Abstract:
The Laplace sequence of the discrete conjugate nets is constructed. The invariants of the nets satisfy, in full analogy to the continuous case, the system of difference equations equivalent to the discrete version of the generalized Toda equation.

Abstract:
We propose the algebro-geometric mothod of construction of solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite field version of the discrete KdV equation. We write down formulas which allow to construct multisoliton solutions of the equations starting from vacuum wave functions on arbitrary non-singular curve.

Abstract:
We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate in detail the procedure on example of a hyperelliptic curve.

Abstract:
We investigate the jump motion among potential energy minima of a Lennard-Jones model glass former by extensive computer simulation. From the time series of minima energies, it becomes clear that the energy landscape is organized in superstructures, called metabasins. We show that diffusion can be pictured as a random walk among metabasins, and that the whole temperature dependence resides in the distribution of waiting times. The waiting time distribution exhibits algebraic decays: $\tau^{-1/2}$ for very short times and $\tau^{-\alpha}$ for longer times, where $\alpha\approx2$ near $T_c$. We demonstrate that solely the waiting times in the very stable basins account for the temperature dependence of the diffusion constant.

Abstract:
The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface approach. We exploit the Bryant-type representation of conformally parametrized surfaces in the hyperbolic space $H^3(\lambda)$ of curvature $-\lambda^2$, which can be interpreted as a 2 by 2 linear problem involving the spectral parameter $\lambda$. In the particular case of constant mean curvature-$\lambda$ surfaces a special limiting procedure $(\lambda\rightarrow 0)$, different from that of Umehara and Yamada [33], allows us to recover the Enneper-Weierstrass representation. Applying such a limiting procedure to the previously known cases, we obtain Sym-type formulas. Finally we exploit the relation between the Bryant representation of constant mean curvature-$\lambda$ surfaces and second-order linear ordinary differential equations. We illustrate this approach by the example of the error function equation.

Abstract:
The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial--boundary data which define the lattice uniquely.

Abstract:
It is analyzed whether the potential energy landscape of a glass-forming system can be effectively mapped on a random model which is described in statistical terms. For this purpose we generalize the simple trap model of Bouchaud and coworkers by dividing the total system into M weakly interacting identical subsystems, each being described in terms of a trap model. The distribution of traps in this extended trap model (ETM) is fully determined by the thermodynamics of the glass-former. The dynamics is described by two adjustable parameters, one characterizing the common energy level of the barriers, the other the strength of the interaction. The comparison is performed for the standard binary mixture Lennard-Jones system with 65 particles. The metabasins, identified in our previous work, are chosen as traps. Comparing molecular dynamics simulations of the Lennard-Jones system with Monte Carlo calculations of the ETM allows one to determine the adjustable parameters. Analysis of the first moment of the waiting distribution yields an optimum agreement when choosing M=3 subsystems. Comparison with the second moment of the waiting time distribution, reflecting dynamic heterogeneities, indicates that the sizes of the subsystems may fluctuate.

Abstract:
An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.

Abstract:
With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrodinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.