Darboux transformation

A simultaneous mapping between solutions and coefficients of a pair of equations (or systems of equations) of the same form. It may be formulated as a covariance principle for the corresponding operators, i.e. the order and form of the operators are saved after the transformation. The important features specifying it are: the Darboux transformation is functionally parametrized by a pair of solutions of the equation and the transform vanishes if the solutions coincide.

The classical form of the Darboux transformation appears in the paper of Th.-F. Moutard (1875) as a specification of the Moutard transformation [a1], [a2], which is connected, in turn, with the Laplace transformation (in geometry).

It may be noticed that the net of points generated by transforms of (Laplace) invariants has two possible symmetry reductions: The first one corresponds to the Moutard case and the second is the case discovered by E. Goursat [a3], [a19], [a20]. Darboux's classical theorem [a1], [a2] was formulated for the equation defined by a second-order differential operator and a parameter $\lambda$.

If $\psi ( x , \lambda ) , \varphi ( x , \mu ) \in C ^ { 2 }$ are solutions of the equation

\begin{equation*} - \psi _ { x x } + u ( x ) \psi = \lambda \psi, \end{equation*}

then

\begin{equation*} \psi [ 1 ] = \psi _ { x } + \sigma \psi ; \quad \sigma = - \varphi _ { x } \varphi ^ { - 1 } \end{equation*}

is the solution of the equation

\begin{equation*} - \psi [ 1 ] _ { xx } + u [ 1 ] \psi [ 1 ] = \lambda \psi [ 1 ], \end{equation*}

where

\begin{equation*} u [ 1 ] = u + 2 \sigma _ { x }. \end{equation*}

Namely, the form of $\psi [ 1 ]$ and $u [ 1 ]$ as a function of the solutions defines the Darboux transformation. Presently (1998), the most general form of Darboux's theorem is given by V.B. Matveev [a4] for associative rings. The formulation of this theorem contains the natural generalization of the Darboux transformation in the spirit of the classical approach of G. Darboux. Start from the class of functional-differential equations for some function $f ( x , t )$ and coefficients $u_m ( x , t )$ from a ring

\begin{equation*} f _ { t } ( x , t ) = \sum _ { m = - M } ^ { m = N } u _ { m } ( x , t ) T ^ { m } ( f ) , \quad t \in \mathbf{R}, \end{equation*}

where $T$ is an automorphism. Then the equation is covariant with respect to the Darboux transformation

\begin{equation*} D ^ { \pm } f = f - \sigma ^ { \pm } T ^ { \pm 1 } ( f ) \end{equation*}

and $\sigma ^ { \pm } = \varphi [ T ^ { \pm 1 } ( \varphi ) ] ^ { - 1 }$. It is possible to reformulate the result for differential-difference or difference-difference equations and give explicit expressions for the transformed coefficients [a4]. This result implies the "older" formulation [a2] for matrix-valued functions and involving

\begin{equation*} T ( f ) ( x , t ) = f ( x + \delta , t ) , \quad x , \delta \in \mathbf{R}, \end{equation*}

or

\begin{equation*} T ( f ) ( x , t ) = f ( q x , t ) , \quad x , q \in \mathbf{R} , q \neq 0. \end{equation*}

It suffices to use the limits

\begin{equation*} \sigma f - f _ { x } = \operatorname { lim } _ { \delta \rightarrow 0 } D ^ { \pm } f = \operatorname { lim } _ { \delta \rightarrow 0 } ( x - x q ) ^ { - 1 } D ^ { \pm } f. \end{equation*}

Being covariant, the Darboux transformation may be iterated. The iterated Darboux transformation is expressed in determinants of Wronskian type (M.M. Crum, 1955). The universal way to generate the transform for different versions of the Darboux transformation, including those involving integral operators, is described in [a2]; the non-Abelian case results in Casorati determinants [a4].

