We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two-level system. Thus, we find three classes of new solutions for the two-level system and the corresponding new potentials that allow such solutions.
The nonlocal Darboux transformation of the two - dimensional stationary Schrödinger equation is considered and its relation to the Moutard transformation is established. It is shown that a special case of the nonlocal Darboux transformation provides the Moutard transformation. New examples of solvable two - dimensional stationary Schrödinger operators with smooth potentials are obtained as an application of the nonlocal Darboux transformation.
In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the "invariance" of the objects of the "Darboux theory of integrability". In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.
It is proved that the Darboux transformation of the system of coherent states of a free particle leads to the states that may be treated as coherent states of soliton-like potentials.
We construct arbitrary-order Darboux transformations for Schroedinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schroedinger case. We apply our method to generate several solvable Dunkl-Schroedinger equations.
Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple systems of Schrodinger equations and provide explicit representation for three classes of such systems. We show also that there is an elegant relationship between these transformations and analytic complex matrix functions.
Recently, Quesne on the one side and on the other side Bougie, Gangopadhyaya and Mallow, have discovered new translational shape invariant potentials not present in previous classifications. By using ordinary Bäcklund--Darboux transformations we propose a new framework in which such shape invariant potentials can be accommodated.
We construct a Darboux transformation for a class of two-dimensional Dirac systems at zero energy. Our starting equation features a position-dependent mass, a matrix potential, and an additional degree of freedom that can be interpreted either as a magnetic field perpendicular to the plane or a generalized Dirac oscillator interaction. We obtain a number of Darbouxtransformed Dirac equations for which the zero energy solutions are exactly known.
The nonlocal Darboux transformation of the stationary axially symmetric Schrödinger equation is considered. It is shown that a special case of the nonlocal Darboux transformation provides the generalization of the Moutard transformation. Formulae for the generalized Moutard transformation are obtained. New examples of two - dimensional potencials and exact solutions for the stationary axially symmetric Schrödinger equation are obtained as an application of the generalized Moutard transformation.
The nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations is considered. Formulae for the nonlocal Darboux transformation are obtained and its relation to the generalized Moutard transformation is established. New examples of two - dimensional potencials and exact solutions for the stationary axially symmetric Schrödinger and Helmholtz equations are obtained as an application of the nonlocal Darboux transformation.
Darboux transformations are non-group type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_1M$ or the intertwining relation $ML=L_1N$ in the cases when the first one is too restrictive. Here we show that Darboux transformations for operators of the form $\partial_x\partial_y +a \partial_x + b\partial_y +c$ (often referred to as 2D Schrödinger operators) are always compositions of atomic Darboux transformations of two different types. This is in contrast with the case of 1D Schrödinger operators and other 1D op…
Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid consists of Darboux transformations of type I. We describe the orbits of this subgroupoid for hyperbolic operators of order three. We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly).
We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.
Darboux transformation is one of the methods used in solving nonlinear evolution equation. Basically, the Darboux transformation is a linear algebra formulation of the solutions of the Zakharov-Shabat system of equations associated with the nonlinear evolution equation. In this work, the evolution of monochromatic electromagnetic wave in a nonlinear-dispersive optical medium is considered. Using the Darboux transformation, explicit multisoliton solutions (one to three soliton solutions) are obtained from a trivial initial solution.
In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.
Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given.
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. 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 introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, …
The Fokker-Planck equation associated with the two - dimensional stationary Schrödinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schrödinger equations. The examples of exactly solvable two - dimensional…
In this paper we construct Darboux transformations for the supersymmetric Two-boson equation. Two Darboux transformations and associated Bäcklund transformations are presented. For one of them, we also obtain the corresponding the nonlinear superposition formula.
The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.
The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems, we consider the Darboux and binary Darboux transformations, expressed in terms of the continuous variable, and obtain exact solutions in Wronskian and Grammian form. We discuss reductions of both systems to the discrete KdV and discrete potential KdV equations, respectively, and exploit this connection to find the Darboux and binary Darboux transformations and exact solutions of these equations.
The extension of Painlevé equations to noncommutative spaces has been considering extensively in the theory of integrable systems and it is also interesting to explore some remarkable aspects of these equations such as Painlevé property, Lax representation, Darboux transformation and their connection to well know integrable equations. This paper is devoted to the Lax formulation, Darboux transformation and Quasideterminant solution of noncommutative Painlevé second equation which is recently introduced by V. Retakh and V. Rubtsov.
Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by $4\times4$ Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scatterin…
In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation. \end{abstract}
We review Darboux-Crum transformation of Heun's differential equation. By rewriting an integral transformation of Heun's differential equation into a form of elliptic functions, we see that the integral representation is a generalization of Darboux-Crum transformation. We also consider conservation of monodromy with respect to the transformations.
For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund-Darboux transformations and the corresponding discrete system are considered for Kupershmidt's super KdV equation. When all the odd variables vanish, a nonlinear superposition formula is obtained for Levi's Bäcklund transformation for the KdV equation.
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the Bäcklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related Bäcklund transformation are used to construct integrable super differential-difference and difference-difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of the solutions of the overdetermined linear systems are derived in the frameworks of the Darboux transformation technique.
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 in…
We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one d…
We develop the method of regularized moving frames of Fels and Olver to obtain explicit general formulas for the basis invariants that generate all the joint differential invariants, under gauge transformations, for the operators [ø{L}=D_xD_y +a(x,y) D_x + b(x,y) D_y +c(x,y)] and an operator of arbitrary order. The problem appeared in connection with invariant construction of Darboux transformations for $ø{L}$.
Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line.
The quantum oscillator thermodynamic actions are the conjugate intensive parameters for the frequency in any frequency changing process. These oscillator actions fulfill simple Riccati equations. Interesting Darboux transformations of the fundamental Planck and pure vacuum actions are discussed here in some detail. It is shown that the one-parameter "Darboux-Transformed-Thermodynamics" refers to superpositions of boson and fermion excitations of positive and negative absolute temperature, respectively. A Darboux generalization of the fluctuation-dissipation theorem is also briefly sketched
In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and Bäcklund transformations are given for the supersymmetric modified Korteweg-de Vries, sinh-Gordon and nonlinear Schrödinger equations. These Darboux and Bäcklund transformations are adopted for the constructions of integrable discrete super systems, and both semi-discrete and fully discrete systems are presented. Also, the continuum limits of the relevant discrete systems are worked out.
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of fractional Darboux transformations which can be recognized also as generalized Cole-Hopf transformations. We apply these transformations to find Schrodinger equations with isospectral potentials and to the linearization of some new classes of nonlinear partial differential equations.
The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on some non-abelian Lie group, is studied and the solutions are shown to be expressed in terms of quasideterminants. As an explicit example, the Darboux transformation on scalar solutions to the system based on the Lie group SU(2) is discussed in detail and the solutions are shown to be expressed as ratios of determinants.
Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.
By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Schödinger equation (DNLS). These transformations are shown to lead to two solution formulas for DNLS in terms of determinants. As applications, several different types of high order solutions are calculated for this equation.
Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function in the case of pure solitonic solution is given and properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as example. This enables formulation of the Darboux transformations in terms of the Cauchy-Jost function and classification of these transformations. Action of Darboux transformations on Grassmanians-i.e., on the space of soliton parameters-is derived and relation of the Darboux transformations with property of total nonnegativity of elements of corresponding Grass…
We obtain isomonodromic transformations for Heun's equation by generalizing Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator which ensures finite-gap property. As an application, we prove conjectures in part III.
The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The expresions of classic Bell polinomils via generalized one is given. The conditions of an exact factorization possibility leads to the intertwine relation and results in some linearizable generalized Burgers equation. An alternative proof of the Matveev theorem is given and Darboux - Matveev transformations formula for coefficients follows from the intertwine rel…
Direct and inverse problems for the Hirota difference equation are considered. Jost solutions and scattering data are introduced and their properties are presented. Darboux transformation in a special case is shown to give evolution with respect to discrete time and a recursion procedure for consequent construction of the Jost solution at arbitrary time, if the initial value is given. Some properties of the soliton solutions are discussed.
We obtain the well-known discrete modified Boussinesq equation in two-component form as well as its Lax pair in $3\times3$ matrix form through a 3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We describe how Darboux transformations and binary Darboux transformations can be constructed for this two-component discrete integrable equation. These transformations are then used to obtain classes of explicit solutions in the form of Casorati- and Gramm-type determinants. $N$-soliton solutions of the discrete modified Boussinesq equation are discussed as well when taking …
Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two.
Potentials of the nonstationary Schrödinger operator constructed by means of $n$ recursive Bäcklund transformations are studied in detail. Corresponding Darboux transformations of the Jost solutions are introduced. We show that these solutions obey modified integral equations and present their analyticity properties. Generated transformations of the spectral data are derived.
In this paper, I construct the Darboux transformations for the non-commutative Toda solutions at $ n=1 $ with the help of linear systems whose compatibility condition yields zero curvature representation of associated systems of non-linear differential equations. I also derive the quasideterminant solutions of the non-commutative Painlevé II equation by taking the Toda solutions at $ n=1 $ as a seed solution in its Darboux transformations. Further by iteration, I generalize the Darboux transformations of the seed solutions to $ N$-th form. At the end I describe the zero curvature representatio…
A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and dressing methods. This transformation is used to find the exact expressions for soliton solutions on zero and nonzero backgrounds. It is shown that the discrete solitons have the same properties as solitons of the continuous massive Thirring model.
The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces is studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutative differential operator expressions related via a Delsarte-Darboux transformation and having a lot of applications in soliton theory.
Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as Gromov-Witten invariants, can be described in terms of bilinear equations to show their connections with the integrable systems. Here in this paper, we mainly discuss the bilinear equations of the transformed tau functions under the successive applications of the Darboux transformations for the KP hierarchy, the modified KP hierarchy (Kupershmidt-Kiso version) and the BKP hierarchy, by the method of the Boson-Fermion correspondence. The Darboux t…
In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the supersymmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.
The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transformations. Finally, as an example, we apply these transformations to the Cartesian background.
Let dμ(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of dμ(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(λ) to λS(λ^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to th…
In this paper, based on the Hirota and Maxwell-Bloch (H-MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell-Bloch (NH-MB) systems, namely, $PT$-symmetric NH-MB system and reverse space-time NH-MB system. Then we construct the Darboux transformations of these NH-MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations.
In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present 6-dimensional YB maps corresponding to Darboux transformations for the Nonlinear Schrödinger (NLS) equation and the derivative Nonlinear Schrödinger (DNLS) equation. These YB maps can be restricted to $4-$dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries \cite{Allan-Pavlos}. We give a $6-$ dimensional YB-map corresponding to t…
In their reply arXiv:1408.2230, the authors corrected some inappropriate sentences and clarified misleading descriptions in their original manuscript arXiv:1407.5194v1.
