Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a matrix constrained CKP hierarchy

Song Li, Kelei Tian, Zhiwei Wu

TL;DR

The paper develops a matrix formulation for the constrained CKP hierarchy via splitting theory, proving its equivalence with the scalar constrained CKP system and linking it to the $\hat{A}_{2n}^{(2)}$-KdV and GD hierarchies. It provides a comprehensive toolkit, including Darboux and scaling transformations, tau-function construction, and explicit Virasoro vector-field actions on $\ln \tau_f$, enabling systematic generation and reduction of solutions. Concrete $m=n=1$ examples illustrate soliton-like solutions and invariance properties under Darboux dressing. Altogether, the work advances the geometric and algebraic understanding of CKP-type hierarchies and their additional symmetries, with potential for broad reductions and applications in soliton theory.

Abstract

The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the $\hat{A}_{2n}^{(2)}$-KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformation and tau functions $\ln τ_f$ for this constrained hierarchy are studied. Moreover, we present formulas for the Virasoro vector fields on $\ln τ_f$ for the $\hat{A}_{2 n}^{(2)}$-KdV hierarchy.

On a matrix constrained CKP hierarchy

TL;DR

The paper develops a matrix formulation for the constrained CKP hierarchy via splitting theory, proving its equivalence with the scalar constrained CKP system and linking it to the -KdV and GD hierarchies. It provides a comprehensive toolkit, including Darboux and scaling transformations, tau-function construction, and explicit Virasoro vector-field actions on , enabling systematic generation and reduction of solutions. Concrete examples illustrate soliton-like solutions and invariance properties under Darboux dressing. Altogether, the work advances the geometric and algebraic understanding of CKP-type hierarchies and their additional symmetries, with potential for broad reductions and applications in soliton theory.

Abstract

The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the -KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformation and tau functions for this constrained hierarchy are studied. Moreover, we present formulas for the Virasoro vector fields on for the -KdV hierarchy.

Paper Structure

This paper contains 6 sections, 27 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Splitting Theory for the Matrix Constrained CKP Hierarchy
  3. Darboux Transformations and Scaling Transformations
  4. Tau Functions
  5. Virasoro action for the $\hat{A}_{2 n}^{(2)}$-KdV hierarchy
  6. Conclusions and Dicsussions

Key Result

Proposition 2.3

Let Let $B^{+}$ and $N^{+}$ be Lie groups associated with the Lie algebras $\mathcal{B}^{+}$ and $\mathcal{N}^{+}$, respectively. Let $N_{n}^{+}$ be the upper triangular matrix of order $n$ with 1 on the diagonal. Given $q \in C^{\infty}\left(\mathbb{R},Y_{m,n}\right)$, there exist a unique $\Delta \in Moreover, We write $u$ as $\Delta\ast q$.

Theorems & Definitions (50)

  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 2.8
  • ...and 40 more