The statement of the classical Darboux theorem is strictly connected with the problem of factorizing a differential operator [a5] and hence with the technique of symbolic manipulation. Namely, the operators in the classical Darboux theorem are factorized as follows. Let $Q ^ { \pm } = \pm D + \sigma$, and

\begin{equation*} H ^ { ( 0 ) } = - D ^ { 2 } + u = Q ^ { - } Q ^ { + }; \end{equation*}

\begin{equation*} H ^ { ( 1 ) } = Q ^ { + } Q ^ { - } = - D ^ { 2 } + u [ 1 ]. \end{equation*}

The operators $H ^ { ( i ) }$ play an important role in quantum mechanics as the one-dimensional energy operators. The spectral parameter $\lambda$ stands for the energy and the relation $Q ^ { + } Q ^ { - } ( Q ^ { + } \psi _ { \lambda } ) = \lambda ( Q ^ { + } \psi _ { \lambda } )$ shows that the Darboux transformations $Q ^ { \pm }$ are ladder operators, or creation-annihilation operators (cf. also Creation operators; Annihilation operators). The majority of explicitly solvable models of quantum mechanics is based on such properties, which make it possible to generate new potentials together with solutions [a6]. The transformed operator saves the discrete spectrum, the Darboux transformation deletes only the level that corresponds to $\varphi$. Conversely, the inverse transformation adds a level. So one can modify the spectrum by a sequence of Darboux transformations. The intertwining relation $H ^ { ( 1 ) } Q ^ { + } = Q ^ { + } H ^ { ( 0 ) }$ gives rise to a supersymmetry algebra, which is an example of an infinite-dimensional graded Lie algebra (cf. also Lie algebra, graded) or, more generally, a Kac–Moody algebra. Naturally, the Darboux transformation (named the "transference" in [a7]) is a useful tool in the problem of commutativity of ordinary differential operators. For applications of the Darboux transformation in multi-dimensional quantum mechanics, see also [a2].

Consideration of pairs of equations (Lax pairs) for a single function leads to a compatibility condition that is, in general, a non-linear equation for their coefficients (called "potentials" ). This means that the non-linear equation is automatically invariant with respect to the potential part of the Darboux transformation, which is a Bäcklund transformation. If there is a reduction compatible with the Darboux transformation in the sense that the transformed potentials satisfy the same constraint as the initial one (a heredity property), the Darboux transformation gives the symmetry for the non-linear equation with constraints. Iteration of the Darboux transformation generates a sequence of potentials which are solutions of the non-linear equation. Excluding the solutions of the linear problems, one obtains a recurrent difference equation (a chain) for the elements $\sigma _ { n }$, which determine factoring of operators. If this chain is non-trivial and infinite, the non-linear equation has an infinite set of solutions (in this case it is said to be integrable with respect to the Darboux transformation). Periodic closures of such chains generate finite-gap potentials of the Schrödinger operator. For example, the closure $n \in \mathbf{Z} _ { 3 }$ provides useful representations of Painlevé transcendents [a17], [a18]. For non-linear partial differential equations, the notion of integrability is strictly connected with the existence of a Lax pair, as constructed in so-called Painlevé analysis or the Painlevé test. The Moutard or Darboux transformations may also appear inside the procedures in the cases of $2 + 1$ and $1 + 1$ dimensions, respectively [a8]. For integrable systems, the method of the inverse-scattering transform was discovered in 1967 [a9] (cf. also Korteweg–de Vries equation). Its application may give a solution of the Cauchy problem; however, it involves the solution of a complicated inverse problem for one of the operators in the Lax pair. Hence alternative direct algebraic methods, such as the method of Darboux transformation (and the related Sato theory), are presently under development (1999).

