Related papers: B\"acklund transformations for fourth Painlev\'e h…
We construct the supersymmetric extensions of the Darboux-Backlund transformations (DBTs) for the Manin-Radul super KdV hierarchy using the super-pseudo-differential operators. The elementary DBTs are triggered by the gauge operators…
Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…
The $\hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $\hat B_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $\hat B_n^{(1)}$-flows along…
The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding of the differential constraints consistent with the…
We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential operators. We identify the elementary DBTs which are triggered by the gauge operators…
In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation.…
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second- order supersymmetric transformations will be used to obtain new…
We study the additional symmetries associated with the $q$-deformed Kadomtsev-Petviashvili ($q$-KP) hierarchy. After identifying the resolvent operator as the generator of the additional symmetries, the $q$-KP hierarchy can be consistently…
A birational transformation is one which leaves invariant an ordinary differential equation, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…
This is a review of recent developments in the theory of beta ensembles of random matrices and their relations with conformal filed theory (CFT). There are (almost) no new results here. This article can serve as a guide on appearances and…
A series of systems of nonlinear equations with affine Weyl group symmetry of type $A^{(1)}_l$ is studied. This series gives a generalization of Painlev\'e equations $P_{IV}$ and $P_{V}$ to higher orders.
In the current paper we study auto-B\"acklund transformations of the non-stationary second Painlev\'e hierarchy $\text{P}_\text{II}^{(n)}$ depending on $n$ parameters: a parameter $\alpha_n$ and times $t_1, \dots, t_{n-1}$. Using generators…
We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional K\"ahler-Einstein equation with symmetry and the one-sided type-D…
The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm…
The Baecklund transformation for the ultradiscrete KP equation is proposed. An algorithm to eliminate variables from the ultradiscrete linear equations is proposed. The consistency condition for the Baecklund transformation equations is…
We provide a method which takes an auto-B\"acklund transformation (auto-BT) and produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions…
Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher…
We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund transformations (BT). Relations among the DS system, the double Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are established. The DS…