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Constraints of the D$Δ$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies

Jin Liu, Da-jun Zhang

Abstract

The paper investigates three eigenfunction constraints of two (2+1)-dimensional differential-difference integrable systems. First, we revisit the known squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$Δ$KP) hierarchy, which gives rise to a semi-discrete Ablowitz-Kaup-Newell-Segur hierarchy. Second, we introduce a linear eigenfunction constraint for the D$Δ$KP system and obtain a combined semi-discrete Burgers (sdBurgers) hierarchy. In the third one, we consider another linear eigenfunction constraint for the modified D$Δ$KP system and obtain the same combined sdBurgers hierarchy. All these constraint results are proved by using recursive algebraic structures of the involved integrable hierarchies generated by their master symmetries.

Constraints of the D$Δ$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies

Abstract

The paper investigates three eigenfunction constraints of two (2+1)-dimensional differential-difference integrable systems. First, we revisit the known squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (DKP) hierarchy, which gives rise to a semi-discrete Ablowitz-Kaup-Newell-Segur hierarchy. Second, we introduce a linear eigenfunction constraint for the DKP system and obtain a combined semi-discrete Burgers (sdBurgers) hierarchy. In the third one, we consider another linear eigenfunction constraint for the modified DKP system and obtain the same combined sdBurgers hierarchy. All these constraint results are proved by using recursive algebraic structures of the involved integrable hierarchies generated by their master symmetries.
Paper Structure (18 sections, 23 theorems, 209 equations)

This paper contains 18 sections, 23 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. D$\Delta$KP hierarchy and algebraic structures
  3. Preliminary
  4. Isospectral and nonisopectral D$\Delta$KP flows
  5. Master symmetry, Lie algebraic structure and recursive structure
  6. Constraint of D$\Delta$KP system to sdAKNS hierarchy
  7. The sdAKNS hierarchy
  8. Squared eigenfunction symmetry, constraint and sdAKNS hierarchy
  9. Main result
  10. Proof of Theorem \ref{['Th-3-1']}
  11. The sdBurgers hierarchies and Lie algebraic structure
  12. Isospectral and nonisopectral sdBurgers hierarchies
  13. Algebraic structure and master symmetry
  14. Constraint of D$\Delta$KP system to sdBurgers hierarchy
  15. Constraint of D$\Delta$mKP system to sdBurgers hierarchy
  16. ...and 3 more sections

Key Result

Proposition 2.1

For the pseudo-difference operator $L$ given in L-dKP and any difference operator $\mathcal{A}_m$ with the form: $\mathcal{A}_m=\sum_{j=0}^m a_j \Delta^{m-j}$, if they satisfy then $\mathcal{A}_m=0$.

Theorems & Definitions (33)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 23 more