Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.     

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.     

A Super mKdV Equation: Bosonization,Painlevé Property and Exact Solutions

The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.     

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.     

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.     

Properties of supersymmetric integrable systems of KP type

The recently proposed supersymmetric extensions of reduced Kadomtsev-Petviashvili (KP) integrable hierarchies in N = 1, 2 superspace are shown to contain in the purely bosonic limit new types of ordinary non-supersymmetric integrable systems. The latter are coupled systems of several multi-component non-linear Schr?dinger-like hierarchies whose basic nonlinear evolution equations contain additional quintic and higher-derivative nonlinear terms. Also, we obtain the N = 2 supersymmetric extension of Toda chain model as Darboux-B?cklund orbit of the simplest reduced N = 2 super-KP hierarchy and find its explicit solution. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: nissimov@inrne.bas.bg RID="b" ID="b"e-mail: svetlana@inrne.bas.bg     

Conformal Invariant Asymptotic Expansion Approach for Solving (3+1)-Dimensional JM Equation

The (3+1)-dimensional Jimbo-Miwa (JM) equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new (3+1)-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.     

Non-local Conservation Laws and Flow Equations¶ for Supersymmetric Integrable Hierarchies

An infinite series of Grassmann-odd and Grassmann-even flow equations is defined for a class of supersymmetric integrable hierarchies associated with loop superalgebras. All these flows commute with the mutually commuting bosonic ones originally considered to define these hierarchies and, hence, provide extra fermionic and bosonic symmetries that include the built-in N=1 supersymmetry transformation. The corresponding non-local conserved quantities are also constructed. As an example, the particular case of the principal supersymmetric hierarchies associated with the affine superalgebras with a fermionic simple root system is discussed in detail.     

Quantization of superparametrized monopoles

Classical finite-energy solutions of the SU(2) Yang-Mills-Higgs system in four-dimensional space-time are embedded in the supersymmetric extension of the theory. Finite supertranslations are constructed and are used to obtain a family of solutions to the supersymmetric field equations, parametrized by fermionic Majorana spinor parameters. The quantum theory around arbitrary classical solutions, parametrized by arbitrary bosonic (global and local) as well as fermionic (global) parameters, is constructed and discussed.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

Bilinearization and soliton solutions of N=1 supersymmetric coupled dispersionless integrable system

An N=1 supersymmetric generalization of coupled dispersionless (SUSY-CD) integrable system has been proposed by writing its superfield Lax representation. It has been shown that under a suitable variable transformation, the SUSY-CD integrable system is equivalent to N=1 supersymmetric sine-Gordon equation. A superfield bilinear form of SUSY-CD integrable system has been proposed by using super Hirota operator. Explicit expressions of superfield soliton solutions of SUSY-CD integrable system have been obtained by using the Hirota method.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

Susy Generalized Harry Dym Equation and Its Bosonic Reduction

We construct the two-component supersymmetric generalized Harry Dym equation which is integrable and study various properties of this model in the bosonic limit. We obtain in this limit a new integrable system which, under a hodograph transformation, reduces to a coupled three-component system. We show how the Hamiltonian structure transforms under a hodograph transformation and study the properties of the model under a further reduction to a two-component system.     

超对称系统的边界条件与可积性

通过研究反射方程的解,构造了一类具有不同边界条件的超对称系统,同时证明了在一维情况下,这类系统是完全可积的.     

Coupled KdV Equations and Their Explicit Solutions Through Two-Dimensional Hamiltonian System with a Quartic Potential

Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

The Neumann Type Systems and Algebro-Geometric Solutions of a System of Coupled Integrable Equations

A system of (1+1)-dimensional coupled integrable equations is decomposed into a pair of new Neumann type systems that separate the spatial and temporal variables for this system over a symplectic submanifold. Then, the Neumann type flows associated with the coupled integrable equations are integrated on the complex tour of a Riemann surface. Finally, the algebro-geometric solutions expressed by Riemann theta functions of the system of coupled integrable equations are obtained by means of the Jacobi inversion.     

Fermion solutions in the supersymmetric CPn−1 model

We find all normalizable fermion solutions of the supersymmetric CP^n?1 model in the background of a general bosonic solution of the equations of motion. The index theorem is seen to be satisfied in a non-minimal way.     

N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.