Supersymmetric Equation Research Articles

Invariant interaction solutions for a supersymmetric mKdV equation

The bosonic supersymmetric modified KdV (BSmKdV) system is obtained by the bosonization approach. The nonlocal symmetry for the BSmKdV equation is obtained by the truncated Painlevé method. By introducing multiple new fields, the finite symmetry transformation for the BSmKdV equation is derived by applying Lie’s first principle to the prolonged systems. The similarity reductions related to the nonlocal symmetry are studied. The interaction solutions among the solitons and other complicated waves, including Painlevé II waves and periodic cnoidal waves, are presented through the reduction theorems. The concrete soliton-cnoidal interaction solutions are illustrated in detail by using the mapping and deformation method.

Constructing new nonlinear evolution equations with supersymmetry

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. Example solutions are given for the nonlinear Schrödinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown ‘dualities’ amongst nonlinear evolution equations.

Algebraic Aspects of the Supersymmetric Minimal Surface Equation

In this paper, a supersymmetric extension of the minimal surface equation is formulated. Based on this formulation, a Lie superalgebra of infinitesimal symmetries of this equation is determined. A classification of the one-dimensional subalgebras is performed, which results in a list of 143 conjugacy classes with respect to action by the supergroup generated by the Lie superalgebra. The symmetry reduction method is used to obtain invariant solutions of the supersymmetric minimal surface equation. The classical minimal surface equation is also examined and its group-theoretical properties are compared with those of the supersymmetric version.

On geometric aspects of the supersymmetric Fokas–Gel’fand immersion formula

In this paper, we develop a new geometric characterization for the supersymmetric versions of the Fokas–Gel’fand formula for the immersion of soliton supermanifolds with two bosonic and two fermionic independent variables into Lie superalgebras. In order to do so, from a linear spectral problem of a supersymmetric integrable system using the covariant fermionic derivative, we provide a technique to obtain two additional linear spectral problems for that integrable system, one using the bosonic variable derivatives and the other using the fermionic variable derivatives. This allows us to investigate, through the first and second fundamental forms, the geometry of the ()-supermanifolds immersed in Lie superalgebras. Whenever possible, the mean and Gaussian curvatures of the supermanifolds are calculated. These theoretical considerations are applied to the supersymmetric sine-Gordon equation.

Nonlocal Symmetry Reductions for Bosonized Supersymmetric Burgers Equation**Supported by the National Natural Science Foundation of China under Grant Nos. 11675146, 11305106, 11472177, 11275129, and the Natural Science Foundation of Zhejiang Province of China under Grant No. LZ15A050001

Based on the bosonization approach, the supersymmetric Burgers (SB) system is transformed to a coupled bosonic system. By solving the bosonized SB (BSB) equation, the difficulties caused by the anticommutative fermionic field of the SB equation can be avoided. The nonlocal symmetry for the BSB equation is obtained by the truncated Painlevé method. By introducing multiple new fields, the finite symmetry transformation for the BSB equation is derived by solving the first Lie’s principle of the prolonged systems. Some group invariant solutions are obtained with the similarity reductions related by the nonlocal symmetry.

Symbolic Representation and Classification of Supersymmetric Evolutionary Equations

We extend the symbolic representation to the ring of supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Fréchet derivative, and super commutator, can be carried out in the symbolic way. Using the symbolic representation, we classify scalar λ‐homogeneous supersymmetric evolutionary equations with nonzero linear term when for arbitrary order and give a comprehensive description of all such integrable equations.

On integrability aspects of the supersymmetric sine-Gordon equation

In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-Bäcklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym–Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.

(2+1)-dimensional supersymmetric integrable equations

By means of two different approaches, we construct the (2+1)-dimensional supersymmetric integrable equations based on the super Lie algebra osp(3/2). We relax the constraint condition of homogenous space of super Lie algebra osp(3/2) in the first approach. In another one, the technique of extending the dimension of the systems is used. Furthermore for the [Formula: see text]-dimensional supersymmetric integrable equations, we also derive their Bäcklund transformations.

Painlevé analysis, nonlocal symmetry and explicit interaction solutions for supersymmetric mKdVB equation

The N=1 supersymmetric mKdVB system is transformed to a coupled bosonic system by using the bosonization approach. By a singularity structure analysis, the bosonized supersymmetric mKdVB (BSmKdVB) equation admits the Painlevé property. Starting from the standard truncated Painlevé method, the nonlocal symmetry for the BSmKdVB equation is obtained. To solve the first Lie’s principle related with the nonlocal symmetry, the nonlocal symmetry is localized to the Lie point symmetry by introducing multiple new fields. Thanks to localization processes, similarity reductions for the prolonged systems are studied by the Lie point symmetry method. The interaction solutions among solitons and other complicated waves including Painlevé II waves and periodic cnoidal waves are given through the reduction theorems. The soliton-cnoidal wave interaction solutions are explicitly given by using the mapping and deformation method. The concrete soliton-cnoidal interaction solutions are displayed both in analytical and graphical ways.

Interaction solutions for supersymmetric mKdV-B equation

The N=1 supersymmetric mKdV-B system is transformed to a system of coupled bosonic equations by using the bosonization approach. The bosonized supersymmetric mKdV-B (BSmKdV-B) equation includes the usual mKdV equation and a linear partial differential equation. The bosonization approach can thus effectively avoid difficulties caused by anticommutative fermionic fields of the supersymmetric systems. The consistent tanh expansion (CTE) method is applied to the BSmKdV-B equation. A nonauto-Bäcklund (BT) theorem is obtained by using CTE method. The interaction solutions among solitons and other complicated waves including Painlevé II waves and periodic cnoidal waves are given through a nonauto-BT theorem. The features of the soliton-cnoidal interaction solutions are investigated both in analytical and graphical ways by combining the mapping and deformation method.

