first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

group invariant classification of separable hamiltonian systems in the euclidean plane and the o 4 symmetric yang mills theories of yatsun we present a new and effective method of determining separable coordinate systems for natural hamiltonians with two degrees of freedom in flat riemannian space the method is based on intrinsic properties of the associated killing tensors and their invariants under the group of rigid motions e 2 applications to the hamiltonian systems derived by the late v a yatsun from o 4 symmetric yang mills theories are presented in addition an equivalence between separability of two dimensional hamiltonian systems and the existence of pfaffian quasi bi hamiltonian representations is specified

orthogonality preservers in c algebras jb algebras and jb triples abstract we study orthogonality preserving operators between c algebras jb algebras and jb triples let t a e be an orthogonality preserving bounded linear operator from a c algebra to a jb triple satisfying that t 1 d is a von neumann regular element then t a e 2 r d every element in t a and d operator commute in the jb algebra e 2 r d and there exists a triple homomorphism s a e 2 r d such that t l d r d s where r d denotes the range tripotent of d in e an analogous result for a being a jb algebra is also obtained when t a b is an operator between two c algebras we show that denoting h t 1 then t orthogonality preserving if and only if there exists a triple homomorphism s a b satisfying h s z s z h h s z s z h and t z l h r h s z 1 2 h r h s z s z r h h this allows us to prove that a bounded linear operator between two c algebras is orthogonality preserving if and only if it preserves zero triple products

a generalized henon heiles system and related integrable newton equations a detailed description is given of integrable cases of the generalized henon heiles systems which differs from the standard h h ones by the term q22 their connection with fifth order one component soliton equations is discussed lax representations are constructed and the bi hamiltonian formulation of dynamics is given it is also shown that the gh h system can be mapped onto another system of newton equations with a nonstandard hamiltonian structure

differential stackel matrices we show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential stackel matrices generalizations of the classical stackel matrices used for multiplicative separation of second order schrodinger equations and additive separation of hamilton jacobi equations we work out the principal properties of these matrices and demonstrate that even for second order laplace equations additive separation may occur when multiplicative separation does not

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

biseparating maps between operator algebras we prove that a biseparating map between spaces b e and some other banach algebras is automatically continuous and a multiple of an algebra isomorphism

dunford pettis properties hilbert spaces and projective tensor products we study complete continuity properties of operators onto 2 and prove several results in the dunford pettis theory of jb triples and their projective tensor products culminating in characterisations of the alternative dunford pettis property for e f where e and f are jb triples

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

on separable schrodinger equations we classify 1 3 dimensional schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables as a result we get 11 classes of the vector potentials of the electromagnetic field a t x a0 t x a t x providing separability of the corresponding schrodinger equations it is established in particular that the necessary condition for the schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables next we prove that any schrodinger equation admitting variable separation into second order ordinary differential equations can be reduced to one of the 11 separable schrodinger equations mentioned above and carry out variable separation in the latter furthermore we apply the results obtained for separating variables in the hamilton jacobi equation

weakly compact orthogonality preservers on c algebras we establish a complete description of weakly compact orthogonality preserving operators from a c algebra a or a jb algebra to a jb triple e we prove that any such operator is the sum of a series of mutually orthogonal triple homomorphisms from a to a certain peirce 2 subspace of e multiplied by a mutually orthogonal norm null sequence in e which enjoy certain operator commutativity relations

integrable theories and generalized graded maillet algebras we present a general formalism to investigate the integrable properties of a large class of non ultralocal models which in principle allows the construction of the corresponding lattice versions our main motivation comes from the subsector of the string theory on ads5 s5 in the uniform gauge where such type of non ultralocality appears in the resulting alday arutyunov frolov aaf model we first show how to account for the second derivative of the delta function in the lax algebra of the aaf model by modifying maillet s r and s matrices formalism and derive a well defined algebra of transition matrices which allows for the lattice formulation of the theory we illustrate our formalism on the examples of the bosonic wadati konno ichikawa shimizu wkis model and the two dimensional free massive dirac fermion model which can be obtained by a consistent reduction of the full aaf model and give the explicit forms of their corresponding r matrices

