相关论文: Separation of variables via integral transformatio…

We consider quantum analogs of the relativistic Toda lattices and give new $2\times 2$ $L$-operators for these models. Making use of the variable separation the spectral problem for the quantum integrals of motion is reduced to solving…

Operator splitting methods allow to split the operator describing a complex dynamical system into a sequence of simpler subsystems and treat each part independently. In the modeling of dynamical problems, systems of (possibly coupled)…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

Separation of variables (SoV) is an extremely efficient and elegant technique for analysing physical systems but its application to integrable spin chains was limited until recently to the simplest su(2) cases. In this paper we continue…

The integrable quantum models, associated to the transfer matrices of the 6-vertex reflection algebra for spin 1/2 representations, are studied in this paper. In the framework of Sklyanin's quantum separation of variables (SOV), we provide…

The paper deals with second order abstract linear partial differential equations (LPDE) over a partial differential field with two commuting differential operators. In terms of usual differential equations the main content can be presented…

Separation of variables by means of the orbit method is implemented to integrable systems on coadjoint orbits in an $\mathfrak{sl}(4)$ loop algebra. This is a development and a kind of explanation for Sklyanin's procedure of separation of…

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the…

The review is based on the author's papers since 1985 in which a new approach to the separation of variables (\SoV) has being developed. It is argued that \SoV, understood generally enough, could be the most universal tool to solve…

Separation of variables (SoV) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher…

A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called…

The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error…

Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable…

The Stochastic Partial Differential Equation (SPDE) approach, now commonly used in spatial statistics to construct Gaussian random fields, is revisited from a mechanistic perspective based on the movement of microscopic particles, thereby…

In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0…

We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in R^n. Practically all previously proposed differential transformations of LPDOs are particular cases of…

We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…

We define, in a consistent way, non-local pseudo-differential operators acting on a space of analytic functionals. These operators include the fractional derivative case. In this context we show how to solve homogeneous and inhomogeneous…

The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV)…

We implement our new Separation of Variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the…