Related papers: Interplay between the Inverse Scattering Method an…
Initial boundary value problem on a half-line for the Modified KdV equation is considered with the boundary conditions equal to zero at the origin and initial condition chosen arbitrary decreasing rapidly enough and this problem is plunged…
The initial-boundary value problem (IBVP) for the nonlinear Schr\"odinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the…
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…
We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
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…
We systematically report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrodinger (DNLS) equation with both zero boundary condition (ZBC)/non-zero boundary conditions (NZBCs) at infinity and…
The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semi-discrete version of the derivative NLS system. The Jost solutions, the scattering coefficients, the bound-state dependency…
We investigate the complete integrability of soliton equations with shifted nonlocal reductions under the rapidly decreasing boundary conditions. The illustrative examples we choose are the Ablowitz-Ladik (AL) system and the…
A special class of multicomponent NLS equations, generalizing the vector NLS and related to the {\bf BD.I}-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic…
Nonlinear integrable models with two spatial and one temporal variables: Kadomtsev-Petviashvili equation and two-dimensional Toda lattice are investigated on the subject of correct formulation for boundary problem that can be solved within…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…
We implement the Numerical Unified Transform Method to solve the Nonlinear Schr\"odinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the…
We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method…
The spectral renormalization method was introduced by Ablowitz and Musslimani in 2005, [Opt. Lett. 30, pp. 2140-2142] as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems.…
The multi-component nonlinear Schrodinger equation related to C.I=Sp(2p)/U(p) and D.III=SO(2p)/U(p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use…
In this paper we present some open problems pertaining to the approximation theory involved in the solution of the important class of Nonlinear Partial Differential Equations (NPDEs) of integrable type. For this class of NPDEs, any Initial…
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…