Pfaffianization of the two-dimensional Toda lattice

Abstract

Pfaffianization procedure due to Hirota and Ohta is applied to the two-dimensional Toda lattice. As a result, a Pfaffianized version of the two-dimensional Toda lattice is found.  2004 Elsevier Inc. All rights reserved.In 1991, Hirota and Ohta [1,2] developed an effective procedure of what we now call Pfaffianization, which allows one to produce new coupled systems of equations with solutions in the form of Pfaffians. Recently, there has been growing interest in this direction. For example, such a procedure has been successfully applied to the Davey-Stewartson equations [3], the discrete KP equation [4] and the self-dual Yang-Mills equation [5]. Besides, the Pfaffianized KP hierarchies have been investigated in [6].The purpose of this paper is to apply the procedure of Hirota and Ohta to the following bilinear two-dimensional Toda lattice:where the bilinear operators D x D s and exp(D n ) are defined by [7,8] D x D s a · b ≡ ∂ ∂x − ∂ ∂x ∂ ∂s − ∂ ∂s a(x, s)b(x , s )| x =x, s =s

On quasiperiodic wave solutions and integrability to a generalized $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional Korteweg–de Vries equation

“…Proof In order to get the conservation laws, it is necessary to rewritten the two-filed condition (22) into the x− and y−derivative of Y -polynomials. First, we need to rewrite the R(υ, ω) as,…”

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

“…Substituting f into the bilinearized Equation (3), and equating the coefficients of different triangle and exponential functions to be zero, we can obtain the following equations, 6 2 4 3 4 2 2 2 6 3 2 2 1 3 1 1 1 3 1 3 3 1 1 3 1 1 1 3 3 3 1 3 3 3 15 5 15 15 15 36 5 5 5 5 3 3 3 2 5 2 3 3 1 3 1 3 1 1 3 1 3 1 3 3 1 3 1 1 3 1 3 3 1 6 20 5 15 6 15 36 5 10 36 5 3 3 3 2 5 2 3 3 2 3 2 3 2 2 3 2 3 2 3 3 2 3 2 2 3 2 3 3 2 6 20 5 15 6 15 36 5 10 36 3 4 2 2 2 6 3 2 2 2 3 2 2 2 3 2 3 3 2 2 3 2 2 2 3 3 3 2 3 3 3 15 5 15 15 15 36 5 5 5 …”

Matrix Integrals and Several Integrable Differential-Difference Systems

“…In [19], we have considered the Pfaffianization of the two-dimensional Toda lattice (27). As a result, the following Pfaffianized two-dimensional Toda lattice is derived:…”