Supersymmetric versions of the Fokas–Gel’fand formula for immersion

In this paper, we construct and investigate two supersymmetric versions of the Fokas–Gel’fand formula for the immersion of 2D surfaces associated with a supersymmetric integrable system. The first version involves an infinitesimal deformation of the zero-curvature condition and the linear spectral problem associated with this system. This deformation leads the surfaces to be represented in terms of a bosonic supermatrix immersed in a Lie superalgebra. The second supersymmetric version is obtained by using a fermionic parameter deformation to construct surfaces expressed in terms of a fermionic supermatrix immersed in a Lie superalgebra. For both extensions, we provide a geometrical characterization of deformed surfaces using the super Killing form as an inner product and a super moving frame formalism. The theoretical results are applied to the supersymmetric sine-Gordon equation in order to construct super soliton surfaces associated with five different symmetries. We find integrated forms of these surfaces which represent constant Gaussian curvature surfaces and nonlinear Weingarten-type surfaces.

New supersymmetric black holes in four-dimensional N = 2 supergravity

In this paper, we consider the four-dimensional N = 2 supergravity theory arising from the compactification of type IIA string theory on a Calabi–Yau manifold. We analyze the supersymmetric flow equations for static, spherically symmetric, single-centered black holes. These flow equations are solved by a set of algebraic equations involving the holomorphic sections and harmonic functions. We examine black hole configurations with D0–D4–D6 charge for which the most general solution of these algebraic equations are considered. Though the black hole solution is unique for a given value of the charges, we find new phases of the black hole solutions upon varying them.

Gazeau-Klauder coherent states for the partner potentials of the trigonometric symmetric scarf type

The supersymmetric partner Schrodinger equations for the superpotential $-A\tan\theta$ with A as non-negative integers are solved by using the Darboux factorization method on the associated Romanovski differential equation. The bound states correspond to a particle which is trapped on the bounded interval $-\pi/2\leq\theta\leq +\pi/2$ and their number is infinite. Its partner Hamiltonians are iso-spectral except for the ground state which, in turn, leads us to associate different Gazeau-Klauder coherent states for each of the partner potentials with different positive definite measures to realize the resolution of identity condition. It is shown that the time evolution does not change the form of the Gazeau-Klauder coherent states. Furthermore, it is discussed that the weight distribution functions of each of the coherent states obey the sub-Poissonian statistics. Finally, Gazeau-Klauder cat states are introduced for one of the partner potentials.

On the uniqueness of supersymmetric attractors

In this paper we discuss the uniqueness of supersymmetric attractors in four-dimensional N=2 supergravity theories coupled to n vector multiplets. We prove that for a given charge configuration the supersymmetry preserving axion free attractors are unique. We generalise the analysis to axionic attractors and state the conditions for uniqueness explicitly. We consider the example of a two-parameter model and find all solutions to the supersymmetric attractor equations and discuss their uniqueness.

Type-II super-Bäcklund transformation and integrable defects for the N = 1 super sinh-Gordon model

A new super-Bäcklund transformation for the N = 1 supersymmetric sinhGordon equation is constructed. Based on this construction we propose a type-II integrable defect for the supersymmetric sinh-Gordon model consistent with this new transformation through the Lagrangian formalism. Explicit expressions for the modified conserved energy, momentum and supercharges are also computed. In addition, we show for the model that the type-II defect can also been regarded as a pair of fused defects of a previously introduced type. The explicit derivation of the associated defect matrices is also presented as a necessary condition for the integrability of the model.

On a supersymmetric nonlinear integrable equation in (2+1) dimensions

A supersymmetric integrable equation in (2+1) dimensions is constructed by means of the approach of the homogenous space of the super Lie algebra, where the super Lie algebra osp(3/2) is considered. For this (2+1) dimensional integrable equation, we also derive its Bäcklund transformation.

Painlevé analysis and exact solutions of bosonized N = 1 supersymmetric Burgers equation

Abstract Based on the bosonization approach, the N = 1 supersymmetric Burgers (SB) system is transformed to a coupled pure bosonic system. The Painlevé property and the Bäcklund transformations (BT) of the bosonized SB (BSB) system are obtained through standard singularity analysis. Explicit solutions such as the muti-solitarywaves and error functionwaves are provided for the BT. The exact solutions of the BSB system are obtained from the generalized tanh expansion method.

Behaviors of N=1 supersymmetric Euler derivatives and Hamiltonian operators under general superconformal transformations

The changing rule of super Hamiltonian operators under general superconformal transformations is established by means of investigating behavior of supersymmetric Euler derivatives under the same kind of changes of variables. In the two particular yet frequent cases such as supersymmetric Miura-type transformations and reciprocal transformations, the results are detailed and applied to construct bi-Hamiltonian structures of some supersymmetric evolution equations. As an interesting example, one of supersymmetric Harry Dym equations is shown to be a bi-Hamiltonian system through its reciprocal link to the classical Harry Dym equation.

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose–Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly \({\mathcal{N} = (1, 0)}\) and \({\mathcal{N} = (2, 0)}\) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.

Solving Supersymmetric Hirota–Satsuma Equation by a Direct Bosonization Approach

The N = 1 supersymmetric Hirota-Satsuma equation is transformed into systems of coupled bosonic equations by expanding fermionic superfield in terms of 2, 3, and n(n ≥ 4) Grassmann parameters, respectively. Taking advantage of the resulting coupled bosonic systems being linear in the undetermined variables, the supersymmetric Hirota-Satsuma equation is solved out by using the mapping and deformation method. Besides, the richness of the localized excitations of the supersymmetric integrable system is discovered.