Differential Equations Invariant Under Conditional Symmetries

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.     

On sufficient conditions of locality for hierarchies of symmetries of evolution systems

We present sufficient conditions ensuring the locality of hierarchies of symmetries generated by repeated commutation of master symmetry with a seed symmetry. These conditions are applicable to a large class of (1+1)-dimensional evolution systems. Our results can also be used for proving that the time-independent part of a suitable linear-in-time symmetry is a nontrivial master symmetry and hence the system in question has infinitely many symmetries and is integrable.     

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.     

Auxiliary ghost fields in statistical dynamics

The path integral formulation of statistical dynamics involves a functional determinant whose role within the theory has remained somewhat unclear. This has occasionally led to incorrect generalizations of the formalism to the case of multiplicative random forces. We present a hidden symmetry of the theory and show how it can be used to clarify these issues. Important further applications are also pointed out.     

Perron–Frobenius Theory and Symmetry of Solutions to Nonlinear PDE's

We suggest a simple but general method of establishing symmetry properties of stable solutions of nonlinear elliptic equations. The method relies on characterization of symmetry breaking with a help of zero modes and on a generalization of the Perron–Frobenius theory.     

On Dirac's Conjecture

The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities, it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system. We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.     

Equilibrium equations and symmetries of classical Coulomb systems

In this paper we establish the validity of the BBGKY equilibrium equations for Coulomb states which have been obtained as thermodynamic limit of finite volume states. We also give a new derivation of thel-sum rules for phases constructed by the cluster expansion. These sum rules are interpreted as Ward identities associated to a symmetry of the screening phase.Supported by the Swiss National Foundation for Scientific Research.     

Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

Finite symmetry transformation group of the Konopelchenkoben Dubrovsky equation from its Lax pair

Starting from a weak Lax pair, the general Lie point symmetry group of the Konopelchenko-Dubrovsky equation is obtained by using the general direct method. And the corresponding Lie algebra structure is proved to be a Kac-Moody-Virasoro type. Furthermore, a new multi-soliton solution for the Konopelchenko-Dubrovsky equation is also given from this symmetry group and a known solution.     

Noether-Type Symmetries and Associated Conservation Laws of Some Systems of Nonlinear PDEs

The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.     

The Virasoro action on the tau function for the constrained discrete KP hierarchy

With the help of the squared eigenfunction potential, the action of the Virasoro symmetry on the tau function of the constrained discrete KP hierarchy is derived.     

On conservation laws and boundary conditions for “short waves” equation

In this paper we study local conservation laws for the equation of short waves in the form of a variational problem. We analyze an infinite symmetry group of the equation and generate a finite number of conservation laws corresponding to given infinite symmetries through appropriate boundary conditions.     

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.     

Reflections on Parity

The selective synthesis and manipulation of chiral molecules is a major part of stereochemistry. Interactions involving components with handedness also occur in nuclear and particle physics. Both fields employ phenomenologies based on group theoretical classifications and symmetry constraints such as the conservation or violation of parity. Smale's discovery of a sphere eversion—homotopic to a central inversion—evades the parity dichotomy by allowing states of opposite chirality to be connected by smooth maps. This transformation demonstrates that parity changes can, in principle, result from the action of sufficiently complex processes, and do not necessarily have to be interpreted as violations of a symmetry.     

Finite Symmetry Transformation Groups and Exact Solutions of Lax Integrable Systems

The general Lie point symmetry groups of the Nizhnik-Novikov-Vesselov (NNV) equation and the asymmetric NNV equation are given by a simple direct method with help of their weak Lax pairs.     

Finite Symmetry Transformation Groups and Exact Solutions of Lax Integrable Systems

The general Lie point symmetry groups of the Nizhnik-Novikov-Vesselov （NNV） equation and the asymmetric NNV equation are given by a simple direct method with help of their weak Lax pairs.     

Scasimir Operator,Scentre and Representations of Uq(osp(1|2))

Recently, Borchers has shown that in a theory of local observables, certain unitary and antiunitary operators, which are obtained from an elementary construction suggested by Bisognano and Wichmann, have the same commutation relations with translation operators as Lorentz boosts and P₁CT operators would have, respectively. It is concluded from this that as soon as the operators considered implement any symmetry, this symmetry can be fixed up to at most some translation. As a symmetry, any unitary or antiunitary operator is admitted under whose adjoint action any algebra of local observables is mapped onto an algebra which can be localized somewhere in Minkowski space.     

On hidden symmetries of extremal Kerr–NUT–AdS–dS black holes

It is well known that the Kerr–NUT–AdS–dS black hole admits two linearly independent Killing vectors and possesses a hidden symmetry generated by a rank-2 Killing tensor. The near-horizon geometry of an extremal Kerr–NUT–AdS–dS black hole admits four linearly independent Killing vectors, and we show how the hidden symmetry of the black hole itself is carried over by means of a modified Killing–Yano potential which is given explicitly. We demonstrate that the corresponding Killing tensor of the near-horizon geometry is reducible as it can be expressed in terms of the Casimir operators formed by the four Killing vectors.     

A Note on the Equivalence of Methods to finding Nonclassical Determining Equations

In this note we prove that the method of Bîlã and Niesen to determine nonclassical determining equations is equivalent to that of Nucci’s method with heir-equations and thus in general is equivalent to using an appropriate form of generalised conditional symmetry.