scholarly journals SUPERSYMMETRIC TWO-BOSON EQUATION, ITS REDUCTIONS AND THE NONSTANDARD SUPERSYMMETRIC KP HIERARCHY

1995 ◽  
Vol 10 (32) ◽  
pp. 4563-4599 ◽  
Author(s):  
J.C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Integrable Models ◽  
Lax Representation ◽  
Kp Equation ◽  
Kp Hierarchy ◽  
Hamiltonian Structures ◽  
Conserved Charges ◽  
Nonlinear Schrödinger ◽  
Field Redefinition

In this paper, we review various properties of the supersymmetric Two-Boson (sTB) system. We discuss the equation and its nonstandard Lax representation. We construct the local conserved charges as well as the Hamiltonian structures of the system, and show how this system leads to various other known supersymmetric integrable models under appropriate field redefinition. We discuss the sTB and the supersymmetric nonlinear Schrödinger (sNLS) equations as constrained, nonstandard supersymmetric Kadomtsev-Petviashvili (sKP) systems and note that the nonstandard sKP systems naturally unify all the KP and mKP flows while leading to a new integrable supersymmetrization of the KP equation. We construct the nonlocal conserved charges associated with the sTB system and show that the algebra of charges corresponds to a graded, cubic algebra. Also, we note that the sTB system has a hidden supersymmetry making it an N=2 extended supersymmetric system.

scholarly journals A NONSTANDARD SUPERSYMMETRIC KP HIERARCHY

1995 ◽  
Vol 07 (08) ◽  
pp. 1181-1194 ◽  
Author(s):  
J.C. BRUNELLI ◽  
A. DAS
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mkdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Kp Equation ◽  
Kp Hierarchy ◽  
Lax Equation ◽  
Conserved Charges ◽  
Nonlinear Schrödinger

We show that the supersymmetric nonlinear Schrödinger equation can be written as a constrained super KP flow in a nonstandard representation of the Lax equation. We construct the conserved charges and show that this system reduces to the super mKdV equation with appropriate identifications. We construct various flows generated by the general nonstandard super Lax equation and show that they contain both the KP and mKP flows in the bosonic limits. This nonstandard supersymmetric KP hierarchy allows us to construct a new super KP equation which is nonlocal.

scholarly journals THE sAKNS HIERARCHY

1998 ◽  
Vol 13 (15) ◽  
pp. 1185-1199 ◽  
Author(s):  
HENRIK ARATYN ◽  
ASHOK DAS
Keyword(s):  
Jacobi Identity ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Lax Representation ◽  
System Of Equations ◽  
Open Problems ◽  
Hamiltonian Structures ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Lax Operator

We study, systematically, the properties of the supersymmetric AKNS (sAKNS) hierarchy. In particular, we discuss the Lax representation in terms of a bosonic Lax operator and some special features of the equations and construct the bosonic local charges as well as the fermionic nonlocal charges associated with the system starting from the Lax operator. We obtain the Hamiltonian structures of the system and check the Jacobi identity through the method of prolongation. We also show that this hierarchy of equations can equivalently be described in terms of a fermionic Lax operator. We obtain the zero curvature formulation as well as the conserved charges of the system starting from this fermionic Lax operator which suggests a connection between the two. Finally, starting from the fermionic description of the system, we construct the soliton solutions for this system of equations through Darboux–Bäcklund transformations and describe some open problems.

scholarly journals BI-HAMILTONIAN STRUCTURE OF THE SUPERSYMMETRIC NONLINEAR SCHRÖDINGER EQUATION

1995 ◽  
Vol 10 (27) ◽  
pp. 2019-2028 ◽  
Author(s):  
J.C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Hamiltonian Reduction ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Supersymmetric Kdv Equation

We show that the supersymmetric nonlinear Schrödinger equation is a bi-Hamiltonian integrable system. We obtain the two Hamiltonian structures of the theory from the ones of the supersymmetric two-boson hierarchy through a field redefinition. We also show how the two Hamiltonian structures of the supersymmetric KdV equation can also be derived from a Hamiltonian reduction of the supersymmetric two-boson hierarchy.

scholarly journals Inner products in integrable Richardson-Gaudin models

2017 ◽  
Vol 3 (4) ◽  
Author(s):  
Pieter W. Claeys ◽  
Dimitri Van Neck ◽  
Stijn De Baerdemacker
Keyword(s):  
Domain Wall ◽  
Integrable Models ◽  
Partition Functions ◽  
Dual Representation ◽  
Determinant Formula ◽  
Quadratic Equations ◽  
Conserved Charges ◽  
Inner Products ◽  
Gaudin Models ◽  
Bethe Equations

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

scholarly journals The Hamiltonian structures of the super‐KP hierarchy associated with an even parity super‐Lax operator

1995 ◽  
Vol 36 (1) ◽  
pp. 258-267 ◽  
Author(s):  
J. Barcelos‐Neto ◽  
Sasanka Ghosh ◽  
Shibaji Roy
Keyword(s):  
Kp Hierarchy ◽  
Hamiltonian Structures ◽  
Lax Operator

The bi-hamiltonian structures of the Manin-Radul super KP hierarchy

1992 ◽  
Vol 291 (1-2) ◽  
pp. 77-84 ◽  
Author(s):  
Sudhakar Panda ◽  
Shibaji Roy
Keyword(s):  
Kp Hierarchy ◽  
Hamiltonian Structures
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