14] C. X. Li and J. J. C. Nimmo. Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466(2120):2471– 2493, 2010. [15] Q. P. Liu. Darboux transformations for supersymmetric Korteweg-de Vries equations. Lett. Math. Phys., 35(2):115–122, 1995. [16] Q. P. Liu and M. Ma˜ nas. Crum transformation and Wronskian type solutions for supersymmetric KdV equation. Phys. Lett. B, 396(1-4):133–140, 1997. [17] Q. P. Liu and M. Ma˜ nas. Darboux transformation for the Manin-Radul supersymmetric KdV equation. Phys. Lett. B, 394(3-4):337–342, 1997. [18] V. B. Matveev. Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters. Letters in Mathematical Physics, 3(3):213–216, 1979. [19] V. B. Matveev and M. A. Salle. Darboux transformations and solitons. Springer Series in Nonlinear Dynamics. Springer-Verlag, Berlin, 1991. [20] S. P. Novikov and I. A. Dynnikov. Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds. Russian Math. Surveys, 52(5(317)):1057–1116, 1997. [21] B. F. Samsonov. On the N -th order Darboux transformation. Russian Math. (Iz. VUZ), 43(6):62–65, 1999. [22] A. Shabat. The infinite-dimensional dressing dynamical system. Inverse Problems, 8(2):303–308, 1992. [23] A. B. Shabat. On the theory of Laplace-Darboux transformations. Theoret. and Math. Phys., 103(1):482– 485, 1995. [24] E. Shemyakova. Proof of the completeness of Darboux Wronskian formulae for order two. Canad. J. Math., 65(3):655–674, 2013. [25] E. Shemyakova. Orbits of Darboux groupoid for hyperbolic operators of order three. In Piotr Kielanowski, S. Twareque Ali, Alexander Odesski, Anatol Odzijewicz, Martin Schlichenmaier, and Theodore Voronov, editors, Geometric Methods in Physics., Trends in Mathematics. Springer, Basel, 2015. [26] E. Shemyakova and E. L. Mansfield. Moving frames for Laplace invariants. In Proceedings of the twentyfirst international symposium on symbolic and algebraic computation, pages 295–302, New York, 2008. ACM. [27] Ekaterina Shemyakova. Invertible Darboux transformations. SIGMA (Symmetry, Integrability and Geometry: Methods and Applications), 9:Paper 002, 10, 2013.

CLASSIFICATION OF DTS: FIRST ORDER AND ITERATED TYPE

23

[28] Ekaterina Shemyakova. Factorization of Darboux transformations of arbitrary order for 2D Schr¨odinger operator. submitted, 2016. arXiv:1304.7063 [math.MP]. [29] S. P. Tsarev. Factorization of linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations. Theoret. and Math. Phys., 122(1):121–133, 2000. [30] S. P. Tsarev and E. Shemyakova. Differential transformations of parabolic second-order operators in the plane. Proceedings of the Steklov Institute of Mathematics, 266(1):219–227, 2009. [31] Sergey P. Tsarev. Factorization of linear differential operators and systems. In Algebraic theory of differential equations, volume 357 of London Math. Soc. Lecture Note Ser., pages 111–131. Cambridge Univ. Press, Cambridge, 2009. [32] A. P. Veselov and A. B. Shabat. A dressing chain and the spectral theory of the Schr¨odinger operator. Funct. Anal. Appl., 27(2):81–96, 1993. 1 Hawk dr., Department of Mathematics, State University of New York at New Paltz, USA 1 Hawk dr., Department of Mathematics, State University of New York at New Paltz, USA E-mail address: [email protected] E-mail address: [email protected]