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

functional separation of variables for laplace equations in two dimensions the authors say that a solution psi of a partial differential equation in two real variables x1 x2 is functionally separable in these variables if psi x1 x2 phi a x1 b x2 for single variable functions phi a b such that phi a b not 0 they classify all possibilities for regular functional separation in local coordinates for equations of the form delta 2 psi f psi x1 x2 where delta 2 is the laplace beltrami operator on a two dimensional riemannian or pseudo riemannian space if the dependence of f on x1 x2 is non trivial then separation can occur for conformal cartesian coordinates on any space if f g psi then for orthogonal coordinates they find that true functional separation i e separation other than additive or multiplicative occurs precisely for cartesian coordinates in the euclidean and pseudo euclidean planes the sine gordon equation provides an example of this separation for many of these cases the separated solutions a b can be expressed in terms of elliptic functions for non orthogonal coordinates and f g psi true functional separation occurs precisely for cartesian coordinates in the pseudo euclidean plane and for a coordinate system on the hyperboloid of one sheet a pseudo riemannian space of constant curvature

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

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

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

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

foundations of the ads5 s5 superstring i we review the recent advances towards finding the spectrum of the ads5 s5 superstring we thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem in certain cases our exposition is original and cannot be found in the existing literature the present part i deals with foundations of classical string theory in ads5 s5 light cone perturbative quantization and the derivation of the exact light cone world sheet scattering matrix

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

novel variable separation solutions and exotic localized excitations via the etm in nonlinear soliton systems in this paper first the etm is applied to obtain variable separation solutions of 2 1 dimensional systems a common formula with some arbitrary functions is derived to describe suitable physical quantities for some 2 1 dimensional models such as the generalized nizhnik novikov veselov davey stewartson broer kaup kupershmidt boiti leon pempinelli integrable kortweg de vries kdv breaking soliton and burgers models the universal formula in tang lou and zhang phys rev e 66 046601 2002 can be simplified to the common formula in the present paper which indicates that redundant process is included there since the easier variable separation form can be employed without loss of generality second this method is successfully generalized to 1 1 dimensional systems such as coupled integrable dispersionless long wave short wave resonance interaction and negative kdv models and obtain another common formula to describe suitable physical fields or potentials of these 1 1 dimensional model

generalized stackel matrices stackel and differential stackel matrices are generalized so that the matrix elements may be functions of the derivatives of the dependent variable as well as the independent variable the inverses of these matrices are characterized and it is shown that for significant classes of linear and nonlinear partial differential equations variable separation is accomplished via this generalized stackel mechanism

application of symmetry analysis to a pde arising in the car windshield design a new approach to parameter identification problems from the point of view of symmetry analysis theory is given a mathematical model that arises in the design of car windshield represented by a linear second order mixed type pde is considered following a particular case of the direct method due to clarkson and kruskal we introduce a method to study the group invariance between the parameter and the data the equivalence transformations associated with this inverse problem are also found as a consequence the symmetry reductions relate the inverse and the direct problem and lead us to a reduced order model

classical and quantum algebras of non local charges in σ models we investigate the algebras of the non local charges and their generating functionals the monodromy matrices in classical and quantum non linear models in the case of the classical chiral models it turns out that there exists no definition of the poisson bracket of two monodromy matrices satisfying antisymmetry and the jacobi identity thus the classical non local charges do not generate a lie algebra in the case of the quantum o n non linear model we explicitly determine the conserved quantum monodromy operator from a factorization principle together withp t and o n invariance we give closed expressions for its matrix elements between asymptotic states in terms of the known two particles matrix the quantumr matrix of the model is found the quantum non local charges obey a quadratic lie algebra governed by a yang baxter equation

on the equivalence theorem for integrable systems we investigate the equivalence theorem for integrable systems using two formulations of the alday arutyunov frolov model we show that the s matrix is invariant under the field transformation which reduces the nonlinear dirac brackets of one formulation into the standard commutation relations in the second formulation we also explain how to perform the direct diagonalization of the transformed hamiltonian by constructing the states corresponding to self adjoint extensions

