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Discretization of the Mikhailov model

Song-lin Zhao, Xiao-gang Mu, Da-jun Zhang

TL;DR

This work addresses discretizing the relativistically invariant Mikhailov model by deploying the Cauchy matrix approach. It constructs two discrete Miura transformations and builds the discrete Mikhailov model from two complementary Cauchy matrix schemes, AKNS-type and KP-type, enabling explicit soliton and multiple-pole solutions. The discrete system manifests as a coupled quadrilateral model for variables $(\mu,\nu)$, with a parallel closed form via an alternative variable pair; straight continuum limits recover both the semi-discrete and continuous Mikhailov models along with their Cauchy matrix structures. The methods connect to the negative-order KN hierarchy (AKNS$(-1)$) and self-dual Yang–Mills reductions, suggesting avenues for discretizing related derivative NLS-type systems and exploring Lax integrability and multidimensional consistency in future work.

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.

Discretization of the Mikhailov model

TL;DR

This work addresses discretizing the relativistically invariant Mikhailov model by deploying the Cauchy matrix approach. It constructs two discrete Miura transformations and builds the discrete Mikhailov model from two complementary Cauchy matrix schemes, AKNS-type and KP-type, enabling explicit soliton and multiple-pole solutions. The discrete system manifests as a coupled quadrilateral model for variables , with a parallel closed form via an alternative variable pair; straight continuum limits recover both the semi-discrete and continuous Mikhailov models along with their Cauchy matrix structures. The methods connect to the negative-order KN hierarchy (AKNS) and self-dual Yang–Mills reductions, suggesting avenues for discretizing related derivative NLS-type systems and exploring Lax integrability and multidimensional consistency in future work.

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.
Paper Structure (16 sections, 3 theorems, 114 equations)

This paper contains 16 sections, 3 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Discretization of the Mikhailov model
  3. Cauchy matrix schemes: KP-type and AKNS-type
  4. Cauchy matrix approach to the discrete Mikhailov model
  5. Dynamics of master functions
  6. Discrete AKNS$(-1)$ system
  7. Miura transformations
  8. Discrete Mikhailov model
  9. Explicit solutions
  10. The AKNS-type scheme
  11. The KP-type scheme
  12. Continuum limits
  13. Semi-discrete Mikhailov model
  14. Continuous Mikhailov model
  15. Concluding remarks
  16. ...and 1 more sections

Key Result

Proposition 2.1

HSZ-TMP-2025LQZ-PD-2023 For the master functions $\{\boldsymbol{S}^{(i,j)}\}$ defined in Sij with $\boldsymbol{M}, \boldsymbol{K}, \boldsymbol{L}, \boldsymbol{r}$ and $\boldsymbol{s}$ satisfying the Sylvester equation SE, the following recurrence relations hold: provided that $\boldsymbol{K}, \boldsymbol{L}$ and $\boldsymbol{C}$ satisfy

Theorems & Definitions (6)

  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2