Discretization of the Mikhailov model

TL;DR

This paper introduces a discretization of the Mikhailov model using the Cauchy matrix approach, constructs discrete Miura transformations, and derives explicit soliton solutions, connecting discrete, semi-discrete, and continuous models.

Contribution

It presents a novel discretization of the Mikhailov model with explicit solutions and links to other integrable systems via Miura transformations.

Findings

Discrete Mikhailov model constructed with Cauchy matrices

Explicit soliton and pole solutions derived

Continuum limits recover semi-discrete and continuous models

Abstract

In this paper the Mikhailov model is discretized by means of the Cauchy matrix approach. A pair of discrete Miura transformations are constructed. The discrete Mikhailov model is a coupled system, in which one equation comes from the compatibility of the two Miura transformations, the other is transformed from the discrete negative order Ablowitz-Kaup-Newell-Segur system by using the Miura transformations. Explicit solutions, including solitons and multiple-pole solutions, are presented via two Cauchy matrix schemes respectively, namely, the Ablowitz-Kaup-Newell-Segur type and the Kadomtsev-Petviashvili type. By straight continuum limits, semi-discrete and continuous Mikhailov models together with their Cauchy matrix structures and solutions are recovered.