We construct the generalized Darboux transformation with arbitrary functions in time t for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-Bäcklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is pr...

According to a theorem of Treves [11], the conserved functionals of the AKNS equation vanish on all pairs of formal Laurent series (q̃, r̃) of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of Bäcklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other p...

It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leads to other hierarchies of integrable equations. In particular, it is shown that stimulated Raman scattering equations generate the hierarchy of flows which include the Heisenberg model equations. Thi...

We propose a scheme for nonlinearizing linear equations to generate integrable nonlinear systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. for the KdV, the m...

Generalized Cauchy matrix approach is used to investigate a discrete negative Ablowitz–Kaup–Newell–Segur (AKNS) equation. Several kinds of solutions more than multi-soliton this equation are derived by solving determining set. Furthermore, applying an appropriate continuum limit we obtain semidiscrete AKNS and after second derive the nonlinear The reductions discrete, semi-discrete continuous s...

In this paper, we are concerned with the integrable decomposition for (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. By utilizing recursive relations and symmetric reductions, propose that (n2 − n1 + 1)-flow of AKNS hierarchy can be decomposed into corresponding n1-flow n2-flow (1+1)-dimensional hierarchy, both in coupled reduced cases. As an appropriate generalization, decompos...

We propose a scheme for nonlinearizing linear equations to generate integrable systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. the KdV, the mKdV, the NLS a...

In the hard pulse approximation, commonly used in nuclear magnetic resonance, one considers potentials for the AKNS system that are sums of δ-functions. The system of differential equations does not, strictly speaking make sense for such potentials. In [8] an analogous discrete forward and inverse problem are analyzed. We review these results and show that pulses obtained using the inverse scat...

New infinite number of oneand two-point Bäcklund transformations (BTs) with explicit expressions are constructed for the high-order constrained flows of the AKNS hierarchy. It is shown that these BTs are canonical transformations including Bäcklund parameter η and a spectrality property holds with respect to η and the ’conjugated’ variable μ for which the point (η, μ) belongs to the spectral cu...