Jonathan Nimmo

On the Combinatorics of the Hirota D-Operators

A generic formula is presented which relates the Hirota D-operators to simple combinatorics. Particular classes of partition polynomials (Bell-polynomials and generalizations) are found to play an important role in the characterization of bilinearizable equations. As a consequence it is shown that bilinear Bäcklund transformations for single-field bilinearizable equations linearize systematically into corresponding Lax-pairs.

An integrable symmetric (2+1)-dimensional Lotka-Volterra equation and a family of its solutions

A symmetric (2+1)-dimensional Lotka–Volterra equation is proposed. By means of a dependent variable transformation, the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables. A bilinear Backlund transformation is found and then the corresponding Lax pair is derived. Explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric (2+1)-dimensional Lotka–Volterra equation are given. As a special case of the pfaffian solutions, we obtain soliton solutions and dromions.

On a non-Abelian Hirota–Miwa equation

A generalization of the Hirota–Miwa equation to an abstract non-Abelian associative algebra is considered. This system is integrable in the sense that it arises as compatibility condition for a linear system and has solutions constructed by means of the application of an arbitrary number of Darboux transformations. These solutions are in general expressed in terms of quasideterminants.

Hall-Littlewood symmetric functions and the BKP equation

The connection between solutions of the BKP equation and Hall-Littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions. This is analogous to the Wronskian formulation of the solution of the KP equation. As a by-product, two novel expressions for certain Hall-Littlewood functions in terms of Pfaffians are derived.

A bilinear approach to a Pfaffian self-dual Yang-Mills equation

By using the bilinear technique of soliton theory, a pfaffian version of the SU(2) self-dual Yang-Mills equation and its solution is constructed.

A Direct Method for Dromion Solutions of the Davey-Stewartson Equations and their Asymptotic Properties

A new direct approach to determining a class of solutions, including the dromion solutions, of the Davey-Stewartson equations is presented. The form of solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined. A class of solutions describing dromion scattering is discussed and computer plots of the interactions of such solutions are presented.

A (2+1)-dimensional generalization of the AKNS shallow water wave equation

An integrable generalization to 2+1 dimensions of the shallow water wave equation of Ablowitz, Kaup, Newell and Segur [Stud. Appl. Math. 53 (1974) 249] is sought through the bilinear approach. This equation is shown to belong to the KP hierarchy and a broad class of solutions including the N-soliton solution is obtained.

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

The linear problem associated with the self-dual Yang-Mills equations is covariant with respect to Darboux and binary Darboux transformations of almost classical type. This technique is used to construct solutions of the problem in the form of Wronskian-like and Gramm-like determinants. The self-dual conditions can be properly realized for only the latter type of solutions.

Yang-Baxter maps and the discrete KP hierarchy

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples.

Wronskian determinants, the KP hierarchy and supersymmetric polynomials

By using the Wronskian representation of the solutions of the bilinear KP hierarchy, a connection between Hirota derivatives and supersymmetric polynomials is brought to light. This correspondence is used in order to give an alternative construction of the hierarchy.