Elliptic N -soliton-type solutions, i.e. solutions emerging from the application of N consecutive Bäcklund transformations to an elliptic seed solution, are constructed for all equations in the ABS list of quadrilateral lattice equations, except for the case of the Q4 equation which is treated elsewhere. The main construction, which is based on an elliptic Cauchy matrix, is performed for the eq...

We classify all integrable 3-dimensional scalar discrete quasilinear equations Q3 = 0 on an elementary cubic cell of the lattice Z 3. An equation Q3 = 0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the lattice Z4. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prov...

In Estonian, disyllabic feet can be in short (Q1), long (Q2), or overlong (Q3) quantity degree. The feet are left headed. Phonologically it is a combination of a stressed vowel and/or following consonant that carries the quantity (see Table 1). The duration of the second syllable compensates for the variation of the first syllable, being long in Q1, short in Q2 and extra-short in Q3. The quanti...

Let q be a prime power and g ∈ {6, 8, 12}. In this paper we obtain (q, g)-graphs on 2qg/2−3(q2 − 1) vertices for g = 6, 8, 12 as subgraphs of known (q + 1, g)-cages. We also obtain (q − r, 6)-graphs on 2(q2 − rq − 1) vertices, and (q − r, 8)-graphs on 2(q3−rq2−q+r) vertices where r is a positive integer such that q ≥ r+3. Furthermore, we find (q − 1, 12)-graphs of order 2q2(q3 − q2 − q + 1). So...

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work central role was played by a Cauchy matrix. In this work we use a different approach, we derive the N -soliton solutions following Hirota’s direct and constructive method. ...

We classify all integrable three-dimensional scalar discrete affine linear equations Q3= 0 on an elementary cubic cell of the lattice Z3. An equation Q3= 0 is called integrable if it may be consistently imposed on all three-dimensional elementary faces of the lattice Z4. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube ...

(Q1). = Q3, (Q2)w = -(Qi + Q2)/V/2, (Q3)D = (Q2 Q1)//2, (Q4)w = Q4. There are similar relations between the Wigner co-ordinates and our co-ordinates of Fig. 4, d. Wigner states that if the number of particles is a power of 2 and the masses are equal, there is a co-ordinate system of high symmetry in which all particles are treated equally. Unfortunately the form of the potential energy in the c...

Chow, T., Distances forbidden by two-colorings of Q3 and A,, Discrete Mathematics 115 (1993) 95-102. For X = Q3 or A, (where A, is the set of points in Q” whose coordinates have odd denominators), we characterize all sets of distances D c R + with the following property: there exists some two-coloring of X such that, for all d ED, no two points in X that arc a distance d apart are the same colo...