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On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

Kelei Tian, Song Li, Ge Yi, Ying Xu, Jipeng Cheng

Abstract

In this paper, we study two generalized constrained integrable hierarchies, which are called the $c$-$k$ constrained KP and BKP hierarchies. The Fermionic picture of the $c$-$k$ constrained KP hierarchy is given. We give some solutions for the $c$-$k$ constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions $q_{i}(t)$ and adjoint eigenfunctions $r_{i}(t)$ of the $c$-$k$ constrained KP hierarchy are presented. Next, we define the $c$-$k$ constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the $c$-$k$ constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the $c$-$k$ constrained KP hierarchy.

On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

Abstract

In this paper, we study two generalized constrained integrable hierarchies, which are called the - constrained KP and BKP hierarchies. The Fermionic picture of the - constrained KP hierarchy is given. We give some solutions for the - constrained KP hierarchy by using the free Fermion operators and define its additional symmetries. Its additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on eigenfunctions and adjoint eigenfunctions of the - constrained KP hierarchy are presented. Next, we define the - constrained BKP hierarchy and obtain its bilinear identity and solutions. The algebra formed by the additional symmetric flow of the - constrained BKP hierarchy that we defined is still a subalgebra of the Virasoro algebra and it is a subalgebra of the algebra formed by the additional flows of the - constrained KP hierarchy.
Paper Structure (15 sections, 41 theorems, 200 equations)

This paper contains 15 sections, 41 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Clifford algebra
  4. The $c$-$k$ constrained KP hierarchy
  5. The $c$-$k$ constrained BKP hierarchy
  6. The Fermionic picture
  7. The case of the $c$-$k$ constrained KP hierarchy
  8. The case of the $c$-$k$ constrained BKP hierarchy
  9. Some solutions
  10. The case of the $c$-$k$ constrained KP hierarchy
  11. The case of the $c$-$k$ constrained BKP hierarchy
  12. Additional symmetries
  13. The case of the $c$-$k$ constrained KP hierarchy
  14. The case of the $c$-$k$ constrained BKP hierarchy
  15. Conclusions

Key Result

Lemma 2.1

Theorems & Definitions (61)

  • Definition 2.1: Clifford algebra $\mathcal{A}$ and $\mathcal{A}_B$
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 3.1
  • Lemma 3.1
  • ...and 51 more