Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

Shangshuai Li, Ken-ichi Maruno, Da-jun Zhang

Abstract

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schrödinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.

A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

Abstract

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schrödinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.
Paper Structure (26 sections, 9 theorems, 156 equations, 1 table)

This paper contains 26 sections, 9 theorems, 156 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary: hierarchy structures of some integrable systems
  3. The ASDYM equations
  4. The AKNS equations
  5. The DNLS equations and gauge transformations
  6. The KP hierarchy and squared eigenfunction symmetry constraint
  7. The mKP hierarchy and squared eigenfunction symmetry constraint
  8. The anti-self-dual Yang-Mills reduction
  9. Dimensional reduction condition
  10. Reduction to the matrix AKNS and GI hierarchies
  11. Reduction to the vector DNLS hierarchies
  12. The scalar case
  13. The vector case
  14. Additional comments on GL(2) gauge group
  15. Lax representation of the AKNS system
  16. ...and 11 more sections

Key Result

Theorem 1

For the GL($M,\mathbb C$) ASDYM hierarchy K-matrix-formulation that satisfies the constraint dimensional-reduction, set $x:=t_1$ as the spacial variable and define Then, $(R,Q)$ satisfy the recursive representation of the matrix AKNS hierarchy where $R$ and $Q^T$ are $m_1\times m_2$ matrix-valued functions.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 8 more