A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. 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-Plücker coordinates ...

The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the τ -function. The so called dispersionless Hirota equations are obtained from the Hirota equations of the τ -function. These dispersionless Hirota equations turn out to be equivalent to a system of Hamilton-Jacobi equations. Other relevant...

Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invari...

In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE...

Polynomials in differentiation operators are considered. Joint covariance with respect to Darboux transformations of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechèt differential (first term of the Taylor series on prolonged space) to establish the operator forms. In the commutative (Abelian) c...

In this paper, we construct Grammian-like quasideterminant solutions of a non-Abelian Hirota–Miwa equation. Through continuum limits of this non-Abelian Hirota–Miwa equation and its quasideterminant solutions, we construct a cascade of noncommutative differential-difference equations ending with the non-commutative KP equation. For each of these systems, the quasideterminant solutions are const...

HE fertility factor, F, controls conjugal fertility of Escherichia coli Kl2 ( LEDERBERG, CAVALLI and LEDERBERG 1952; HAYES 1953). F also determines functions such as immunity to superinfection by a second F factor (MAAS 1963) ; growth inhibition of certain phages such as tau ( 7 ) (HAKURA, OTSUJI and HIROTA 1964) , T7 ( MAKEL;~, MAKEL;~ and SOIKKELI 1964), and T3 ( SCHELL et al. 1963) ; and for...