Related papers: B\"acklund transformations between the AKNS and DN…
The two matrix spectral problems of Ablowitz-Kaup-Newell-Segur (AKNS) and Kaup-Newell (KN) types associated with so(3,R) are generalized. The corresponding hierarchies of generalized soliton equations are derived by the standard procedure…
The ``truncation procedure'' initiated by Weiss et al. is best understood as a Darboux transformation. If it leads to the Lax pair of the PDE under study, the B\"acklund transformation follows by an elimination, thus proving the…
We summarize the results of our recent work on B\"acklund transformations (BTs), particularly focusing on the relationship of BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its…
We study additional non-isospectral symmetries of constrained (reduced) N=2 supersymmetric KP hierarchies of integrable ``soliton''-like evolution equations. These symmetries are shown to form an infinite-dimensional non-Abelian superloop…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund…
While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schr\"odinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a "continuous…
What types of numeric representations emerge in neural systems, and what would a satisfying answer to this question look like? In this work, we interpret Neural Network (NN) solutions to sequence based number tasks using a variety of…
We consider an integrable generalization of the nonlinear Schr\"odinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the…
This paper is concerned with the construction of new solutions in terms of birational maps to the functional tetrahedron equation and parametric tetrahedron equation. We present a method for constructing solutions to the parametric…
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ B\"acklund transformation relating them? We express some obstructions to such existence in terms of…
We extend the reduction group method to the Lax-Darboux schemes associated with nonlinear Schr\"odinger type equations. We consider all possible finite reduction groups and construct corresponding Lax operators, Darboux transformations,…
We study the B\"acklund symmetry for the Moyal Korteweg-de Vries (KdV) hierarchy based on the Kuperschmidt-Wilson Theorem associated with second Gelfand-Dickey structure with respect to the Moyal bracket, which generalizes the result of…
It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the B\"acklund transformations. Several new examples of such transformations are found. In particular we obtained the B\"acklund…
In our earlier papers we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the generalized KdV and modified KdV (mKdV)…
A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the…
The geometry of an admissible B\"acklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal…
Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein-Gordon (NLKG), the Korteweg-deVries (KdV) and water wave equations, it is shown that the leading…
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by…
A new method of determining B\"acklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding B\"acklund transformations is brought down to…
It is shown that there exists two inner authomorpism which lead to different form of the sistems equations of integrable hierarchy. We present discrete and Backlund transformation connected with such systems and a general formula for…