When searching for an alternative formulation of the method while retaining the principal ideas in Darboux's approach, one may introduce an elementary Darboux transformation on a differential ring [a10], [a11]. A particular case of this not depending on solutions (but only on potentials) is known as the Schlesinger transformation [a11], [a12]. The combination of Darboux transformations or elementary Darboux transformations for direct and conjugate equations gives rise to a binary Darboux transformation [a2], [a10], which realizes the dressing method for the solution of integrable non-linear equations (cf. also Soliton) and has a symmetric form useful for constraints heredity. The dressing method appears in the development of the method of inverse problems for the matrix case of Lax pairs and involves a Riemann–Hilbert problem [a9]. It may be shown that the binary Darboux transformation solves the matrix Riemann–Hilbert problem with zeros (compare the expressions in [a10] and [a9]).

The general idea of the Darboux transformation allows one to embed the Combesqure and Levy transformations of conjugate nets and congruences into classical differential geometry [a13].

Recently (1998), vectorial Darboux transformations for quadrilateral lattices have appeared [a14].

Important aspects concerning operator algebras can be found in [a15], while interesting applications of the theory of Darboux transformations to matrix factorization may be found via [a16].

References

[a1]	G. Darboux, "Leçons sur la théorie général des surfaces" , Gauthier-Villars (1912) (Edition: Second)
[a2]	V.B. Matveev, M.A. Salle, "Darboux transformations and solitons" , Springer (1991)
[a3]	E. Ganzha, "On one analogue of the Moutard transformation for the Goursat equation" Theor. Math. Phys. (1999)
[a4]	V.B. Matveev, "Darboux transformations in differential rings and functional-difference equations" A. Karman (ed.) , Proc. CRM Workshop Bispectral Problems (Monreal March, 1997) , Amer. Math. Soc. (1998)
[a5]	S.P. Tsarev, "An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator" , Proc. 1996 Internat. Symp. Symbolic Algebraic Computation 24-26 July 1996, Zürich, Switzerland , ACM (1996) pp. 226–231
[a6]	L. Infeld, T. Hull, "The factorization method" Rev. Mod. Phys. , 23 (1951) pp. 21–68
[a7]	G.A. Latham, E. Previato, "Darboux transformations for higher rank Kadomtsev–Petviashvili and Krichever–Novikov equations" Acta Applic. Math. , 39 (1995) pp. 405–433
[a8]	P.G. Estevez, S.B. Leble, "A KdV equation in $2 + 1$ dimensions: Painlevé analysis, solutions and similarity reductions" Acta Applic. Math. , 39 (1995) pp. 277–294
[a9]	S.P. Novikov, et al., "Theory of solitons. The inverse scattering method" , Plenum (1984)
[a10]	S.B. Leble, N.V. Ustinov, "Deep reductions for matrix Lax system, invariant forms and elementary Darboux transforms" J. Phys. A: Math. Gen. , 26 (1993) pp. 5007–5016
[a11]	S.B. Leble, "Elementary and binary Darboux transformations at rings" Comput. Math. Appl. , 35 (1998) pp. 73–81
[a12]	L.J. Schlesinger, J. Reine Angew. Math. , 141 (1912) pp. 96
[a13]	L. Eisenhart, "Transformations of surfaces" , Chelsea (1962)
[a14]	M. Manas, A. Doliwa, P.M. Santini, Phys. Lett. , A232 (1997) pp. 99–105
[a15]	V.A. Marchenko, "Nonlinear equations and operator algebras" , Reidel (1988)
[a16]	P.A. Deift, L.C. Li, C. Tomei, "Matrix factorizations and integrable systems" Commun. Pure Appl. Math. , 42 (1989) pp. 443–521
[a17]	A.P. Veselov, A.B. Shabat, "Dressing chain and spectral theory of the Schrödinger operator" Funkts. Anal. Prilozh. , 23 : 2 (1993) pp. 1–21 (In Russian)
[a18]	A.B. Shabat, "Third dressing scheme" to appear
[a19]	É. Goursat, "Sur une équation aux dérivées partielles" Bull. Soc. Math. France , 25 (1897) pp. 36–48
[a20]	É. Goursat, "Sur une transformation de léquation $s ^ { 2 = 4 \lambda } ( x , y ) p q$" Bull. Soc. Math. France , 28 (1900) pp. 1–6