Related papers: Toda Darboux transformations and vacuum expectatio…
We propose a new multi-component two-dimensional Toda lattice hierarchy (mc2dTLH) which includes two-dimensional Toda lattice equation with self-consistent sources (2dTLSCS) as the first non-trivial equation. The Lax representations for…
The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$.…
Toda lattice hierarchy and the associated matrix formulation of the $2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working…
In this paper we review two concepts directly related to the Lax representations: Darboux transformations and Recursion operators for integrable systems. We then present an extensive list of integrable differential-difference equations…
For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion…
Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$, are invesitigated by Darboux transformations $T_D(f)=f^{[1]}\cdot\Delta\cdot…
We extend one component Gross-Pitaevskii equation to two component coupled case with the damping term, linear and parabolic density profiles, then give the Lax pair and infinitely-many conservations laws of this coupled system. The system…
We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian-Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order - this…
Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schr\"odinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized…
The stationary Schroedinger equation of the harmonic oscillator is deformed by a Darboux transformation to construct time-dependent potentials with the oscillator profile. The Darboux (supersymmetric or factorization) method is usually…
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…
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem.…
For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding…
We generalize the determinant representation of the KP $\tau$ functions to the case of the 2D Toda $\tau$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions; for which we…
The classical Darboux system governing rotation coefficients of three-dimensional metrics of diagonal curvature possesses an equivalent formulation as a sixth-order PDE for a scalar potential (related to the corresponding $\tau$-function).…
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…
Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have…
We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian…
The use of effective Darboux transformations for general classes Lax pairs is discussed. The general construction of ``binary'' Darboux transformations preserving certain properties of the operator, such as self-adjointness, is given. The…
We present an unifying description of the graded $SL(p,q)$ KP-KdV hierarchies, including the Wronskian construction of their tau-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of…