Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.     

New potential symmetries for some evolution equations

In this paper we derive new potential symmetries that seem not to be recorded in the literature. These potential symmetries are determined by considering a generalized potential system, rather than the natural potential system or a general integral variable. An inhomogeneous diffusion equation, a porous medium equation and the Fokker-Planck equation have been considered as application of this procedure.     

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.     

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.     

Continuous symmetries of certain nonlinear partial difference equations and their reductions

In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible.     

Conditional Lie-Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations

The second-order conditional Lie-Bäcklund symmetries of nonlinear diffusion equations with variable coefficients are studied. A number of examples are considered and some exact solutions are constructed via the compatibility of conditional Lie-Bäcklund symmetries and the governing equations. These solutions possess the extended forms of the separation of variables, including the extensions of the instantaneous source solutions of the porous medium equations.     

The reduction of the Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries

We prove a general theorem which allows the determination of Lie symmetries of the Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of the Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of the Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of the Laplace equation (i.e., the wave equation) in the Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non-gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to general relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein’s equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.     

On the symmetries of integrable systems with boundaries

We employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the case. With the help of linear intertwining relations involving the aforementioned solutions of the reflection equation, the symmetry of the open spin chain with a particular choice of the left boundary is exhibited. The symmetry of the corresponding local Hamiltonian is also explored. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

广义非完整力学系统的Lie对称性与Noether守恒量

研究广义非完整力学系统的Lie对称性与Noether守恒量,建立Lie对称性的确定方程、限制方程和附加限制方程,给出结构方程和Noether守恒量的形式,研究Lie对称性的逆问题,并举算例说明结果的应用.     

Conserved quantities and symmetries related to stochastic Hamiltonian systems

In this paper symmetries and conservation laws for stochastic dynamical systems are discussed in detail. Determining equations for infinitesimal approximate symmetries of Ito and Stratonovich dynamical systems are derived. It shows how to derive conserved quantities for stochastic dynamical systems by using their symmetries without recourse to Noether's theorem.     

On diffusion equations for dynamical systems driven by noise

A unified formalism is presented to study Hamiltonian linear systems driven by noise. With this formalism, the phase averaging approximation, valid at weak noise, is easily performed. Already known results are straightforwardly recovered and new ones are obtained. After introducing this formalism on the exactly solvable one-degree-of-freedom problem with uncorrelated noise, one studies the corresponding exponentially correlated case. The validity of the approximate results thus obtained is considered by investigating the systematic weak-disorder expansion beyond the quasilinear approximation. In particular, it is argued that this expansion behaves uniformly for weak and large correlation time. The two-degrees-of-freedom problem is completely solved at the low-disorder approximation and this result is applied to the two-channel Anderson localization problem. The invariant measure and the two positive Lyapunov exponents are computed at all coupling between the channels. For systems withn degree of freedom the phase averaging leads to a Fokker-Planck equation for the measure in action space describing the system. However, it is argued that it is not solvable except in a special case which is explicitly displayed and solved. Nevertheless, in the large-n limit, it is possible to compute the largest Lyapunov exponent. Moreover, generalized Lyapunov exponents are calculated in this limit, and they do not exhibit a dispersion: in particular, log/log1, where is the energy of the system and where the brackets denote averaging over the noise. On the other hand, it is possible to compute at weak noise the sum of all the positive Lyapunov exponents. Taking into account all these results allows more insight on the whole spectrum of Lyapunov exponents.     

Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations

In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266–281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question.     

Perfectly matched layers for the heat and advection–diffusion equations

We design a perfectly matched layer for the advection–diffusion equation. We show that the reflection coefficient is exponentially small with respect to the damping parameter and the width of the PML and this independently of the advection and of the viscosity. Numerical tests assess the efficiency of the approach.     

Lie symmetries and conserved quantities for a two-dimensional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

Lie symmetries and conserved quantities for a two-dimentional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

Time relaxation of the solutions of master equations for large systems

The time relaxation behavior of the solutions of certain classes of discrete master equations is studied in the limit of an infinite number of states. Depending on the range of the transition matrix, a relaxation behavior is found reaching from at ^–1/2 law for short range, over enhanced relaxation to an exponential relaxation for the extreme long-range case. The behavior in the limit of a continuous family of states is also discussed.     

一类动力学方程的Mei对称性

研究一类动力学方程的Mei对称性的定义和判据，由Mei对称性通过Noether对称性可找到Noether守恒量.由Mei对称性通过Lie对称性可找到Hojman守恒量.同时，也可找到一类新型守恒量.     

A nonextensive maximum entropy approach to a family of nonlinear reaction–diffusion equations

We consider a monoparametric family of reaction–diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time-dependent particular solutions of the Tsallis’ maximum entropy (MaxEnt) form. The evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically. A simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q-MaxEnt solutions. When the reaction term vanishes the time-dependent distributions studied here reduce to the previously known Tsallis’ MaxEnt solutions for the nonlinear diffusion equation.