scholarly journals LAX PAIRS FOR N=2,3 SUPERSYMMETRIC KdV EQUATIONS AND THEIR EXTENSIONS

1998 ◽  
Vol 13 (18) ◽  
pp. 1435-1443 ◽  
Author(s):  
S. KRIVONOS ◽  
A. PASHNEV ◽  
Z. POPOWICZ
Keyword(s):  
Infinite Family ◽  
Lax Pairs ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Lax Operator

We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin-1 and fermionic spin-1/2, is discussed.

scholarly journals Negative Times of the Davey-Stewartson Integrable Hierarchy

2021 ◽  
Author(s):  
Andrei K. Pogrebkov ◽  
Keyword(s):  
Integrable Hierarchy ◽  
Lax Pairs ◽  
Negative Numbers ◽  
Integrable Equations ◽  
Lax Operator

We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

scholarly journals Kadomtsev–Petviashvili Hierarchy: Negative Times

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1988
Author(s):  
Andrei K. Pogrebkov
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Linear Differential Equations ◽  
Lax Pairs ◽  
Negative Numbers ◽  
Infinite Hierarchy ◽  
Petviashvili Equation ◽  
Lax Operator ◽  
Leading Term ◽  
Linear Differential

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.

scholarly journals THE LAX OPERATOR APPROACH FOR THE VIRASORO AND THE W-CONSTRAINTS IN THE GENERALIZED KdV HIERARCHY

1993 ◽  
Vol 08 (20) ◽  
pp. 3457-3478 ◽  
Author(s):  
SUDHAKAR PANDA ◽  
SHIBAJI ROY
Keyword(s):  
Symmetry Operator ◽  
String Equation ◽  
Kp Hierarchy ◽  
Operator Approach ◽  
Additional Symmetry ◽  
Kdv Hierarchy ◽  
Lax Operator

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.

scholarly journals Genus expansion of open free energy in 2d topological gravity

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Kazumi Okuyama ◽  
Kazuhiro Sakai
Keyword(s):  
Free Energy ◽  
Riemann Surfaces ◽  
Two Dimensions ◽  
Intersection Theory ◽  
Genus Zero ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Higher Genus ◽  
Topological Gravity ◽  
Genus Expansion

Abstract We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak’s differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak’s equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.

scholarly journals Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Binlu Feng ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Loop Algebras ◽  
Negative Order ◽  
Integrable Couplings ◽  
De Vries ◽  
Integrable Coupling

Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.

scholarly journals SELF-DUALITY AND THE SUPERSYMMETRIC KdV HIERARCHY

1993 ◽  
Vol 08 (15) ◽  
pp. 1399-1406 ◽  
Author(s):  
ASHOK DAS ◽  
C. A. P. GALVÃO
Keyword(s):  
Kdv Equation ◽  
Curvature Condition ◽  
Four Dimensions ◽  
Yang Mills ◽  
Kdv Hierarchy ◽  
Self Duality ◽  
Kdv Equations ◽  
Duality Condition ◽  
Graded Lie Algebra ◽  
Supersymmetric Kdv Equation

We show how the supersymmetric KdV equation can be obtained from the self-duality condition on Yang-Mills fields in four dimensions associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of SUSY KdV equations as well as the s-KdV equations from such a condition. We formulate the SUSY KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self-duality condition in four dimensions can also lead to these equations.

scholarly journals Two Kinds of Darboux-Bäcklund Transformations for theq-Deformed KdV Hierarchy with Self-Consistent Sources

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Hongxia Wu ◽  
Liangjuan Gao ◽  
Jingxin Liu ◽  
Yunbo Zeng
Keyword(s):  
Soliton Solution ◽  
Pseudodifferential Operators ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Backlund Transformations ◽  
Self Consistent

Two kinds of Darboux-Bäcklund transformations (DBTs) are constructed for theq-deformedNth KdV hierarchy with self-consistent sources (q-NKdVHSCS) by using theq-deformed pseudodifferential operators. Note that one of the DBTs provides a nonauto Bäcklund transformation for twoq-deformedNth KdV equations with self-consistent sources (q-NKdVESCS) with different degree. In addition, the soliton solution to the first nontrivial equation ofq-KdVHSCS is also obtained.

scholarly journals On tau-functions for the KdV hierarchy

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Boris Dubrovin ◽  
Di Yang ◽  
Don Zagier
Keyword(s):  
Tau Functions ◽  
Kdv Hierarchy
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