Matrix Darboux Transformation Research Articles

Novel solutions of general and reverse space-time nonlocal coupled integrable dispersionless systems

In this paper, we have studied a general coupled integrable dispersionless (CID) system and obtained nontrivial solutions in terms of the ratio of determinants by using matrix Darboux transformation. We obtain symmetry non-preserving oscillating solutions of the general CID system in zero background. By using suitable reduction conditions we obtain symmetry preserving unstable soliton solutions for reverse space-time nonlocal CID system. We also recovered stable soliton solutions for the classical CID system under local symmetry reduction condition. Furthermore, the dynamics of symmetry breaking and symmetry preserving first- and second-order nontrivial solutions for generalized, reverse space-time nonlocal and classical CID systems have been presented.

Symmetry broken and symmetry preserving multi-soliton solutions for nonlocal complex short pulse equation

Abstract In this article we consider a general system of complex short pulse equation (CSPE) that under certain nonlocal symmetry reduction yields a reverse space-time nonlocal complex short pulse equation (NL-CSPE). We apply matrix Darboux transformation to the associated Lax pair and construct multi-soliton solutions. K-soliton solution is expressed in terms of quasideterminant formula which enable us to compute explicit expressions of symmetry broken and symmetry preserving one- and two-soliton solutions for NL-CSPE. In addition to these solutions, we also obtain one-bright and interaction of two-bright solitons for classical CSPE. All these investigations end up with the conclusion that both symmetry preserving and broken solutions exist for NL-CSPE.

Quasideterminant solutions of 2-component non–commutative complex coupled integrable dispersionless system

We present a non–commutative extension of two-component complex coupled integrable disspersionless (NC-CCID) system. We studied matrix Darboux transformation and generate multiple soliton solutions by using the notation of quasideterminants. Further, we obtain single and double soliton solutions in terms of quasideterminants for NC-CCID system. In the commutative limit, we obtain soliton solutions for CCID system as ratio of ordinary determinants.

Darboux transformation and multi-soliton solutions of local/nonlocal N-wave interactions

We study matrix Darboux transformation (MDT) for the generalized N-wave interactions (local/nonlocal) and express multi-soliton solutions in terms of ratios of determinants. In order to explain our construction, we obtain explicit expressions of single and double soliton solutions for local/nonlocal 3-wave and 4-wave interactions.

Exact solutions of the noncommutative and commutative nonlinear Schrödinger equation in 2 + 1 dimensions

In this paper, we presented a noncommutative (NC) generalization of nonlinear Schrödinger equation (NLSE) in 2 + 1 dimensions. A matrix Darboux transformation (MDT) is used to generate multiple soliton solutions for NC-NLSE and commutative NLSE in 2 + 1 dimensions. We expressed multiple soliton solutions in terms of quasideterminants and as ratios of ordinary determinants for NC and commutative NLSE in 2 + 1 dimensions, respectively. The quasideterminant formula for K-times repeated MDT enables us to compute single, double and triple soliton solutions for NC and commutative (2 + 1)-dimensional NLSE. Some interesting localized solutions are obtained for the NC and commutative NLSE in 2 + 1 dimensions.

Exact Solutions of Hermitian Symmetric Space Derivative Nonlinear Schrödinger Equations

A generalized matrix Darboux transformation (MDT) is constructed for the HSS-DNLSEs. Using the MDT, exact expressions of multiple solutions are constructed in the form of quasideterminants formula. By using particular solutions to the associated Lax pair, explicit expressions for one- and two-soliton solutions for different HSS-DNLSEs are computed. For each HSS-DNLSE we obtain constant, periodic and soliton solutions.

Quasideterminant solutions of nonlinear Schrödinger equations based on Hermitian symmetric spaces

Quasideterminant solutions of nonlinear Schrödinger (NLS) equations based on Hermitian symmetric spaces, have been investigated. Matrix Darboux transformation method has been employed to construct quasideterminant multi-soliton solutions of the system. By using properties of quasideterminants, a general expression of multi-soliton solutions of the symmetric space NLS equations is obtained. Finally, explicit expressions of one and two soliton solutions of NLS equations based on Hermitian symmetric spaces of types AIII,CI and DIII have been calculated.

ON THE DRESSING METHOD FOR THE GENERALIZED COUPLED DISPERSIONLESS INTEGRABLE SYSTEM

The dressing method of Zakharov and Shabat [Funct. Anal. Appl.8, 226 (1974) and ibid.13, 166 (1980)] has been employed to the generalized coupled dispersionless integrable system in two dimensions. The dressed solutions to the Lax pair and to the nonlinear matrix equation have been obtained in terms of Hermitian projectors. The dressing method has been related with the quasi-determinant solutions obtained by using the standard matrix Darboux transformation. The iteration of dressing procedure has been shown to give N-soliton solutions of the system. At the end, the explicit soliton solution has been obtained for the system based on Lie group SU(2).

Supersymmetric twisting of carbon nanotubes

We construct exactly solvable models of twisted carbon nanotubes via supersymmetry, by applying the matrix Darboux transformation. We derive the Green's function for these systems and compute the local density of states. Explicit examples of twisted carbon nanotubes are produced, where the back-scattering is suppressed and bound states are present. We find that the local density of states decreases in the regions where the bound states are localized. Dependence of bound-state energies on the asymptotic twist of the nanotubes is determined. We also show that each of the constructed unextended first order matrix systems possesses a proper nonlinear hidden supersymmetric structure with a nontrivial grading operator.

