Related papers: Non-trivial Darboux solutions of Classical Painlev…
We study the degenerate Garnier system which generalizes the fifth Painlev\'{e} equation. We present two classes of particular solutions, classical transcendental and algebraic ones. Their coalescence structure is also investigated.
By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…
We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko…
The Lax representation and Backlund transformations for the systems similar to WZNW (Wess-Zumino-Novicov-Witten) systems and non-abelian affine Toda models are obtained in present paper. One of these systems is a new integrable extension of…
In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlev\'e VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux…
We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schr\"odinger equations that admit Lax representations. The…
In this paper, we construct a generalized recursive Darboux transformation of a focusing vector nonlinear Schr\"odinger equation known as the Manakov system. We apply this generalized recursive Darboux transformation to the Lax-pairs of…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
All Painlev\'e equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic).…
We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the…
We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear $2\times 2$ system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the…
In this article, we present a new quantum Painlev\'e II Lax pair which explicitly involves the Planck constant $ \hbar $ and an arbitrary field variable $v$ so these two objects make this new pair different from Flaschka-Newell Painlev\'e…
It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at…
Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the…
We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform…
In the work we use integral formulas for calculating the monodromy data for the Painlev\'e-2 equation. The perturbation theory for the auxiliary linear system is constructed and formulas for the variation of the monodromy data are obtained.…
A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…
The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions…
This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable…
We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux…