Bilinear Form Approach to the Self-Dual Yang-Mills Equations and Integrable Systems in (2+1)-Dimension

Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

“…Furthermore investigation of the noncommutative extension of a bilinear form approach to the ASDYM equation (Gilson, Nimmo and Ohta (1998) ;Sasa, Ohta and Matsukidaira (1998); Wang and Wadati (2004)) would be beneficial because many aspects in these paper are close to ours. The relationship with noncommutative Darboux and noncommutative binary Darboux transformations (Salaam, Hassan and Siddiq (2007)) is also interesting.…”

Applications of Darboux transformations to the self-dual Yang-Mills equations

“…As well as arising through Darboux transformations, such determinants are familiar as ansatze used in Hirota's method such as the recent work of Sasa et al [7].…”

“…This situation arises here if wish to construct solutions J in SU(iV). Before showing how this done, we first comment that recently Sasa et al [7] were able to construct solutions similar to the wronskian like ones given in Corollary 2 but only for the case of SU (2). This was done by building the SU(2) symmetry into J from the start but it not clear how one could do this in any other case.…”

“…In doing this we give a simple way of understanding the form of the solutions of these equations to readers familiar with Darboux transformations or Hirota's method but not with the more sophisticated methods that have been used in the literature. The original motivation of this work was to find a generalisation to the general case of the work ofSasa et al [7] on the Hirota form ofSU (2) SDYM. In independent work, Darboux transformations have been used recently to study a reduced version of the SDYM equations [8].…”

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

“…Equation (1.2) is related to the self-dual Yang-Mills (SDYM) equation, and has been studied by several researchers from various viewpoints: Lax pairs [Bo2,Sc,St1], Hirota bilinear method [SOM,St2], twistor approach [St2], Painlevé analysis [JB], and so on. In the case X = Y , this equation is reduced to iu T + u XX + 2|u| 2 u = 0, (1.3) which is the celebrated Nonlinear Schrödinger (NLS) equation.…”