A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems rationally depending on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Diractype systems, including systems with singularities, and for the system auxiliary to the N -wave equation are reviewed. New results on ex...

allowed for determined boundary conditions corresponding to the standard potential . With the very useful Darboux transformation (DT) [3-6] is possible to generalize any specific standard potential and thus generate new interaction models with the same energy levels. The DT is related to the Sturm-Liouville theory [7-10], and it is easy to see the implicit presence of DT in supersymmetric quant...

The first Painlevé equation is expressed as a vector field on the three dimensional weighted projective space CP (3, 2, 4, 5). With the aid of the dynamical systems theory and the orbifold structure of CP (3, 2, 4, 5), a simple proof of the Painlevé property and the construction of the space of initial conditions are given. In particular, Painlevé’s transformation is geometrically derived, whic...

The analytical solutions for a (2+1)-dimensional breaking solution equation is proposed in this paper by using mapping and projective method darboux transformation, and Some exact propagating solutions are constructed for this Breaking equation, and the M × N multi-soliton could be obtained by using Weierstrassp function and setting the perfect parameters. The potential application of breaking ...

We construct the Lax pair for a higher-order nonlinear Schrödinger equation that includes terms accounting for the third-order dispersion, the self-steepening effect, and the delayed nonlinear response effect. Two exact analytic solutions that describe (i) modulation instability and (ii) soliton propagation on a continuous wave background are obtained by using the Darboux transformation. In add...

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plücker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W–congruences...

We consider the averaged dispersion-managed (DM) fiber system equation, which governs the dynamics of the core of the DM solitons. For a special case of such a system equation, we derive the exact soliton solutions using the Darboux transformation. Further, we discuss the interaction scenario between two neighboring solitons. Finally, we derive a dark soliton solution for such a system by assum...

A series of exactly solvable non-trivial complex potentials (possessing real spectra) are generated by applying the Darboux transformation to the excited eigenstates of a non-Hermitian potential V (x). This method yields an infinite number of non-trivial partner potentials, defined over the whole real line, whose spectra are nearly exactly identical to the original potential. e-mail : anjana23@...