Integrable discretization of the modified KdV equation and applications

Abstract

An integrable discretization of the modified Korteweg-de Vries (mKdV) equation is considered. The mKdV algorithm is designed in terms of the resulting discrete mKdV equation. It is shown that a class of algebraic equations can be solved numerically in terms of the algorithm.

Schur flows and orthogonal polynomials on the unit circle

“…In [13,14] the Bessel modification of measures appeared and a part of our main result which concerns (3) ⇒ (1) is proved. In a recent paper [15] the author deals with the Poisson structure and Lax pairs for the Ablowits-Ladik systems closely related to the Schur flows.…”

“…x f. In [8] the authors deal with finite real Schur flows and suggest two more distinct Lax equations based on the Hessenberg matrix representation of the multiplication operator (see also [3]). In [13,14] the Bessel modification of measures appeared and a part of our main result which concerns (3) ⇒ (1) is proved. In a recent paper [15] the author deals with the Poisson structure and Lax pairs for the Ablowits-Ladik systems closely related to the Schur flows.…”

Breather lattice and its stabilization for the modified Korteweg–de Vries equation

“…In the numerical simulation of the MKdV problem, we have found that the direct center-difference discretization does a poor job in adequately following the MKdV equation (and in conserving the corresponding integrals of motion). While one can also use the integrable scheme of Ablowitz-Ladik (see e.g., [2,18,19] and references therein), we have followed a different path here in spatially discretizing the partial differential equation and following the integrable discretization scheme of [20] for KdV and adapting it to the case of the MKdV. In particular, the spatially discrete version of our equation reads (with lattice spacing h):…”

A new approach to numerical algorithms in terms of integrable systems

“…The latter algorithm is formulated by discretizing the Schur flow which is an integrable system defined by a one-parameter deformation equation of the Szegö OPs. The discrete Schur flow has applications not only to a continued fraction expansion but also to a computation of zeros of polynomials [25]. The discrete Schur flow will see applications to various problems in applied science.…”