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

a class of liouville integrable hamiltonian systems with two degrees of freedom a class of two dimensional liouville integrable hamiltonian systems is studied separability of the corresponding hamilton jacobi equation for these systems is shown to be equivalent to other fundamental properties of hamiltonian systems such as the existence of the lax and bi hamiltonian representations of certain fixed types applications to physical models including the calogero moser model an integrable case of the henon heiles potential and the nonperiodic toda lattice are presented

separate weak continuity of the triple product in dual real jb triples we prove that if e is a real jb triple having a predual e then e is the unique predual of e and the triple product on e is separately sigma e e continuous

the algebraic curve of classical superstrings on ads5 s5 we investigate the monodromy of the lax connection for classical iib superstrings on ads5 s5 for any solution of the equations of motion we derive a spectral curve of degree 4 4 the curve consists purely of conserved quantities all gauge degrees of freedom have been eliminated in this form the most relevant quantities of the solution such as its energy can be expressed through certain holomorphic integrals on the curve this allows for a classification of finite gap solutions analogous to the general solution of strings in flat space the role of fermions in the context of the algebraic curve is clarified finally we derive a set of integral equations which reformulates the algebraic curve as a riemann hilbert problem they agree with the planar one loop open image in new window supersymmetric gauge theory proving the complete agreement of spectra in this approximation

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

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

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

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

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

separation of variables for scalar evolution equations in one space dimension the flow of an autonomous kth order evolution equation in one space dimension is generated by a kth order ordinary differential operator on the space of fields the characteristic fields are the solutions of the characteristic equation of the operator a solution of the evolution equation is separable if and only if it is a curve in the k c 1 parameter space of characteristic fields the non stationary characteristic fields with separable evolution are those which remain characteristic under small dilation every characteristic field has local separable evolution if and only if the characteristic equation is infinitesimally dilation invariant this is the case when the evolutionary generator is an infinitesimal symmetry of the characteristic equation so that its flow stabilizes the space of characteristic fields the evolution equation is then locally homogeneous if and only if the characteristic value is an invariant of the restricted flow which is a non generic property the separable kth order evolution equations are parametrized by the kth order invariants of the locally homogeneous k c 1 th order ordinary differential equations

the off shell symmetry algebra of the light cone ads5x s5 superstring we analyze the psu 2 2 4 supersymmetry algebra of a superstring prop agating in the ads5 s 5 background in the uniform light cone gauge we consider the off shell theory by relaxing the level matching condition and take the limit of infinite light cone momentum which decompactifies the string world sheet we fo cus on the psu 2 2 psu 2 2 subalgebra which leaves the light cone hamiltonian invariant and show that it undergoes extension by a central element which is ex pressed in terms of the level matching operator this result is in agreement with the conjectured symmetry algebra of the dynamic s matrix in the dual n 4 gauge theory

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

on grassmannians associated with jb triples then is a complex lie group that has precisely n 1 orbits e0 en in e each er is the locally closed complex submanifold of all matrices of rank r clearly er 1 is in the closure of er for every 1 r n and en is the unique open orbit for every pair of integers p q 0 let gp q be the grassmannian of all p planes in cp q then gp q is a compact symmetric hermitian manifold of dimension pq holomorphically equivalent to gq p every er is in a canonical way a equivariant holomorphic fibre bundle

approximate similarity reduction for singularly perturbed boussinesq equation via symmetry perturbation and direct method we investigate the singularly perturbed boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method the similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods series reduction solutions are consequently derived for the approximate symmetry perturbation method similarity reduction equations of the zero order are equivalent to the painleve iv painleve i and weierstrass elliptic equations for the approximate direct method similarity reduction equations of the zero order are equivalent to the painleve iv painleve ii painleve i or weierstrass elliptic equations the approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

