Two new supersymmetric equations of Harry Dym type and their supersymmetric reciprocal transformations

A new N=1$N=1$ supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed.     

A supersymmetric Sawada-Kotera equation

A new supersymmetric equation is proposed for the Sawada-Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator.     

Odd Hamiltonian structure for supersymmetric Sawada-Kotera equation

We study the supersymmetric N=1 hierarchy connected with the Lax operator of the supersymmetric Sawada-Kotera equation. This operator produces the physical equations as well as the exotic equations with odd time. The odd Bi-Hamiltonian structure for the N=1 supersymmetric Sawada-Kotera equation is defined. The product of the symplectic and implectic Hamiltonian operator gives us the recursion operator. In that way we prove the integrability of the supersymmetric Sawada-Kotera equation in the sense that it has the Bi-Hamiltonian structure. The so-called “quadratic” Hamiltonian operator of even order generates the exotic equations while the “cubic” odd Hamiltonian operator generates the physical equations.     

Some remarks on a model of supersymmetric quantum mechanics

We develop a recently proposed model within supersymmetric quantum mechanics that puts a group structure on the creation and annihilation operators. We apply the scheme to a variety of quantum mechanical problems and work out a two-term energy recursion equation when the overall group structure isU(1, 1).     

谐振子薛定谔方程的超对称解法

通过构造哈密顿量与谐振子系统哈密顿量对易的超对称系统,量子谐振子的性质就可以通过对超对称系统的研究来得到.利用超对称系统的性质,在没有用到厄米多项式的情况下,给出了谐振子本征函数中展开系数间的递推关系,由递推关系可以直接得到本征函数.此方法下得到的归一化本征函数与用厄米多项式表达的本征函数完全相同,并且本征函数的宇称可以明显的显示出来.     

Gelfand-Dikii analysis forN=2 supersymmetric KdV equations

We generalize the resolvent approach of Gelfand and Dikii to the KdV equation to study theN=2 supersymmetric KdV equations of Laberge and Mathieu. For the associated Lax operators, we study the coincidence limits of the resolvent kernel and its derivatives, and obtain differential equations which they satisfy. These allow us to obtain recursion relations for the analogues of the Gelfand-Dikii polynomials and to obtain a proof of Hamiltonian integrability of the supersymmetric KdV equations. We are also able to write the Lax equations for the corresponding hierarchies in terms of these polynomials.Address after January 1, 1993: Department of Physics, University of Western Australia, Nedlands, Australia 6009     

An introduction to on-shell recursion relations

This article provides an introduction to on-shell recursion relations for calculations of tree-level amplitudes. Starting with the basics, such as spinor notations and color decompositions, we expose analytic properties of gauge-boson amplitudes, BCFW-deformations, the large z-behavior of amplitudes, and on-shell recursion relations of gluons. We discuss further developments of on-shell recursion relations, including generalization to other quantum field theories, supersymmetric theories in particular, recursion relations for off-shell currents, recursion relation with nonzero boundary contributions, bonus relations, relations for rational parts of one-loop amplitudes, recursion relations in 3D and a proof of CSW rules. Finally, we present samples of applications, including solutions of split helicity amplitudes and of N = 4 SYM theories, consequences of consistent conditions under recursion relation, Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations for color-ordered gluon tree amplitudes, Kawai-Lewellen-Tye (KLT) relations.     

The N = 2 supersymmetric Toda lattice hierarchy

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax-pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N = 2 supersymmetric generalized Toda lattice hierarchies.     

The even and odd supersymmetric Hunter-Saxton and Liouville equations

It is shown that two different supersymmetric extensions of the Harry Dym equation lead to two different negative hierarchies of the supersymmetric integrable equations. While the first one yields the known even supersymmetric Hunter-Saxton equation, the second one is a new odd supersymmetric Hunter-Saxton equation. It is further proved that these two supersymmetric extensions of the Hunter-Saxton equation are reciprocally transformed to two different supersymmetric extensions of the Liouville equation.     

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the super-symmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.     

Dynamics of loops in superspace

The loop equation, analogous to the Makeenko-Migdal equation, is derived for the supersymmetric generalization of the Wilson loop for the supersymmetric (N = 1) gauge theory.     

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlev\'{e test and we also calculate its one- and two-soliton solutions.     

A new recursion operator for Adler?s equation in the Viallet form

For Adler?s equation in the Viallet form and Yamilov?s discretisation of the Krichever-Novikov equation we present new recursion and Hamiltonian operators. This new recursion operator and the recursion operator found in [A.V. Mikhailov, et al., Theor. Math. Phys. 167 (2011) 421, arXiv:1004.5346] satisfy the spectral curve associated with the equation.     

A new supersymmetric classical Boussinesq equation

In this paper, we obtain a supersymmetric generalization for the classical Boussinesq equation. We show that the supersymmetric equation system passes the Painlev\'{e test and we also calculate its one- and two-soliton solutions.     

On the theory of recursion operator

The general structure and properties of recursion operators for Hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered. Weak and strong recursion operators are introduced. The conditions which determine weak and strong recursion operators are found.In the theory of nonlinear waves a method for the calculation of the recursion operator, which is based on the use of expansion into a power series over the fields and the momentum representation, is proposed. Within the framework of this method a recursion operator is easily calculated via the Hamiltonian of a given equation. It is shown that only the one-dimensional nonlinear evolution equations can posses a regular recursion operator. In particular, the Kadomtsev-Petviashvili equation has no regular recursion operator.     

Construction of Recursion Formulas for Lax Pairs of (2+1)-Dimensional Equation:Modified Generalized Dispersive Long Wave Equation

In this article, we study the Lax pairs of (2+1)-dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.     

Generalized recursion relations for correlators in the gauge-gravity correspondence

We show that a generalization of the Britto-Cachazo-Feng-Witten recursion relations gives a new and efficient method of computing correlation functions of the stress tensor or conserved currents in conformal field theories with an (d+1)-dimensional anti-de Sitter space dual, for d≥4, in the limit where the bulk theory is approximated by tree-level Yang-Mills theory or gravity. In supersymmetric theories, additional correlators of operators that live in the same multiplet as a conserved current or stress tensor can be computed by these means.     

Nonlinear superposition formula for SUSY SG/MKdV equations revisited

The derivation of nonlinear superposition formula is reexamined for the N=1 supersymmetric sinh-Gordon equation. It is shown that this formula results from the spatial part of Bäcklund transformation only and therefore can be associated with the supersymmetric MKdV equation.     

Seiberg-Witten/Whitham Equations and Instanton Corrections in N=2 Supersymmetric Yang-Mills Theory

We obtain the instanton correction recursion relations for the low energy effective prepotential in pure N=2 SU(n) supersymmetric Yang-Mills gauge theory from Whitham hierarchy and Seiberg-Witten/Whitham equations. These formulae provide us a powerful tool to calculate arbitrary order instanton corrections coefficients from the perturbative contributions of the effective prepotential in Seiberg-Witten gauge theory. We apply this idea to evaluate one- and two- order instanton corrections coefficients explicitly in SU(n) case in detail through the dynamical scale parameter expressed in terms of Riemann's theta-function.     

Transformation of recursion relations for generalized random walks

It is shown that recursion relation for the generalized random walks (GRW) or correlated random walks can be directly transformed into the recursion relation for the usual random walks. The recursion relation for the GRW is expressed by a non-linear difference equation. To transform the non-linear difference equation, the Hopf-Cole transformation is modified and expressed in a discrete form. Formal solution of the GRW is obtained in an integral representation.