Related papers: Initial-boundary value problems associated with th…
We show how to solve initial-boundary value problems for integrable nonlinear differential-difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The…
This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz-Ladik system posed on a…
We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for…
We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems…
We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve $3 \times 3$ matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical…
We obtain solution representation formulas for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first…
We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation…
We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically…
In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with $3 \times 3$ Lax pairs. The solution can be expressed in terms of…
In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system…
In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential…
It is known that the initial-boundary value problem for certain integrable partial differential equations (PDEs) on the half-line with integrable boundary conditions can be mapped to a special case of the Inverse Scattering Method (ISM) on…
We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the…
By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ matrices.…
We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS…
We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With…
For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schr\"odinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik system…