Scattering data computation for the Zakharov-Shabat system

Abstract

Abstract. A numerical method to solve the direct scattering problem for the Zakharov-Shabat system associated to the initial value problem for the nonlinear Schrödinger equation is proposed. The method involves the numerical solution of Volterra integral systems with structured kernels and the identification of coefficients and parameters appearing in monomial-exponential sums. Numerical experiments confirm the effectiveness of the proposed technique.

Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation

“…The matrix triplet method is explicit in terms of matrix exponentials and inverse matrices, where a proof of the existence of the matrix inverses is available in the literature [4]. The solution formulas are amenable to using matrix algebra methods and can be (and have been) used to test the accuracy of numerical methods to solve integrable nonlinear evolution equations [26]. Also, explicit solution formulas obtained by means of the matrix triplet method for the nonlinear Schrödinger and modified Korteweg-de Vries equations have been verified by direct substitution, disregarding entirely the IST method to derive them [4,19].…”

A new quadrature scheme based on an Extended Lagrange Interpolation process

“…We point out that the efficient computation of integrals ( 1) is needed in many contexts, as well in numerical methods for approximating the solution of integral (systems of integral) equations [18], [20], [17], [11], [7]. For instance, the Marchenko system in [7] is connected to inverse and direct scattering problems extensively treated in [32], [10]. The Wiener-Hopf integral equations, in connection with problems in radiative transfer, and related to the solution of boundary integral equations for planar problems (see [1] and the references therein, [18], [12]), are another remarkable class of integral equation for which the quadrature rules we treat here can be useful.…”

“…Then, taking into account Theorem 3.2 and that under its assumption it follows k y ∈ L 1 (0, +∞), it is and by (10), estimate (22) follows.…”

Numerical methods for the inverse nonlinear fourier transform

“…Comparison between the exact solution(8) and the solution obtained with the NT, NCG, and IC numerical methods.…”