We consider two different models of randomly coupled N>>1 autonomous differential equationswith the aim of counting fixed points (aka equilibria), and classifying them by their ''instability index'', i.e. the number of unstable directions. In the first model, characterized by both translational and rotational statistical symmetry of the vector field, we estimate the probability of an equilibrium to have a given index in a phase with exponentially many equilibria. In the second model, characterized by only rotational statistical symmetry around a chosen stable equilibrium, we find a characteristic distance beyond which the multitude of equilibria prevents a trajectory to go towards the stable equilibrium. This may shed light on ''resilience'' mechanisms of complex ecosystems.

We consider kinetic equation for quintic-NLSE, corresponding to 6-wave nonlinear waves interaction. Kinetic equation for the periodic boundary conditions was derived. We propose conditions on equation parameters which guarantee reasonably accurate description of the wave field by wave kinetic equation. Direct numerical simulations in both dynamical equation and wave kinetic equation frameworks demonstrate acceptable correspondence between these significantly different models when proposed conditions for parameters are satisfied.

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-Hübsch-Krawitz (BHK) mirror symmetry. We show that the periods of the holomor phic nonvanishing form of different Calabi-Yau orbifolds, which are BHK mirrors of the the same CY family, coincide.
Alexander Belavin, Vladimir Belavin, Gleb Koshevoy, "Periods of Berglund–Hübsch–Krawitz mirrors", arXiv: hep-th-2012.03320

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.

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.

The multifractal superconducting state originates from the interplay of Anderson localization and interaction effects. In this article we overview the recent theory of the superconductivity enhancement by multifractality and extend it to describe the spectral properties of superconductors on the scales of the order of the superconducting gap. Specifically, using the approach based on renormalization group within the nonlinear sigma model, we develop the theory of a multifractal superconducting state in thin films. We derive a modified Usadel equation that incorporates the interplay of disorder and interactions at energy scales larger than the spectral gap and study the effect of such an interplay on the low-energy physics. We determine the spectral gap at zero temperature which occurs to be proportional to the multifracally enhanced superconducting transition temperature. The modified Usadel equation results in the disorder-averaged density of states that, near the spectral gap, resembles the one obtained in the model of a spatially random superconducting order parameter. We reveal strong mesoscopic fluctuations of the local density of states in the superconducting state. Such strong mesoscopic fluctuations imply that the interval of energies in which the superconducting gap establishes is parametrically large in systems with multifractally-enhanced superconductivity.

Making use of the gauge/string duality, it is possible to study some aspects of the string breaking phenomenon in the three quark system. Our results point out that the string breaking distance is not universal and depends on quark geometry. The estimates of the ratio of the string breaking distance in the three quark system to that in the quark-antiquark system would range approximately from 2/3 to 1. In addition, it is shown that there are special geometries which allow more than one breaking distance.

The solution of the Schrödinger equation for the ground state of a particle in a potential field is investigated. Since the wave functions of the ground state have no nodes, it turns out to be possible to unambiguously determine potentials of different types. It turned out that for a wide range of model potentials the ground state energy is zero. Moreover, the zero level may be the only level at the edge of the continuous spectrum. The crater-type potentials, which have a monotonic dependence on coordinates, are considered for the case of one, two and three dimensions. In the one-dimensional case, of interest are potentials of the "instanton" type with two equilibrium points of the particle. For the Coulomb potential, the ground state energy is stable to its screening, both at large and small distances. Two-soliton solutions of the nonlinear Schrödinger equation are found. The effectiveness of the proposed "inverse problem method" for investigating the solutions of differential equations is argued.
We found a wide class of potentials for which the energy of the ground state is stationary during the translations of wave function for the dimension d = 1,2,3. For dimension d = 1, these functions are known from the theory of nonlinear equations. Examples of nonlinear equations with zero ground state energy are given. Functions localized on a sphere and the crater-like potentials are investigated.

The thermodynamic properties of superconducting granules with a small parameter $ \delta = \varepsilon_0 / Tc $ are determined, where $ \varepsilon_0 $ is the distance between the levels of dimensional quantization, $ Tc $ is the temperature of the transition to the superconducting state of the bulk sample. It is shown that the order parameter $ \Delta (T) $ does not vanish at $T > Tc$ and even passes through a minimum at $T \approx eTc $; where $ \Delta (eTc) / \Delta (0) \approx \delta ^{1/2} $, and at the transition point $ \Delta (Tc) / \Delta (0) \approx \delta ^ {1/4 } $. The parameter $ \Delta (T) \gg \varepsilon_0 $ in a wide temperature range $ T> Tc $, for example, $ \Delta (eTc) / \varepsilon_0 \sim \delta ^ {- 1/2} $.
The temperature dependence of the excess heat capacity of granules $ \Delta C = [C_S (T) -C_N (T)] / C_N (Tc) $ is determined, where $ C_S $ and $ C_N $ are the heat capacities of the superconducting and normal phases. At low T, the heat capacity of the granules does not depend on their size. At $T \rightarrow Tc \; \; \Delta C (Tc) = A (1-2 / \pi) $, where $ A $ is the jump in the heat capacity of the bulk sample. In the region $ T> Tc$, $\Delta C (T) $ depends only on the parameter $z = ln (T/Tc) / \delta^{1/2} $; for $z> 1$ $\Delta C = 3/ (2\pi ^2 z^2)$. The slow logarithmic dependence $\Delta C (T)$ can be experimentally observed at $T \sim 2Tc$.