nonlocal symmetries and formulae for generation of solutions for a class of diffusion convection equations abstract new formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation we study in what particular ways additional lie symmetries of diffusion convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations the formulae derived are used for the construction of exact solutions

classification of reduction operators and exact solutions of variable coefficient newell whitehead segel equations abstract a class of the newell whitehead segel equations also known as generalized fisher equations and newell whitehead equations is studied with lie and nonclassical symmetry points of view the classifications of lie reduction operators and of regular nonclassical reduction operators are performed the set of admissible transformations the equivalence groupoid of the class is described exhaustively the criterion of reducibility of variable coefficient newell whitehead segel equations to their constant coefficient counterparts is derived wide families of exact solutions for such variable coefficient equations are constructed

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

optimal system invariant solutions and evolution of weak discontinuity for isentropic drift flux model abstract inequivalent classes of similarity solutions for isentropic drift flux two phase flow model are obtained by using one dimensional optimal classifications the optimality of this classification is shown in a constructive manner i e by constructing various invariant functions including conditional and numerical invariants further with the help of invariant solution we discuss the evolution of weak discontinuity which displays the nature of the general solution

nonequilibrium thermodynamics of stochastic systems with odd and even variables the total entropy production of stochastic systems can be divided into three quantities the first corresponds to the excess heat while the second two comprise the housekeeping heat we denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter valid for all markovian stochastic dynamics a previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity

symmetry groups and fundamental solutions for systems of parabolic equations in this paper the relationship between lie point symmetry and fundamental solution for systems of parabolic equations is explored it is shown that the fundamental solutions of the systems of parabolic equations admitting certain symmetries can be obtained by inverting the laplace transformation of the corresponding group invariant solutions several examples are presented to illustrate the approach furthermore the relationship between fundamental solutions for two systems of parabolic equations related by an equivalence transformation is identified

the entropy concept for non equilibrium states in earlier work we presented a foundation for the second law of classical thermodynamics in terms of the entropy principle more precisely we provided an empirically accessible axiomatic derivati

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

symmetries of the fisher kolmogorov petrovskii piskunov equation with a nonlocal nonlinearity in a semiclassical approximation abstract the classical group analysis approach used to study the symmetries of integro differential equations in a semiclassical approximation is considered for a class of nearly linear integro differential equations in a semiclassical approximation an original integro differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis the approach is illustrated by the calculation of the lie symmetries in explicit form for a special case of the one dimensional nonlocal fisher kolmogorov petrovskii piskunov population equation

symmetries for certain coupled nonlinear schrodinger equations generalized symmetries are sought for certain coupled nonlinear schrodinger equations it is shown that all lie backlund symmetries but one are equivalent to lie point symmetries

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

simple waves and simple states inr2 abstract the so called solutions constructed by means of riemann invariants for quasi linear systems of pdes play an important role in gas dynamics since they describe the propagation and the superposition of waves the existing methods for finding solutions simple waves and simple states of this type usually deal with systems whose coefficients do not depend on the independent variables and have a very limited application to non homogeneous systems in this paper a method for finding such solutions in the case of general quasi linear systems of pdes inr2is presented this method is based on a detailed investigation of the respective overdetermined system and on a special construction involving pfaff systems

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

new exact solutions of burgers s equation an extension to the direct method of clarkson and kruskal new exact solutions of burgers s equation which is the simplest evolution equation to embody nonlinearity and dissipation are obtained these solutions are obtained by extending the method for finding symmetry reductions due to clarkson and kruskal new similarity solutions of the boussinesq equation j math phys 30 2201 2213 1989 the novel feature of this extended method is that one can seek reductions to a system of ordinary differential equations rather than the usual single equation and this leads to a wider class of solutions one is able to complete the calculations necessary for burgers s equation in all but a single case where it is necessary to integrate an abel equation of the second kind this indicates that the method is practicable in contrast it is often the case that new methods are of limited use in practice in particular solutions in terms of a class of confluent hypergeometric functions are computed by comparison solutions found by the original method due to clarkson and kruskal and other reduction methods are in terms of parabolic cylinder functions or airy functions which are special cases of this class

simple waves and invariant solutions of quasilinear systems the solutions of a system of homogeneous first order quasilinear evolution equations for which the propagation velocity depends only on the field value are characterized by their invariance under a particular type of flow all simple wave solutions are shown to have this property the local images of simple wave solutions are the integral curves of a system of ordinary differential equations on the field space the model for barotropic compressible flow is an example of a system that has invariant solutions of higher rank a solution of this system is invariant under the flow of the convective derivative if and only if it is divergence free the evolution of initial data is shown to be divergence free if and only if its differential is nilpotent

conditional lie backlund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source abstract the functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional lie backlund symmetry method the variant forms of the considered equations which admit the linear conditional lie backlund symmetries are identified a number of examples are considered and some exact solutions defined on the polynomial trigonometric and exponential invariant subspaces determined by the linear conditional lie backlund symmetries are constructed

