Integrability and Solutions of the(2+1)-dimensional Hunter–Saxton Equation

In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.     

The matrix Lax representation of the generalized Riemann equations and its conservation laws

It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter-Saxton equation. New matrix and scalar Lax representation are presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.     

N = 2 superconformal algebra and integrable O(2) fermionic extensions of the Korteweg-de Vries equation

The N = 2 superconformal algebra is shown to be related to the second hamiltonian structure of three integrable fermionic extensions of the Korteweg-de Vries equation. One of these systems is bi-hamiltonian but not supersymmetric while the reverse is true for the other two.     

Integrable extensions of N = 2 supersymmetric KdV hierarchy associated with the nonuniqueness of the roots of the Lax operator

We present new supersymmetric integrable extensions of the a = 4, N = 2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generates a new hierarchy of integrable equations, for which we investigate the Hamiltonian structure. In a special case our system describes the interaction of the KdV equation with the two MKdV equations.     

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.     

Towards the construction of N = 2 supersymmetric integrable hierarchies

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super-W_n second Hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct Hamiltonians we are led to a new definition of super-residues for degenerate N = 2 supersymmetric pseudo-differential operators. We have found a new non-polynomials Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.     

Second-Order Linear Differential Equations for Effective Barotropic FRW Cosmologies

In 1999, Faraoni wrote a simple second-order linear differential equation for FRW cosmologies with barotropic fluids. His results have been extended by Rosu, who employed techniques belonging to nonrelativistic supersymmetry to obtain time-dependent effective adiabatic indices. Further extensions are presented here using the known connection between the linear second-order differential equations and Dirac-like equations in the same supersymmetric context. These extensions are equivalent to adding an imaginary part to the effective adiabatic index, which is proportional to the mass parameter of the Dirac spinor. The natural physical interpretation of the imaginary part is related to the particular dissipation and instabilities of the effective barotropic FRW hydrodynamics that are introduced by means of this supersymmetric scheme.     

Contraction-based classification of supersymmetric extensions of kinematical lie algebras

We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry—Lévy-Leblond classification.     

Supersymmetric descendants of self-adjointly extended quantum mechanical Hamiltonians

We consider the descendants of self-adjointly extended Hamiltonians in supersymmetric quantum mechanics on a half-line, on an interval, and on a punctured line or interval. While there is a 4-parameter family of self-adjointly extended Hamiltonians on a punctured line, only a 3-parameter sub-family has supersymmetric descendants that are themselves self-adjoint. We also address the self-adjointness of an operator related to the supercharge, and point out that only a sub-class of its most general self-adjoint extensions is physical. Besides a general characterization of self-adjoint extensions and their supersymmetric descendants, we explicitly consider concrete examples, including a particle in a box with general boundary conditions, with and without an additional point interaction. We also discuss bulk-boundary resonances and their manifestation in the supersymmetric descendant.     

15顶角模型反射方程的常数解

得到了15顶角模型A2(1)模型和超对称t–J模型反射方程的非对角解,结果发现,A2(1)模型具有三种形式的非对角解,超对称t–J模型具有一种形式的非对角解,每种形式的非对角解均含有两个解,每个非对角解中均含有三个任意参数.关于对角解也得到了一些新的形式的解.     

A two dimensional Lagrangian model with extended supersymmetry and infinitely many constants of motion

The reduction of the supersymmetric gradedSU(2|1) /S(U ₂×U ₁) -model is discussed. If no extra constraint is imposed, one gets a set of two coupled equations (involving two scalar superfields) which generalizes the supersymmetric sine-Gordon equation. It is shown that these equations, which can be derived by a supersymmetric Lagrangian, reproduce in the bosonic limit the reduced version of theO(4) -model (Pohlmeyer, Lund Regge, Getmanov model). Moreover the associate linear set and an infinite set of local conservation laws for this new supersymmetric model are exhibited. It turns out that, beyond the spinorial charge which generates the supersymmetry transformations, another unexpected spinorial charge is conserved; then the model appears to be invariant underN=2 extended supersymmetry.     

Semiclassical Method to Schroedinger Equation with Position-Dependent Effective Mass

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schroedinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schroedinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.     

The Higgs sector of supersymmetric theories and the implications for high-energy colliders

One of the main motivations for low-energy supersymmetric theories is their ability to address the hierarchy and naturalness problems in the Higgs sector of the standard model. In these theories, at least two doublets of scalar fields are required to break the electroweak symmetry and to generate the masses of the elementary particles, resulting in a rather rich Higgs spectrum. The search for the Higgs bosons of supersymmetry and the determination of their basic properties is one of the major goals of high-energy colliders and, in particular, the LHC, which will soon start operation. We review the salient features of the Higgs sector of the minimal supersymmetric standard model and of some of its extensions and summarize the prospects for probing them at the LHC and at the future ILC. In memoriam of Julius Wess, 1934–2007.     

CP Violating Phenomena and Theoretical Results

An introduction to CP violating phenomena is given and the standard model and its most popular low energy extensions are reviewed in this context. The discussion comprises the minimal supersymmetric extension of the standard model, left-right symmetry, the standard model with more than one Higgs doublet and gauged horizontal symmetries.     

Higgs bosons in the standard model,the MSSM and beyond

I summarize the basic theory and selected phenomenology for the Higgs boson(s) of the standard model, the minimal supersymmetric model and some extensions thereof, including the next-to-minimal supersymmetric model.     

Electroweak breaking and supersymmetry breaking

We discuss the clash between the absence of fine tuning in the Higgs potential and a sufficient suppression of flavour changing neutral current transitions in supersymmetric extensions of the standard model. It is pointed out that horizontalU(1) symmetry combined with theD-term supersymmetry breaking provides a realistic framework for solving both problems.     

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.     

Energy-Levels Crossing and Radial Dirac Equation: Supersymmetry and Quasi-Parity Spectral Signatures

The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energy-levels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the related interactions’ way-of-coupling into Dirac equation, it is observed that the two ultimate/effective descendents, Dirac-Coulomb and Dirac-oscillator, exhibit different conditions on the energy-levels crossings.     

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.     

Geometric Integrability of Two-Component Camassa-Holm and Hunter-Saxton Systems

It is shown that the two-component Camassa-Holm and Hunter-Saxton systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws are directly constructed. In addition, a class of nonlocal symmetries depending on the pseudo-potentials are obtained.