Jon Nimmo

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

A noncommutative version of the modified KP equation and a family of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux transformations and the solutions are verified directly. We also verify directly an explicit connection between quasideterminant solutions of the noncommutative mKP equation and the noncommutative KP equation arising from the Miura transformation.

Pfaffianization of the discrete KP equation

We use the procedure of Ohta and Hirota to generate an integrable, coupled system of discrete equations from the discrete KP equation.

Pfaffianization of the Davey-Stewartson equations

We use the bilinear method from soliton theory to produce a Pfaffian version of the Davey–Stewartson equations. The solutions of this new system of equations are Pfaffians.

Quasideterminant solutions of a non-Abelian Hirota?Miwa equation

A non-Abelian version of the Hirota–Miwa equation is considered. In an earlier paper of Nimmo (2006 J. Phys. A: Math. Gen. 39 5053–65) 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-Plucker coordinates and that the non-Abelian Hirota–Miwa equation may be written as a quasi-Plucker relation. The special case of the matrix Hirota–Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared.

Darboux transformations and the discrete KP equation

This paper presents two results. First it is shown how the discrete KP equation arises from a superposition principle associated with the Darboux transformation of the two-dimensional Toda system. Then Darboux transformations and binary Darboux transformations are derived for the discrete KP equation and it is shown how these may be used to construct exact solutions.

Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation

We consider matrix solutions of a noncommutative KP and a noncommutative mKP equation; these solutions can be expressed as quasideterminants. In particular, we investigate the interaction of two-soliton solutions.

A direct approach to the ultradiscrete KdV equation with negative and non-integer site values

A generalization of the ultra-discrete KdV equation is investigated using a direct approach. We show that evolution through one time step serves to reveal the entire solitonic content of the system.