Affine Solitons: A Relation Between Tau Functions, Dressing and B\"acklund Transformations

We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the $N$-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to built the $N$-soliton solutions from the vacuum solution. The dressed $\tau$-functions can be computed in two different ways~: either using adjoint actions in the affine Lie algebra $\hat {sl_2}$, and this gives the relation with the B\"acklund transformations, or using the level one representations of the affine Lie algebra $\widehat{sl_2}$, and this directly gives the formulae for the $\tau$-functions in terms of vertex operators.Comment: 33 pages. SPhT-92-055; LPTHE-92-1

A Relation Between Approaches to Integrability in Superconformal Yang-Mills Theory

We make contact between the infinite-dimensional non-local symmetry of the typeIIB superstring on AdS5xS5 worldsheet theory and a non-abelian infinite-dimensional symmetry algebra for the weakly coupled superconformal gauge theory. We explain why the planar limit of the one-loop dilatation operator is the Hamiltonian of a spin chain, and show that it commutes with the g*2 N = 0 limit of the non-abelian charges.Comment: 19 pages, harvma

Symmetries in the Lorenz-96 model

The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing $F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is $\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$ using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces, that play an important role in this model. We exploit them in order to generalise results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for $F<0$ in specific dimensions $n$: In all even dimensions, the equilibrium $(F,\ldots,F)$ exhibits a supercritical pitchfork bifurcation. In dimensions $n=4k$, $k\in\mathbb{N}$, a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension $n=2^qp$, where $q\in\mathbb{N}\cup\{0\}$ and $p$ is odd, there is a finite cascade of exactly $q$ subsequent pitchfork bifurcations, whose bifurcation values are independent of $n$. This structure is discussed and interpreted in light of the symmetries of the model.Comment: 31 pages, 9 figures and 3 table