Related papers: New differential equations in the six-vertex model

A Markov process is constructed to numerically study the phase separation in the 6-vertex model with domain wall boundary conditions. It is a random walk on the graph where vertices are states and edges are elementary moves. It converges to…

The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued…

We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is…

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex SOS model, we obtain the explicit determinant expression of the partition function of the eight-vertex model with a generic non-diagonal reflecting end and…

Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…

Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent…

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general…

A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss…

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically…

In the first part of the thesis we construct models, called integrable, in which we can perform exact computations of physical quantities. We introduce several new out-of-equilibrium models that are obtained by solving, in specific cases,…

We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. The Bethe Ansatz equations have many solutions, so we can consider a wide variety of…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

In this paper we take the first step toward a classification of the approximation complexity of the six-vertex model, an object of extensive research in statistical physics. Our complexity results conform to the phase transition phenomenon…

We consider the homogeneous five-vertex model on a rectangle domain of the square lattice with so-called scalar-product boundary conditions. Peculiarity of these boundary conditions is that the configurations of the model are in an…

One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local…

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical…

Data driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data driven techniques face the dilemma of balancing search space with…

In this paper, we present a framework for Stability Analysis of Systems of Coupled Linear Partial-Differential Equations. The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet,…

We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…