Claire R. Gilson

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.

Lump solutions of the BKP equation

The BKP equation (Date, Jimbo, Kashiwara and Miwa 1981, Jimbo and Miwa 1983)

Pfaffianization of the discrete KP equation

{({{\rm{u}}_{\rm{t}}} + 15{\rm{u}}{{\rm{u}}_{{\rm{3x}}}} + 15{\rm{u}}_{\rm{x}}^{\rm{3}} - 15{{\rm{u}}_{\rm{x}}}{{\rm{u}}_{\rm{y}}} + {{\rm{u}}_{{\rm{5x}}}})_{\rm{x}}} - 5{{\rm{u}}_{{\rm{3x,y}}}} - 5{{\rm{u}}_{{\rm{yy}}}} - 0,

Pfaffianization of the Davey-Stewartson equations

(1) is a 2+1 dimensional generalisation of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation (Caudrey, Dodd and Gibbon 1976, Sawada and Kotera 1974). This equation arises from the B-type Lie algebras as opposed to the KP equation which arises from the A-type algebras.

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

Resonant behaviour in the Davey-Stewartson equation

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

Minimum-error discrimination between three mirror-symmetric states

A class of fully nonlinear third-order partial differential equations (PDES) is considered. This class contains several examples which have recently appeared in the literature and for which rather unusual travelling-wave solutions have been given. These solutions consist essentially of sums of exponentials; we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling-wave reduction. In addition, we consider the Painleve analysis of this set of equations, both for the original PDE and also for reductions to ordinary differential equations (ODES). No equation in the class considered survives the combination of PDE and ODE tests. Also, an equation in the class considered which is known to be integrable is shown to possess only the weak Painleve property. Our analysis, therefore, confirms the limitations of the Painleve test as a test for complete integrability when applied to fully nonlinear PDES.

Quasideterminant solutions of a non-Abelian Hirota?Miwa equation

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.