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Long range correlations in two-dimensional (2D) systems are significantly altered by disorder potentials. Theory has predicted the existence of disorder induced phenomena such as Anderson localization and the emergence of novel glass and insulating phases as the Bose glass. More recently, it has been shown that disorder breaking the 2D continuous symmetry, such as a one dimensional (1D) modulation, can enhance long range correlations. Experimentally, developments in quantum gases have allowed the observation of a wealth of phenomena induced by the competition between interaction and disorder. However, there are no experiments exploring the effect of symmetry-breaking disorder. Here, we create a 2D vortex lattice at 0.1 K in a superconducting thin film with a well-defined 1D thickness modulation and track the field induced modification using scanning tunneling microscopy. We find that the 1D modulation becomes incommensurate to the vortex lattice and drives an order-disorder transition, behaving as a scale-invariant disorder potential. We show that the transition occurs in two steps and is mediated by the proliferation of topological defects. We find that critical exponents determining the loss of positional and orientational order are far above theoretical expectations for scale-invariant disorder and follow instead the critical behaviour which describes dislocation unbinding melting. Our data show for the first time that randomness disorders a 2D crystal, and evidence enhanced long range correlations in presence of a 1D modulation demonstrating the transformation induced by symmetry breaking disorder in interactions and the critical behaviour of the transition., Comment: 7 pages, 3 figures and supplementary information (11 pages, 9 figures)

Solution of Monge equation of arbitrary degree (non linear differential equation n-orden) is connected with solution of functional equation for 4 functions with 4 different arguments. Some number solutions of this equation is represented in explicit form., Comment: 16 pages

A way to obtain the series solutions of the 1 + 2 dimensional continuous Toda chain is presented., Comment: 10 pges, no figures One co-authors full name is given

We study the behavior of bundles of superconducting vortices when increasing the magnetic field using scanning tunneling microscopy and spectroscopy (STM/S) at 100 mK. Pinning centers are given by features on the surface corrugation. We find strong net vortex motion in a bundle towards a well defined direction. We observe continuos changes of the vortex arrangements, and identify small displacements, which stress and deform the vortex bundle, separated by larger re-arrangements or avalanches, which release accumulated stress., Comment: 4 pages, 4 figures, and 2 EPAPS figures

Topological defects such as dislocations and disclinations are predicted to determine the twodimensional (2-D) melting transition. In 2-D superconducting vortex lattices, macroscopic measurements evidence melting close to the transition to the normal state. However, the direct observation at the scale of individual vortices of the melting sequence has never been performed. Here we provide step by step imaging through scanning tunneling spectroscopy of a 2-D system of vortices up to the melting transition in a focused-ion-beam nanodeposited W-based superconducting thin film. We show directly the transition into an isotropic liquid below the superconducting critical temperature. Before that, we find a hexatic phase, characterized by the appearance of free dislocations, and a smectic-like phase, possibly originated through partial disclination unbinding. These results represent a significant step in the understanding of melting of 2-D systems, with impact across several research fields, such as liquid crystal molecules, or lipids in membranes., Comment: Submitted to Nature Physics

We present very low temperature Scanning Tunneling Microscopy and Spectroscopy (STM/S) measurements of a W based amorphous thin film grown with focused-ion-beam. In particular, we address the superconducting properties close to the border, where the thickness of the superconducting film decreases, and the Au substrate emerges. When approaching the Au substrate, the superconducting tunneling conductance strongly increases around the Fermi level, and the quasiparticle peaks do not significantly change its position. Under magnetic fields, the vortex lattice is observed, with vortices positioned very close to the Au substrate., Comment: To appear in Journal of Physics: Conference series

We present very low temperature Scanning Tunneling Microscopy and Spectroscopy (STM/S) measurements in W-based amorphous superconducting nanodeposits grown using a metal-organic precursor and focused-ion-beam. The superconducting gap closely follows s-wave BCS theory, and STS images under magnetic fields show a hexagonal vortex lattice whose orientation is related to features observed in the topography through STM. Our results demonstrate that the superconducting properties at the surface of these deposits are very homogeneous, down to atomic scale. This, combined with the huge nanofabrication possibilities of the focused-ion-beam technique, paves the way to use focused-ion-beam to make superconducting circuitry of many different geometries.

Multi-soliton solution of the 3-waves problem is represented in explicit determinative form., Comment: 12 pages

Discrete transformation for 3- waves problem is constructed in explicit form. Generalization of this system on the matrix case in three dimensional space together with corresponding discrete transformation is presented also., Comment: LaTeX, 16 pages

It is shown that in the case of multicomponent integrable systems connected with algebras $A_n$, the discrete transformation $T$ possesses the fine structure and can be represented in the form $T=\prod T_i^{l_i}$, where $T_i$ are n commuting basis discrete transformations, and $l_i$ are arbitrary natural numbers. All the calculations are conducted in detail for the case of a 3-wave interacting system., Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP/