Quasideterminant solutions of a non-Abelian Hirota-Miwa equation

A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. \textbf{39}, 5053-5065] it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper we discuss these solutions from a different perspective and show that the solutions are quasi-Pl\"{u}cker coordinates and that the non-Abelian Hirota-Miwa equation may be written as a quasi-Pl\"{u}cker relation. The special case of the matrix Hirota-Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared

On a direct approach to quasideterminant solutions of a noncommutative KP equation

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the regular, commutative KP equation but, in the noncommutative case, no bilinearising transformation is available.Comment: 11 page

The Sasa-Satsuma higher order nonlinear Schrodinger equation and its bilinearization and multi-soliton solutions

Higher order and multicomponent generalizations of the nonlinear Schrodinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately the construction of multi-soliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petvishvili hierarchy. In the process we also get bilinearizations and multi-soliton formulae for a two component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional generalization.Comment: 13 pages in RevTex, added reference