Volume 3

1. Full-deautonomisation of a class of second-order mappings in ancillary form

Basil Grammaticos ; Ralph Willox.
We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

2. The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds: Bilinearisation and reduction approach

Da-jun Zhang ; Shi-min Liu ; Xiao Deng.
In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schrödinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.

3. Integrable Ito equations and properties of the associated Fokker-Planck equations

Giuseppe Gaeta ; Miguel Angel Rodriguez.
In a recent paper we have classified scalar Ito equations which admits a standard symmetry; these are also directly integrable by the Kozlov substitution. In the present work, we consider the diffusion (Fokker-Planck) equations associated to such symmetric Ito equations.

4. Derivation of Painlevé type system with $D_4^{(1)}$ affine Weyl group symmetry in a self-similarity limit

H. Aratyn ; J. F. Gomes ; G. V. Lobo ; A. H. Zimerman.
We show how the zero-curvature equations based on a loop algebra of $D_4$ with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlevé III models with four canonical variables and $D_4^{(1)}$ affine Weyl group symmetry.

5. Geometrical description of equations related to the E8 affine Weyl group

Alfred Ramani ; Basil Grammaticos ; Thamizharasi Tamizhmani.
We present a method for the construction of the trajectory of a discrete Painlevé equation associated with the affine Weyl group E$_8^{(1)}$ on the weight lattice of said group. The method is based on the geometrical description of the lattice and the construction of the fundamental Miura relation. To this end we introduce the relation between the nonlinear variables and the corresponding $\tau$ functions. Our approach is heuristic and makes use of some simple rules of thumb in order to derive the result. Once the latter is obtained, verifying that it does indeed correspond to the equation at hand is elementary. We apply our approach to the explicit construction of the trajectory of well-known, E$_8^{(1)}$ associated, discrete Painlevé equations derived in previous works of ours. For each of them we investigate the possibility of defining an evolution by periodically skipping up to four intermediate points in the trajectory and identifying the resulting equation to one previously obtained, whenever the latter exists.