This paper shows that several integrable lattices can be transformed into coupled bilinear differential-difference equations by introducing auxiliary variables. By testing the Bäcklund transformations for this type of coupled bilinear equations, a new integrable lattice is found. By using the Bäcklund transformation, soliton solutions are obtained. By the dependent variable transformation, this...

The Knizhnik-Zamolodchikov (KZ ) equations is a holonomic system of differential equations for correlation functions in conformal field theory on the sphere [KZ]. The KZ equations play an important role in representation theory of affine Lie algebras and quantum groups, see for example [EFK]. There are rational, trigonometric and elliptic versions of KZ equations, depending on what kind of coef...

Formulating a Schubert problem as the solutions to a system of equations in either Plücker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation en...

one of the ecient and powerful schemes to solve linear and nonlinear equationsis homotopy analysis method (ham). in this work, we obtain the approximate solution ofa system of partial dierential equations (pdes) by means of ham. for this purpose, wedevelop the concept of ham for a system of pdes as a matrix form. then, we prove theconvergence theorem and apply the proposed method to nd the a...

‎in this paper‎, ‎we propose two preconditioned aor iterative methods to solve systems of linear equations whose coefficient matrices are z-matrix‎. ‎these methods can be considered as improvements of two previously presented ones in the literature‎. ‎finally some numerical experiments are given to show the effectiveness of the proposed preconditioners‎.‎

Finite state bilinear systems are described by a system of differential equations which are linear in state, linear in control, but not jointly linear in both as are traditional linear systems. An introduction to the physical relevance of these systems, their controllability and optimal control is analyzed in references [i] and [2]. It is shown that linear system controllability and performance...

The main purpose of the paper is to summarize some fundamental results concerning the matrix impulsive differential equations with impulses at fixed points. The theory of impulsive matrix differential equations is interesting in itself and it will assume greater importance in the near future, since the application of the theory to various fields of science is also increasing, especially in the ...

Nonlinear evolution wave equations (NEEs) are partial differential equations (PDEs) involving first or second order derivatives with respect to time. Such equations have been intensively studied for the past few decades [1-3] and several new methods to solve nonlinear PDEs either numerically or analytically are now available. Hirota's bilinear method is a powerful tool for obtaining a wide clas...