Auxiliary systems for matrix nonisospectral equations, including coupled NLS with external potential and KdV with variable coefficients, were introduced. Explicit solutions of nonisospectral equations were constructed using the GBDT version of the Bäcklund-Darboux transformation. PACS numbers: 02.30.Ik, 02.30Yy, 03.65.Ge

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.

Classical Bianchi-Lie, Bäcklund and Darboux transformations are considered. Their generalizations for the dynamical systems are discussed. For the transformation being the generalization of the normal shift the special class of dynamical systems is defined. The effective criterion for separating such systems in form of partial differential equations is found.

The tri-Hamiltonian method is applied to the Merola-Ragnisco-Tu equation. This enables us construct a new integrable system, whose continuum limit related AKNS system. Moreover, system proved have linear spectral problems (Lax pair), bi-Hamiltonian structure and Darboux-Bäcklund transformation. Through transformation, we some exact solutions for

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux trans...

In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.

In this paper, we construct the Lax pair for a soliton transmission system in nonuniform optical fibers and give N-soliton solution using the Darboux transformation. The explicit one-soliton and two-soliton solutions are presented. Further, we discuss the interaction scenario between two neighboring solitons and the effect of the inhomogeneities of the fiber (z0) on the interaction between two ...

Transformations of coherent states of the free particle by bounded and semibounded symmetry operators are considered. Resolution of the identity operator in terms of the transformed states is analyzed. A generalized identity resolution is formulated. Darboux transformation operators are analyzed as operators defined in a Hilbert space. Coherent states of multisoliton potentials are studied.

Using the Darboux method and its relation with supersymmetric quantum mechanics we construct all SUSY partners of the harmonic oscillator. With the help of the SUSY transformation we introduce ladder operators for these partner Hamiltonians and shown that they close a quadratic algebra. The associated coherent states are constructed and discussed in some detail.