Related papers: On dbar-problem and integrable equations
A bilinear quadrature numerically evaluates a continuous bilinear map, such as the $L^2$ inner product, on continuous $f$ and $g$ belonging to known finite-dimensional function spaces. Such maps arise in Galerkin methods for differential…
Some new developments in constrained Lax integrable systems and their applications to physics are reviewed. After summarizing the tau function construction of the KP hierarchy and the basic concepts of the symmetry of nonlinear equations,…
Recently, Lembert, Gilson et al proposed a lucid and systematic approach to obtain bilinear B\"{a}cklund transformations and Lax pairs for constant-coefficient soliton equations based on the use of binary Bell polynomials. In this paper, we…
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
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…
Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of…
The quasi-classical limit of the scalar nonlocal dbar-problem is derived and a quasi-classical version of the dbar-dressing method is presented. Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative examples. It is shown…
In this paper, we present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota's bilinear method. This approach is mainly based on the compatibility between an integrable system and its B\"acklund…
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems, we consider the Darboux and binary…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a…
The aim of this paper is to summarize some recently obtained relations between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It has been shown that solutions of finite subsystems of the ALH can be used to derive a wide…
An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem…
It is shown that the matrix KP hierarchy can yield new integrable equations in $(2+1)$-dimensions along with the corresponding Lax pair. For particular gauge choice this may result derivative and also a higher order nonlinear extension of…
Additional non-isospectral symmetries are formulated for the constrained Kadomtsev-Petviashvili (\cKP) integrable hierarchies. The problem of compatibility of additional symmetries with the underlying constraints is solved explicitly for…
In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic…
We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper…
A class of nonlinear problems on the plane, described by nonlinear inhomogeneous $\bar{\partial}$-equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources) is described…