Related papers: Rational Solutions to the ABS List: Transformation…
We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced…
The BT of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Backlund…
A new form of Darboux-B\"acklund transformation and its higher order form is derived for Derivative Nonlinear Schrodinger Equation(DNLS). The new form arises due to the different form of Lax pair. It is observed that by a special choice of…
In this paper we revisit the Adler-Bobenko-Suris H2 equation. The H2 equation is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and $S^{(1,0)}$…
Some results about non-bijective quadratic transformations generalizing the Kustaanheimo-Stiefel and the Levi-Civita transformations are reviewed in \S 1. The three remaining sections are devoted to new results: \S 2 deals with the Lie…
It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry…
Combining recent results on rational solutions of the Riccati-Schr\"odinger equations for shape invariant potentials to the finite difference B\"acklund algorithm and specific symmetries of the isotonic potential, we show that it is…
We consider 3D consistent systems of six independent quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of…
Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations ($RS$-transformations) are used to construct algebraic solutions of the sixth…
We construct N-soliton solutions to the equation called Q3 in the recent Adler-Bobenko-Suris classification. An essential ingredient in the construction is the relationship of $(Q3)_{\delta=0}$ to the equation proposed by Nijhoff, Quispel…
We introduce the concept of $\omega$-lattice, constructed from $\tau$ functions of Painlev\'e systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlev\'e systems…
We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this…
The classification of lattice equations that are integrable in the sense of higher-dimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the 'admissible transformations' to the…
We prove the existence of a solution to the semirelativistic Hartree equation $$\sqrt{-\Delta+m^2}u+ V(x) u = A(x)\left( W * |u|^p \right) |u|^{p-2}u $$ under suitable growth assumption on the potential functions $V$ and $A$. In particular,…
A lattice system is derived which amounts to a higher-rank analogue of the Q3 equation, the latter being an integrable partial difference equation which has appeared in the ABS list of multidimensionally consistent quadrilateral lattice…
Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Backlund transformation, and the method of solving a Riccati map by…
A general Casoratian formulation is proposed for the 2D Toda lattice equation, which involves coupled eigenfunction systems. Various Casoratian type solutions are generated, through solving the resulting linear conditions and using a…
We obtain novel periodic as well as hyperbolic solutions of an Ablowitz-Musslimani variant of the coupled nonlocal, nonlinear Schr\"odinger equation (NLS) as well as a coupled nonlocal modified Korteweg-de Vries (mKdV) equation which can be…
We search and classify two-component versions of the quad equations in the ABS list, under certain assumptions. The independent variables will be called $y,z$ and in addition to multilinearity and irreducibility the equation pair is…
In the paper we derive infinitely many conservation laws for the ABS lattice equations from their Lax pairs. These conservation laws can algebraically be expressed by means of some known polynomials. We also show that H1, H2, H3, Q1, Q2, Q3…