We analyze the spatial distribution of the order parameter \Delta(r) in strongly disordered superconductors close to Superconductor-Insulator Transition. The analysis is based on a model of a superconductor on a locally loopless lattice with a pseudogap. We derive and solve a set of equations for the local distribution function P(\Delta) at zero temperature. The results are applicable both in the region of relatively small disorder corresponding to a Gaussian profile of P(\Delta) and in the region of strong fluctuation. The analytical results are in excellent agreement with the direct numerical solution of the self-consistency equations.

We study determininstic power-law quantum hopping model local Gaussian disorder in low dimensions d = 1, 2 under the condition d < β < 3d/2. We demonstrate unusual combination of exponentially decreasing density of the ”tail states” and localization-delocalization transition (as function of disorder strength w) pertinent to a small (vanishing in thermodynamic limit) fraction of eigenstates. In a broad range of parameters density of states ν(E) decays into the tail region E < 0 as simple exponential. We develop simple analytic theory which describes E0 dependence on power-law exponent β, dimensionality d and disorder strength W , and compare its predictions with exact diagonalization results. At low energies within the bare ”conduction band”, all eigenstates are localized due to strong quantum interference at d = 1, 2; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder.

Hydrodynamics is a powerful theory for the emergent behaviour at large wavelengths and low frequencies in many-body systems. The theory says that only few degrees of freedom are sufficient in order to describe what is observed at large scales of space and time, and it provides equations for the dynamics of these degrees of freedom. It is strongly based on the presence of microscopic conservation laws in the many-body model, such as conservation of energy, momentum and mass. But the standard equations of hydrodynamics fail to describe cold atom experiments in low dimensions. It is now understood that this is because the model accurately describing these experiments, the Lieb-Liniger model, is integrable. Integrable systems admit an extensive number of conservation laws, which must be taken into account in the emergent hydrodynamic theory. Recently this hydrodynamic theory, dubbed ``generalised hydrodynamics”, has been developed. In this colloquium, I will review fundamental aspects of hydrodynamics and the main idea and equations of generalised hydrodynamics, with the simple example of the quantum Lieb-Liniger model. I will discuss recent cold-atom experiments that confirm the theory, and some of the exact results that can be obtained with this formalism, such as exact nonequilibrium steady states and exact asymptotic of correlation functions at large space-time separations in Gibbs and generalised Gibbs states.

The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to construct isomonodromic Lax pairs for them.