Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

Based on the recent formulation of a general scheme to construct boundary Lax pairs,we develop this systematic construction for the A (1) n affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sineGordon (A (1) 1 ) and the A (1) 2 models. The A (1) 2 Toda theory is the first non trivial example of the hierarchy that exhibits two distinct type...

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than o...

In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and some special types of high dimensional models which possess Schwarz variants may have a second order Lax pair. The explicit Lax pairs for (1+1)-dimensional Kor...

We consistently develop a recently proposed scheme of matrix extension dispersionless integrable systems for the general case multidimensional hierarchies, concentrating on dimension $d\geqslant 4$. present extended Lax pairs, Lax-Sato equations, equations background vector fields and dressing scheme. Reductions, construction solutions connections to geometry are discussed. consider separately ...

Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are nonautonomous versions of the chiral model in (2 + 1) dimensions, generalized nonlinear Schrödinger, Korteweg-de Vries, Toda lattice, Garnier and Euler-Arnold equations. The Lax pairs for all of these equations are derived by the symmetry reductions of ...

The paper contains five examples of using cyclic bases of zero-curvature representations in studies of weak and strong Lax pairs, hierarchies of evolution systems, and recursion operators.