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

nonlinear evolution equations admitting multilinear variable separable solutions based on the hirota s bilinear transformation method it is shown that a family of nonlinear partial differential equations admit multilinear variable separation solutions this extends the class of nonlinear partial differential equations which admit the multilinear variable separation solutions

the prolongation formula for tensor fields we derive a generalization of the prolongation formula for tensor valued functions by taking into account the natural action of diffeomorphisms on tensor fields this differs from the usual procedure where such fields are viewed as defining multi component scalars it replaces the ordinary derivative in the expression of the characteristic by a lie derivative the resulting version of noether s theorem takes a form more familiar to theoretical physicists a very simple proof of the conformal invariance of the maxwell lagrangian also follows from this procedure the eshelby tensor is also readily obtained as a further illustration of the practical use of this new prolongation formula

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

linearization of second order ordinary differential equations via cartan s equivalence method abstract we use cartan s equivalence method to study the differential invariants of a single second order ordinary differential equation relative to the pseudo group of point transformations as a result of the analysis a simple characterization is given of those second order equations which are linearizable by a point transformation

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

non linear reaction diffusion systems with variable diffusivities lie symmetries ansatze and exact solutions abstract this work first considers the classical lie symmetry analysis of a class of systems of two quasilinear reaction diffusion equations having variable diffusivities subsequently non lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems in particular families of exact solutions of a diffusive lotka volterra type system are constructed

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

automatic continuity of biorthogonality preservers between compact c algebras and von neumann algebras abstract we prove that every biorthogonality preserving linear surjection between two dual or compact c algebras or between two von neumann algebras is automatically continuous

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

multi dimensional quasi simple waves in weakly dissipative flows abstract a multi dimensional simple wave formalism is employed to formulate a multi dimensional quasi simple wave for a weakly dissipative fluid this is a natural but nontrivial generalization of the so called unidirectional quasi simple wave the method is more close to multi orders analysis which is different from the standard perturbation method detailed solving up to the second order is presented for a 2d sound simple wave a new 2d burgers equation is derived for the wave phase it essentially differs from the known 2d generalizations of the burgers equation e g from the zabolotskaya khokhlov equation the derived equation is investigated in the framework of group analysis of differential equations multi parameter families of its exact solutions are constructed simplest solutions are chosen for analysis of their physical relevance in the initial variables

bi hamiltonian structure of the bi dimensional superintegrable nonlinear isotonic oscillator the higher order superintegrability of the two dimensional isotonic oscillator noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator with rational ratio of frequencies is directly related with the existence of some complex functions with interesting poisson bracket properties first the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi hamiltonian complex structure in the second part several real symplectic structures are obtained and the properties of the recursion operators are studied

variable separation theory for the null hamilton jacobi equation the theory of the separation of variables for the null hamilton jacobi equation h 0 is systematically revisited and based on levi civita separability conditions with lagrangian multipliers the separation of the null equation is shown to be equivalent to the ordinary separation of the image of the original hamiltonian under a generalized jacobi maupertuis transformation the general results are applied to the special but fundamental case of the orthogonal separation of a natural hamiltonian with a fixed value of the energy the separation is then related to conditions which extend those of stackel and kalnins and miller for the null geodesic case and it is characterized by the existence of conformal killing two tensors of special kind

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

geometrical and gauge equivalence of the generalized hirota heisenberg and wkis equations with linear inhomogeneities integrable evolution equations can take several equivalent forms in a geometrical sense here we consider the equivalence of generalized versions involving linear inhomogeneities of three important nonlinear evolution equations namely the hirota heisenberg ferromagnetic spin and wadati konno ichikawa shimizu wkis equation through a moving helical space curve formalism and stereographic representation from the geometrical consideration we also construct suitable 2 2 matrix linear eigenvalue equations involving however non isospectral flow the eigenvalues evolve in time however these systems are also gauge equivalent we briefly analyse the scattering problem and show that infinite number of constants of motion can exist for these systems

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

the nonclassical group analysis of the heat equation abstract the nonclassical method of reduction was devised originally by bluman and cole in 1969 to find new exact solutions of the heat equation much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance however the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved although particular solutions have been given recently arrigo goard and broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical lie symmetry reduction solutions their arguments leave open the problem of what is the general nonclassical group action and its effect on the relevant solution of the heat equation in this article both these problems are solved in the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of burgers equation