Ralph Willox

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.

BILINEAR FORM AND SOLUTIONS OF THE K-CONSTRAINED KADOMTSEV-PETVIASHVILI HIERARCHY

We show how to derive an alternative bilinear formulation for the k-constrained Kadomtsev - Petviashvili hierarchy. This Hirota form allows for the easy identification of a broad class of solutions to these equations.

Symmetry reductions of the BKP hierarchy

A general symmetry of the bilinear BKP hierarchy is studied in terms of tau functions. We use this symmetry to define reductions of the BKP hierarchy, among which new integrable systems can be found. The reductions are connected to constraints on the Lax operator as well as on the bilinear formulation. A class of solutions for the reduced equations is derived.

Binary Darboux transformations for constrained KP hierarchies

We describe how Darboux transformations and binary Darboux transformations can be constructed for (vector-) constrained KP hierarchies. These transformations are then used to obtain explicit classes of Wronskian and Grammian solutions for these hierarchies. The relationship between these two types of solutions is also discussed.

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.

On the Hirota Representation of Soliton Equations with One Tau-Function

Alternative Hirota representations in terms of a single tau-function are derived for a variety of soliton equations, including the sine-Gordon and the Tzitzeica equations. The relevance of these representations with respect to known bilinear representations of integrable hierarchies is briefly discussed. The essentials of the derivation method are presented.

KP symmetry reductions and a generalized constraint

We study the link between symmetry reductions and constraints of the Kadomtsev - Petviashvili equations in terms of the tau function. We propose a generalization - adapted to non-zero boundary conditions - of the standard constraints, and show a particular class of solutions (solitons).

Some new exact solutions of the Novikov - Veselov equation

The Novikov - Veselov (NV) equation is considered. Using Backlund transformations and a nonlinear superposition formula for the NV equation, some new exact solutions to this equation are found, among which are the so-called one- and two-lump solutions.

Complete integrability of a modified vector derivative nonlinear Schrodinger equation

Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schrodinger equation, if charge separation in Poissons equation and the displacement current in Amperes law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schrodinger equation and hence its possible integrability and related properties need to be established afresh. Indeed, the new equation is shown to be integrable by the existence of a bi-Hamiltonian structure, which yields the recursion operator needed to generate an infinite sequence of conserved densities. Some of these have been found explicitly by symbolic computations based on the symmetry properties of the new equation. Since the new equation includes as a special case the derivative nonlinear Schrodinger equation, the recursion operator for the latter one is now readily available.

Soliton Solutions of Wronskian Form to the Nonlocal Boussinesq Equation

We investigate how one may write the soliton solutions of a nonlocal Boussinesq equation in Wronskian form, and subsequently prove the existence of N -soliton solutions making use of the bilinear form of this equation. This technique also allows us to construct a bilinear Backlund transformation for this equation, mapping N -soliton solutions on ( N +1)-soliton solutions. Our analysis extends the results previously obtained by Hirota for the Classical Boussinesq system to actual ( c ≠0) “ p q = c ”-reductions performed on Wronskians.