We show that the quantum field theoretical formulation of the τ function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Lévy and adjoint Lévy transformations and iii) the diagonal s...

The structure of integrable field theories in the presence of jump defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the boundary functions for the N = 1 and N = 2 super sinh-Gordon models are constructed and shown to generate the Backlund transformations for its soliton solution...

The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim σ-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups F ⊃ G ⊃ H . We show that for every symmetric space F/G, the generalized sine-Gordon models can be derived from the G/H WZW action, p...

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painleve equation. One obtains a Riemann–Hilbert correspondence between moduli spaces rank two connections on ℙ1 and for monodromy data. these identified with Okamoto–Painleve varieties property follows. For an explicit computation full group Backlund transformations, three introduced, inspired by symmetric form PIV, studied M. Noumi...

This paper shows that several integrable lattices can be transformed into coupled bilinear differential-difference equations by introducing auxiliary variables. By testing the Bäcklund transformations for this type of coupled bilinear equations, a new integrable lattice is found. By using the Bäcklund transformation, soliton solutions are obtained. By the dependent variable transformation, this...

Recent algorithms for sparse coding and independent component analysis (ICA) have demonstrated how localized features can be learned from natural images. However, these approaches do not take image transformations into account. We describe an unsupervised algorithm for learning both localized features and their transformations directly from images using a sparse bilinear generative model. We sh...

Based on the Bell polynomials scheme, this paper presents a Maple computer algebra program PDEBellII which can automatically construct the bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type soliton equations. Some examples are given to verify the validity of our program.