Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources

We present integral-type Darboux transformation for the mKdV hierarchy and for the mKdV hierarchy withself-consistent sources. In contrast with the normal Darboux transformation, the integral-type Darboux transformationscan offer non-auto-Backlund transformation between two (2n 1)-th mKdV equations with self-consistent sources withdifferent degrees. This kind of Darboux transformation enables us to construct the N-soliton solution for the mKdVhierarchy with self-consistent sources. We also propose the formulas for the m times repeated integral-type Darbouxtransformations for both mKdV hierarchy and mKdV hierarchy with self-consistent sources.     

BINARY DARBOUX TRANSFORMATION AND QUASIDETERMINANT SOLUTIONS OF THE CHIRAL FIELD

The standard binary Darboux transformation is composed and is used to obtain exact multisoliton solutions of the chiral field model in two dimensions. The solutions are expressed in terms of quasideterminants. It has been shown that the standard binary Darboux transformation is equivalent to the elementary binary Darboux transformation.     

Darboux transformations for a system of coupled discrete Schrödinger equations

Darboux transformations and a factorization procedure are presented for a system of coupled finite-difference Schrödinger equations. The conformity between generalized Darboux transformations and the factorization method is established. Factorization chains and consequences of Darboux transformations are obtained for a system of coupled discrete Schrödinger equations. The proposed approach permits constructing a new series of potential matrices with known spectral characteristics for which coupled-channel discrete Schrödinger equations have exact solutions.     

Line soliton solutions for a generalized Davey-Stewartson equation with variable coefficients

The Darboux transformation for a generalized Davey-Stewartson equation is constructed. Expressions for line solitons are obtained explicitly by one and two applications of Darboux transformations.     

DARBOUX POLYNOMIALS FOR LOTKA–VOLTERRA SYSTEMS IN THREE DIMENSIONS

We consider Lotka–Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.     

Darboux Transformation and Explicit Solutions for Drinfel＇d-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.     

Modified Darboux transformations with foreign auxiliary equations

We construct a new type of first-order Darboux transformations for the stationary Schrödinger equation. In contrast to the conventional case, our Darboux transformations support arbitrary (foreign) auxiliary equations. We show that among other applications, our formalism can be used to systematically construct Darboux transformations for Schrödinger equations with energy-dependent potentials, including a recent result (Lin et al., 2007) [16] as a special case.     

Darboux Transformations of Bispectral Quantum Integrable Systems

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 nontrivial 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 dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero–Moser.     

Binary Darboux Transformation for the Modified Kadomtsev--Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.     

Darboux transformation of coherent states for a Lobachevski plane

A Darboux transformation of coherent states is considered for a singular oscillator. A coordinate representation and a holomorphic representation are obtained for the Darboux transformation operator and the coherent states. The Hermitian metric and the Kaler potential of the transformed system are calculated. It is established that a Darboux transformation corresponds to curvature of the phase space of the classical system. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 29–36, September, 1997.     

Supersymmetry and Darboux transformations for the generalized Schrödinger equation

We construct Darboux transformations for a generalized Schrödinger equation by means of the intertwining operator method. We establish a relation between first-order Darboux transformations, supersymmetry, and factorization of the Hamiltonians that are associated with our generalized Schrödinger equation. Furthermore, our methods allow for the generation of isospectral potentials, where one of the potentials has additional or less bound states than its partner. In the particular case of a conventional Schrödinger equation our generalized Darboux transformations reduce correctly to the well-known expressions.     

Some properties of higher-order Darboux transformations

We introduce the operator of the Darboux transformation of order n as an nth-order differential operator transforming the solution of the one-dimensional Schrödinger equation with a given potential into the solution corresponding to a different potential. For n=1 we obtain the usual Darboux transformation. It is shown that when n1 this transformation can always be represented as a product of n first-order Darboux transformations. We discuss the relationship between higher-order Darboux transformations and the supersymmetric approach to quantum mechanics and also the algebra of the dynamical symmetry of the system.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1995.     

A Unified Explicit Construction of 2N-Soliton Solutions for Evolution Equations Determined by 2×2 AKNS System

An explicit N-fold Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving alinear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.     

Method of Darboux Transformation Matrix for the KdV Equation

For the KdV equation, the method based upon the Darboux transformation matrix is developed. All the conditions are shown to hold by suitably choosing the constants involved. Exphiit expressions of Darboux matrices are determined in a recursive manner.     

On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.     

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations, we derive new vector Yang–Baxter map and integrable discrete vector sine-Gordon equation on a sphere.     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

LAX PAIR,BINARY DARBOUX TRANSFORMATION AND NEW GRAMMIAN SOLUTIONS OF NONISOSPECTRAL KADOMTSEV–PETVIASHVILI EQUATION WITH THE TWO-SINGULAR-MANIFOLD METHOD

In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the Nth iterated transformation formula in the form of Grammian is also presented. By using these Darboux transformations, we obtain some new grammian solutions.     

Generalized linear Schrödinger equations and their Darboux transformations

We construct explicit Darboux transformations of arbitrary order for a class of generalized, linear Schrödinger equations. Our construction contains the well-known Darboux transformations for Schrödinger equations with position-dependent mass, Schrödinger equations coupled to a vector potential and Schrödinger equations for weighted energy.     

Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrödinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrödinger equations is constructed with the help of a gauge transformation of spectral problems. As a reduction, the Darboux transformation for well-known Gerdjikov-Ivanov equation is further obtained, from which a general form of N-soliton solutions for Gerdjikov-Ivanov equation is given.