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

point symmetries of conditionally integrable nonlinear evolution equations the lie point symmetries of the first two equations in the kadomtsev petviashvili kp hierarchy introduced by jimbo and miwa are investigated the first is the potential kp equation the second involves four independent variables and is called the jimbo miwa jm equation the joint symmetry algebra for the two equations is shown to have a kac moody virasoro structure whereas the symmetry algebra of the jm equation alone does not subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions

a nonequilibrium extension of the clausius heat theorem we generalize the clausius in equality to overdamped mesoscopic and macro scopic diffusions in the presence of nonconservative forces in contrast to previous frame works we use a decomposition scheme for heat which is based on an exact variant of the minimum entropy production principle as obtained from dynamical fluctuation theory this new extended heat theorem holds true for arbitrary driving and does not require assumptions of local or close to equilibrium the argument remains exactly intact for diffusing fields where the fields correspond to macroscopic profiles of interacting particles under hydro dynamic fluctuations we also show that the change of shannon entropy is related to the antisymmetric part under a modified time reversal of the time integrated entropy flux

thermodynamic transformations of nonequilibrium states we consider a macroscopic system in contact with boundary reservoirs and or under the action of an external field we discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one in this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered following the general proposal by oono and paniconi and using results of the macroscopic fluctuation theory we give a natural definition of a renormalized work we then discuss its thermodynamic relevance by showing that it satisfies a clausius inequality and that quasi static transformations minimize the renormalized work in addition we connect the renormalized work to the quasi potential describing the fluctuations in the stationary nonequilibrium ensemble the latter result provides a characterization of the quasi potential that does not involve rare fluctuations

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

time reversal and entropy there is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time reversal symmetry and the entropy production in such processes we explain on an elementary mathematical level the relations between entropy production phase space contraction and time reversal starting from a deterministic dynamics both closed and open systems in the transient and in the steady regime are considered the main result identifies under general conditions the statistical mechanical entropy production as the source term of time reversal breaking in the path space measure for the evolution of reduced variables this provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful in equalities we also discuss the markov approximation important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

a nash game approach for osnr optimization with capacity constraint in optical links this paper develops a nash game towards optimizing optical signal to noise ratio osnr in the presence of link capacity constraint in optical wavelength division multiplexing wdm networks all wavelength multiplexed channels in a link share the optical fiber in order to limit nonlinear effects the total power launched into a fiber has to be below a nonlinearity threshold this can be regarded as the optical link capacity constraint we formulate an extended osnr nash game that incorporates this underlying system constraint the status of an optical link is considered directly in the cost function sufficient conditions for existence and uniqueness of the nash equilibrium ne solution are given two iterative algorithms for channel power control are proposed to compute the ne solution a parallel update algorithm pua and a relaxed pua r pua their convergence is studied under different conditions both theoretically and numerically

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

exact solutions of semilinear radial wave equations in n dimensions abstract exact solutions are derived for an n dimensional radial wave equation with a general power nonlinearity the method which is applicable more generally to other nonlinear pdes involves an ansatz technique to solve a first order pde system of group invariant variables given by group foliations of the wave equation using the one dimensional admitted point symmetry groups these groups comprise scalings and time translations admitted for any nonlinearity power in addition to space time inversions admitted for a particular conformal nonlinearity power this is shown to yield not only group invariant solutions as derived by standard symmetry reduction but also other exact solutions of a more general form in particular solutions with interesting analytical behavior connected with blow ups as well as static monopoles are obtained

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

hidden symmetries associated with the projective group of nonlinear first order ordinary differential equations hidden symmetries those not found by the classical lie group method for point symmetries are reported for nonlinear first order ordinary differential equations odes which arise frequently in physical problems these are for the special class of the eight nonabelian two parameter subgroups of the eight parameter projective group the first order odes can be transformed by non local transformations to new separable first order odes which then can be reduced to quadratures the first order odes include riccati equations and equations which in particular cases are of the form of abel s equation the procedure demonstrates the feasibility of integrating nonlinear odes that do not show any apparent lie group point symmetry applications to the vlasov characteristic equation and the reaction diffusion equation are given

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

non classical symmetries and the singular manifold method a further two examples this paper discusses two equations with the conditional painleve property the usefulness of the singular manifold method as a tool for determining the non classical symmetries that reduce the equations to ordinary differential equations with the painleve property is confirmed once more the examples considered in this paper are particularly interesting because they have recently been proposed by other authors as counterexamples of the conjecture made by the authors that the singular manifold method allows us to identify non classical symmetries we demonstrate here that the conjecture still holds for these two cases as well a detailed study of the way of solving this apparent contradiction is offered

entropy and nonlinear nonequilibrium thermodynamic relation for heat conducting steady states among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states ness we take the one which is guided by operational thermodynamics and the clausius relation in our previous study we derived the extended clausius relation for ness where the heat in the original relation is replaced by its renormalized counterpart called the excess heat and the gibbs shannon expression for the entropy by a new symmetrized gibbs shannon like expression here we concentrate on markov processes describing heat conducting systems and develop a new method for deriving thermodynamic relations we first present a new simpler derivation of the extended clausius relation and clarify its close relation with the linear response theory we then derive a new improved extended clausius relation with a nonlinear nonequilibrium contribution which is written as a correlation between work and heat we argue that the nonlinear nonequilibrium contribution is unavoidable and is determined uniquely once we accept the very natural definition of the excess heat moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference one may only use the previous clausius relation without a nonlinear term or must use the new relation depending on the operation i e the path in the parameter space this peculiar twist may be a clue to a better understanding of thermodynamics and statistical mechanics of ness

exact equalities and thermodynamic relations for nonequilibrium steady states we study thermodynamic operations which bring a nonequilibrium steady state ness to another ness in physical systems under nonequilibrium conditions we model the system by a suitable markov jump process and treat thermodynamic operations as protocols according to which the external agent varies parameters of the markov process then we prove among other relations a ness version of the jarzynski equality and the extended clausius relation the latter can be a starting point of thermodynamics for ness we also find that the corresponding nonequilibrium entropy has a microscopic representation in terms of symmetrized shannon entropy in systems where the microscopic description of states involves momenta all the results in the present paper are mathematically rigorous

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant