Hierarchy of (2 + 1)-Dimensional Nonlinear Schrödinger Equation, Self-Dual Yang-Mills Equation, and Toroidal Lie Algebras

Abstract

The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schrödinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed. 818, x = ix 1 , y = y 0 , t = −y 1 . In this sense, the evolution equations (2.5) and 822

On the (2+1)-Dimensional Extension of 1-Dimensional Toda Lattice Hierarchy

“…We obtain the bilinear identity slightly different from that of Theorem 2, which enables us to relate our system to the sl tor 2 -hierarchy of [8].…”

“…Substituting the expressions (2.7) into (2.3), we obtain "dressing relations" for the Lax operators 8) and evolution equations for the Sato-Wilson operators…”

“…On these backgrounds, in [8], Kakei and two authors in [5] developed their previous work to a hierarchy associated with the (2+1)-dimensional nonlinear Schrödinger (NLS) equation [2,11,12] iu x + u xy + 2u The aim of the present paper, motivated by the work of Ikeda, Takasaki and Kakei, is to develop the theory of the Toda lattice (TL) hierarchy to treat (2+1)-dimensional soliton equations. Namely we consider a (2+1)-dimensional extension of the so-called 1-dimensional Toda lattice (1-D TL) hierarchy [13] of nonlinear differential-difference equations associated with the Toda lattice equation The resulting is a system of nonlinear differential-differential-difference equations, which we call the (2+1)-dimensional Toda lattice ((2+1)-D TL) hierarchy.…”

“…For the second formulation, we obtain the bilinear identity slightly different from the previous one. This bilinear identity can be identified with that of the sl tor 2 -hierarchy of [8] arising from the homogeneous vertex representation of sl tor 2 . This paper is organized as follows.…”

Solutions of a Derivative Nonlinear Schrödinger Hierarchy and Its Similarity Reduction

“…We apply a modified version of Date's method [7,8] to construct a special class of solutions for (2.6), which we shall seek in the form…”