an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

analytical and numerical studying of the perturbed korteweg de vries equation the perturbed korteweg de vries equation is considered this equation is used for the description of one dimensional viscous gas dynamics nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves the integrability of this equation is investigated using the painleve approach the condition on parameters for the integrability of the perturbed korteweg de vries equation equation is established new classical and nonclassical symmetries admitted by this equation are found all corresponding symmetry reductions are obtained new exact solutions of these reductions are constructed they are expressed via trigonometric and airy functions stability of the exact solutions of the perturbed korteweg de vries equation is investigated numerically

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

symmetry properties of a system of euler type equations for magnetized plasmas we provide all lie point symmetries of a system of partial differential equations of great interest to plasma physics we deduce the corresponding invariant solutions and discuss their physical interpretation we also consider some of its conditional and partial symmetries and obtain other interesting solutions finally we point out that although the system can be cast in divergence form and admits conserved currents it does not admit potential symmetries

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

perceptual separation of concurrent speech sounds absence of across frequency grouping by common interaural delay three experiments and a computational model explored the role of within channel and across channel processes in the perceptual separation of competing complex broadband sounds which differed in their interaural phase spectra in each experiment two competing vowels whose first and second formants were represented by two discrete bands of noise were presented concurrently for identification experiments 1 and 2 showed that listeners were able to identify the vowels accurately when each was presented to a different ear but were unable to identify the vowels when they were presented with different interaural time delays itds i e listeners could not group the noisebands in different frequency regions with the same itd and thereby separate them from bands in other frequency regions with a different itd experiment 3 demonstrated that while listeners were unable to exploit a difference in interaural delay between the pairs of noisebands listeners could identify a vowel defined by interaurally decorrelated noisebands when the other two noisebands were interaurally correlated a computational model based upon that of durlach j acoust soc am 32 1075 1076 1960 showed that the results of these and other experiments can be interpreted in terms of a within channel mechanism which is sensitive to interaural decorrelation thus the across frequency integration which occurs in the lateralization of complex sounds may play little role in segregating concurrent sounds

conditional symmetry and new classical solutions of the yang mills equations we suggest an effective method for reducing the yang mills equations to systems of ordinary differential equations with the use of this method we construct extensive families of new exact solutions of the yang mills equations analysis of the solutions thus obtained shows that they correspond to the conditional non classical symmetry of the equations under study

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

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

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

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

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

macroscopic limits for stochastic partial differential equations of mckean vlasov type a class of quasilinear stochastic partial differential equations spdes driven by spatially correlated brownian noise is shown to become macroscopic i e deterministic as the length of the correlations tends to 0 the limit is the solution of a quasilinear partial differential equation the quasilinear spdes are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations sodes coupled though a mean field interaction and driven by correlated brownian noise the limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes we also compare our approach to sodes with the one introduced by kunita

new conditional symmetries and exact solutions of the nonlinear wave equation we have constructed the new class of conditional symmetries of the nonlinear wave equation in 1 3 dimensions these symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

similarity reductions of the 1 3 dimensional burgers equation we consider the 1 3 dimensional burgers equation u t u x x u y y u z z uu x which has considerable interest in mathematical physics we complete the list of similarity reductions that are obtained from the lie symmetries admitted by this equation to achieve this goal we employ two and three dimensional subalgebras of the lie symmetry algebra in addition to the one dimensional subalgebras that appears in the literature we solve analytically where it is possible the reduced ordinary differential equations and hence we obtain closed form solutions for the original equation finally we derive certain non lie solutions

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

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

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

risk sensitive mean field games in this paper we study a class of risk sensitive mean field stochastic differential games we show that under appropriate regularity conditions the mean field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a hamilton jacobi bellman hjb equation with an additional quadratic term we provide an explicit solution of the mean field best response when the instantaneous cost functions are log quadratic and the state dynamics are affine in the control an equivalent mean field risk neutral problem is formulated and the corresponding mean field equilibria are characterized in terms of backward forward macroscopic mckean vlasov equations fokker planck kolmogorov equations and hjb equations we provide numerical examples on the mean field behavior to illustrate both linear and mckean vlasov dynamics

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

separation of variables solutions of nonlinear reaction diffusion systems sign invariance plays an important role in the study of an explicit solution to single partial differential equations in this paper we extend the sign invariant theory to study the nonlinear reaction diffusion system as a consequence we obtain some new explicit solutions to the resulting systems and we illustrate the exact solutions for a nonlinear parabolic system

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

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

large population cost coupled lqg problems with nonuniform agents individual mass behavior and decentralized varepsilon nash equilibria we consider linear quadratic gaussian lqg games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs a state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv nash equilibrium property a stability property of the mass behavior is established and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques

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