change of type in three phase flow a simple analogue abstract we consider a system of two conservation laws in one space variable which exhibits change of type hyperbolic elliptic the characteristics are complex in a strip in phase space this example has a structure similar to model systems which appear in three phase porous medium flow for example in the saturation equations obtained employing stone s model for the relative permeabilities we solve the riemann problem analytically for initial states in and near the elliptic strip

the jacobi rosochatius problem on an ellipsoid the lax representations and billiards in this article we construct the lax representations of the geodesic flow the jacobi rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid we prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces also we consider billiards within an ellipsoid under the influence of the hook and rosochatius potentials between the impacts a geometric interpretation of the integrability analogous to the classical chasles and poncelet theorems is given

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

crossing the elliptic region in a hyperbolic system with change of type behavior arising in flow between two parallel plates change of type behavior from hyperbolic to elliptic is common to quasilinear hyperbolic systems this issue is addressed here for the particular case of miscible flow of three fluids between two parallel plates change of type occurs at the leading edge of the displacement front and reflects the failing of the equilibrium assumption necessary for the quasilinear hyperbolic formalism at the front to cross the elliptic region requires the solution of the full higher dimensionality problem obtained here using lattice gas simulations for the specific example it is found that the system self selects a front structure independent of injection conditions

elliptical billiard systems and the full poncelet s theorem in n dimensions in this work is presented a generalization of poncelet s theorem to n dimensions which is refered to as the full poncelet s theorem the theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon then there exists an n 1 parameter family of polygons having the same property a physical realization and a projective geometrical proof of this theorem are given if all the reflecting quadrics coincide the above theorem reduces to the n dimensional poncelet s theorem presented by chang and friedberg the geometrical proof is a finite construction based on a preliminary theorem which extends hart s lemma the full poncelet s theorem may thus be extended to projective geometries over most fields including discrete ones

weak asymptotic methods for scalar equations and systems abstract in this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions this construction produces weak asymptotic methods which are issued from maslov asymptotic analysis we obtain a sequence of functions which tend to satisfy the equation s in the weak sense in the space variable and in the strong sense in the time variable to this end we reduce the partial differential equations to a family of ordinary differential equations in a classical banach space for scalar equations we prove that the initial value problem is well posed in the l 1 sense for the approximate solutions we construct then we prove that this method gives back the widely accepted solutions when they are known for systems we obtain existence in the general case and uniqueness in the analytic case using an abstract cauchy kovalevska theorem

a numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting the present treatment of elliptic regions via hyperbolic flux splitting and high order methods proposes a flux splitting in which the corresponding jacobians have real and positive negative eigenvalues while resembling the flux splitting used in hyperbolic systems the present generalization of such splitting to elliptic regions allows the handling of mixed type systems in a unified and heuristically stable fashion the van der waals fluid dynamics equation is used convergence with good resolution to weak solutions for various riemann problems are observed

existence and uniqueness of the riemann problem for a nonlinear system of conservation laws of mixed type we study the system of conservation laws given by ut u 1 v 0 vt v a u x 0 a 1 is a constant with any riemann initial data ut v the system is elliptic in the domain where v u a 1 2 4 a 1 u 0 we combine and generalize lax criterion and oleinik liu criterion to introduce the generalized entropy condition g e c by which we can show that the riemann problem always has a weak solution any discontinuity satisfies the g e c for any initial data however not necessarily unique we introduce the minimum principle then in the definition of an admissible weak solution for the riemann problem and the existence and uniqueness of the solution for any riemann data

variationally consistent domain integration for isogeometric analysis abstract spline type approximations for solving partial differential equations are the basis of isogeometric analysis while the common approach of using integration cells defined by single knot spans using standard e g gaussian quadrature rules is sufficient for accuracy more efficient domain integration is still in high demand the recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques in this work variationally consistent domain integration is proposed for isogeometric analysis test function gradients are constructed to meet the consistency conditions which only requires solving small linear systems of equations the proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability accuracy and optimal convergence properties of higher order quadrature rules several numerical examples are provided to illustrate the performance of the proposed domain integration technique

on integrable potential perturbations of the billiard system within an ellipsoid generalizing kozlov s method of integrable potential perturbations for systems with three degrees of freedom new integrable billiard systems within ellipsoid in r3 are constructed two countable families of the basic solutions in the class of the laurent polynomials are obtained explicit formulas are given

when is a hamiltonian system separable the hamiltonian system given by h 1 2 p2 v q with v c rn is considered a method for integrating such a system is that of separating the variables in the hamilton jacobi equation it is known that if such a separation is possible then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these a criterion is proposed for deciding if separation is possible and if it is in which degeneration of elliptic coordinates it takes place

conservation laws of mixed type describing three phase flow in porous media in this paper we examine the mathematical structure of a model for three phase incompressible flow in a porous medium we show that in the absence of diffusive forces the system of conservation laws describing the flow is not necessarily hyperbolic we present an example in which there is an elliptic region in saturation space for reasonable relative permeability data a linearized analysis shows that in nonhyperbolic regions solutions grow exponentially however the nonhyperbolic region if present will be of limited extent which inherently limits the exponential growth to examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method these experiments indicate that the solutions of riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space in particular when initial states are outside the elliptic region the riemann problem solution appears to stay outside the region further

on the riemann problem for a prototype of a mixed type conservation law we study the system of conservation laws given by with initial value the system is elliptic when u2 v2 2 and hyperbolic when u2 v2 2 following liu s construction it is found that the system always has a weak solution which however is not necessarily unique

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

the appell hypergeometric functions and classical separable mechanical systems a relationship between two old mathematical subjects is observed the theory of hypergeometric functions and the separability in classical mechanics separable potential perturbations of integrable billiard systems and the jacobi problem for geodesics on an ellipsoid are expressed through the appell hypergeometric functions f4 of two variables even when the number of degrees of freedom increases if an ellipsoid is symmetric the number of variables in the hypergeometric functions does not increase wider classes of separable potentials are given by the obtained new formulae automatically

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

geodesic compatibility and integrability of geodesic flows we give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a pseudo riemannian metric we prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition a lot of metrics studed in riemannian and kahlerian geometry satisfy such conditions we apply our results for obtaining an infinite family hierarchy of completely integrable flows on the complex projective plane cpn