# Analysis of Malmquist-Takenaka-Christov rational approximations with applications to the nonlinear Benjamin equation

## TL;DR

This paper investigates the approximation capabilities of Malmquist-Takenaka-Christov systems, demonstrating their rapid convergence and stability for solving nonlinear equations like the Benjamin equation, with practical efficiency comparable to spectral methods.

## Contribution

It introduces a new application of MTC approximations to nonlinear PDEs and provides rigorous convergence and stability analysis for the Benjamin equation.

## Key findings

- MTC approximations converge rapidly for functions with mild asymptotic conditions.
- The collocation MTC scheme is stable and convergent for the nonlinear Benjamin equation.
- The method's efficiency is comparable to established spectral and hybrid spectral methods.

## Abstract

In the paper, we study approximation properties of the Malmquist-Takenaka-Christov (MTC) system. We show that the discrete MTC approximations converge rapidly under mild restrictions on functions asymptotic at infinity. This makes them particularly suitable for solving semi- and quasi-linear problems containing Fourier multipliers, whose symbols are not smooth at the origin. As a concrete application, we provide rigorous convergence and stability analyses of a collocation MTC scheme for solving the nonlinear Benjamin equation. We demonstrate that the method converges rapidly and admits an efficient implementation, comparable to the best spectral Fourier and hybrid spectral Fourier/finite-element methods described in the literature.