Related papers: B\"acklund Transformations of MKdV and Painlev\'e …

Starting from nonlocal symmetries related to B\"acklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact…

We present certain general structures related to the solutions of Painlev\'e equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking…

A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space…

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…

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also…

General $N$-solitons in three recently-proposed nonlocal nonlinear Schr\"odinger equations are presented. These nonlocal equations include the reverse-space, reverse-time, and reverse-space-time nonlinear Schr\"odinger equations, which are…

We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to…

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…

We study the low-energy effective actions for gauge superfields induced by quantum N=2 and N=4 supersymmetric matter fields in three-dimensional Minkowski space. Analyzing the superconformal invariants in the N=2 superspace we propose a…

Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order…

Starting from the Akulov-Volkov (AV) action, we compute a finite-dimensional Lie group G of all field transformations of the form \lambda -> \lambda ' = \lambda + O(\lambda ^3) which preserve the functional structure of low-energy…

In this paper we extend the novel approach to discrete Painlev\'e equations initiated in our previous work [2]. A classification scheme for discrete Painlev\'e equations proposed by Sakai interprets them as birational isomorphisms between…

\hspace{.2in}We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of…

We systematically study Darboux-type transformations for the KdV and AKNS hierarchies and provide a complete account of their effects on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies.

In this note, we review the notion of Painlev\'e scheme of the sixth Painlev\'e equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${\Sigma_2}$. The key method is Painlev\'e…

Super-matrix KdV and super-generalized non-linear Schrodinger equations are shown to arise from a symmetry reduction of ordinary self-dual Yang-Mills equations with supergauge groups

We propose a discrete Darboux-Lax scheme for deriving auto-B\"acklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the…

In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in…

In this short note we give two examples of using the algebro-geometric theory of Painlev\'e equations to solve the Painlev\'e identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in…

A seventh order ordinary differential equation (ODE) arising by reduction of the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its link with a…