Singular matrix Darboux transformations in the inverse-scattering method

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.

Discrete symmetries, Darboux transformation, and exact solutions of the Wess–Zumino–Novikov–Witten model

The matrix Darboux transformation is applied to an auxiliary problem of the classical Wess–Zumino–Novikov–Witten model. One- and two-soliton solutions are written explicitly, and a matrix expression for the N-soliton solution is given. Discrete symmetries of the WZNW model are analyzed, and a solution of the linearized equation of motion is obtained. Bibliography: 19 titles.

Soliton dynamics in the Wess-Zumino-Novikov-Witten model

Exact solutions for the classical Wess-Zumino-Novikov-Witten model are constructed by the matrix Darboux transformation method. The behavior of one- and two-soliton solutions is analyzed. The formula for an N-soliton solution is given. The corresponding linearized problem is considered.

Matrix-valued bispectral operators and quasideterminants

We consider a matrix-valued version of the bispectral problem, that is, find differential operators and with matrix coefficients such that there exists a family of matrix-valued common eigenfunctions ψ(x, z): where f and Θ are matrix-valued functions. Using quasideterminants, we prove that the operators L obtained by non-degenerated rational matrix Darboux transformations from are bispectral operators, where and D is a diagonal matrix. We also give a procedure to find an explicit formula for the operator B extending previous results in the scalar case.

Discrete Chebyshev nets and a universal permutability theorem

The Pohlmeyer–Lund–Regge system which was set down independently in the contexts of Lagrangian field theories and the relativistic motion of a string and which played a key role in the development of a geometric interpretation of soliton theory is known to appear in a variety of important guises such as the vectorial Lund–Regge equation, the O(4) nonlinear σ-model and the SU(2) chiral model. Here, it is demonstrated that these avatars may be discretized in such a manner that both integrability and equivalence are preserved. The corresponding discretization procedure is geometric and algebraic in nature and based on discrete Chebyshev nets and generalized discrete Lelieuvre formulae. In connection with the derivation of associated Bäcklund transformations, it is shown that a generalized discrete Lund–Regge equation may be interpreted as a universal permutability theorem for integrable equations which admit commuting matrix Darboux transformations acting on su(2) linear representations. Three-dimensional coordinate systems and lattices of ‘Lund–Regge’ type related to particular continuous and discrete Zakharov–Manakov systems are obtained as a by-product of this analysis.

Intertwining technique for the matrix Schrödinger equation

The intertwining operator technique is applied to the matrix Schrödinger equation. The following related aspects are investigated: the first- and second-order matrix Darboux transformations, factorization, supersymmetry, chains of transformations. An interrelation is established between the differential and integral transformation operators. It is shown that in a particular case the second-order integral transformations turn into ordinary expressions for matrix solutions and potentials obtained by the inverse problem. The method is illustrated by some examples.

Darboux transformations of the Jaynes–Cummings Hamiltonian

A detailed analysis of matrix Darboux transformations under the condition that the derivative of the superpotential be self-adjoint is given. As a onsequence, a class of the symmetries associated to Schr\"odinger matrix Hamiltonians is characterized. The applications are oriented towards the Jaynes-Cummings eigenvalue problem, so that exactly solvable $2\times 2$ matrix Hamiltonians of the Jaynes-Cummings type are obtained. It is also established that the Jaynes-Cummings Hamiltonian is a quadratic function of a Dirac-type Hamiltonian.

Factorization of second-order matrix differential operators and a matrix Darboux transformation

It is shown that a class of matrix Schrödinger operators can be factored into a product of two first-order matrix operators. The equations that relate the elements in these first-order operators to the elements of the potential matrix of the Schrödinger operator are obtained. They are found to be coupled first-order differential equations in the variables of the first-order matrix operators. Finally, an example of a factorization of a matrix operator is obtained, and a general solution associated to a value of the spectral parameter is given. PACS Nos.: 02.30.Mq, 12.39.Pn

Analytic dynamics of coupled two-level systems

We solve a problem of quantum dynamics of two coupled Schr\"odinger equations. Utilizing a matrix transformation analogous to the Darboux method we find analytic solutions valid in special cases, ``quasiexact solutions.'' The general conditions for matrix Darboux transformations are formulated. The method is applied to a special case of wave-packet dynamics.

Multisoliton Solutions of the Matrix KdV Equation

We consider multisoliton solutions of the matrix KdV equation. We obtain the formulas for changing phases and amplitudes during the interaction of two solitons and prove that no multiparticle effects appear during the multisoliton interaction. We find the conditions ensuring the symmetry of the corresponding solutions of the matrix KdV equation if they are constructed by the matrix Darboux transformation applied to the Schrodinger operator with zero potential.

Differential equations in the spectral parameter for matrix differential operators

We use matrix Darboux transformation to generate a class of matrix differential operators L that have the following property: There exist a family ψ( x,k) of eigenfunctions of L also satisfying a differential equation in the spectral parameter k of the form B( k, ∂ k ) ψ = Θ( x) ψ, where B( ∂ k , k is a diffrential and Θ is a non-constant function of x. We also obtain Nth-order scalar operators which have this property.