discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

on the well posedness of weakly hyperbolic equations with time dependent coefficients in this paper we analyse the gevrey well posedness of the cauchy problem for weakly hyperbolic equations of general form with time dependent coefficients the results involve the order of lower order terms and the number of multiple roots we also derive the corresponding well posedness results in the space of gevrey beurling ultradistributions

rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

double vogan diagrams and semisimple symmetric spaces a vogan diagram is a set of involution and painting on a dynkin diagram it selects a real form or equivalently an involution from a complex simple lie algebra we introduce the double vogan diagram which is two sets of vogan diagrams superimposed on an affine dynkin diagram they correspond to pairs of commuting involutions on complex simple lie algebras and therefore provide an independent classification of the simple locally symmetric pairs

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws we introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space besides classical shock waves it also exhibits overcompressive marginal overcompressive and marginal undercompressive shock waves our purpose is to study the behavior of the corresponding viscous waves in particular the manner in which these waves are stable there are several basic differences between classical shock waves and other types of shock waves a perturbation of an overcompressive shock wave gives rise to a new wave monotone marginal overcompressive waves behave distinctly from the nonmonotone ones analytical techniques used in our study include characteristic energy and weighted energy methods and nonlinear superposition through time invariants although we carry out our analysis for a simple model the general phenomena would hold for overcompressive waves which occur in other physical models

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

ultradiscrete painleve ii equation and a special function solution ultradiscretization with parity variables which keeps the information of original variables sign is applied to the q painleve ii equation of type a6 q pii a special solution of the resulting ultradiscrete system which corresponds to the special function solution of q pii is constructed ultradiscrete analogues of the q airy equation and its special solutions are also discussed in the process

vanishing viscosity with short wave long wave interactions for systems of conservation laws motivated by benney s general theory we propose new models for short wave long wave interactions when the long waves are described by nonlinear systems of conservation laws we prove the strong convergence of the solutions of the vanishing viscosity and short wave long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems we analyze several of the representative examples such as scalar conservation laws general symmetric systems nonlinear elasticity and nonlinear electromagnetism

the cauchy problem for hyperbolic systems with holder continuous coefficients with respect to time abstract we discuss the local existance and uniqueness of solutions of certain nonstrictly hyperbolic systems with holder continuous coefficients with respect to time variable we reduce the nonstrictly hyperbolic systems to the parabolic ones then we shall prove them by use of tanabe sobolevski s method

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

on a nonstrictly hyperbolic system of conservation laws a system of quasi linear hyperbolic conservation laws which is hyperbolic but not strictly hyperbolic is studied the system was derived as a model for the elastic string b keyfitz and kranzer arch rational mech anal 72 1980 210 241 and is assumed to be diagonizable interest is mainly in the large time behavior of the solution due to the nonlinearity of the system and the entropy condition solutions converge to very simple elementary waves nonstrict hyperbolicity of the system may cause stronger nonlinear interactions between waves pertaining to different families in particular such interactions may regularize linear waves in the solution the solutions are constructed using the random choice method

weakly hyperbolic equations with non analytic coefficients and lower order terms in this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time dependent coefficients and lower order terms we prove the gevrey well posedness of the cauchy problem under c k regularity of coefficients of the principal part and natural levi conditions on lower order terms which may be only continuous in the case of analytic coefficients in the principal part we establish the c infty well posedness the proofs are based on using the quasi symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions the main novelty compared to the existing literature is the possibility to include lower order terms to the equation which have been untreatable until now in these problems as well as considering any space dimensions we also give results on the ultradistributional and distributional well posedness of the problem and we look at new effects for equations with discontinuous lower order terms

affine kac moody symmetric spaces related with a1 1 a2 1 a2 2 symmetric spaces associated with lie algebras and lie groups which are riemannian manifolds have recently got a lot of attention in various branches of physics for their role in classical quantum integrable systems transport phenomena etc their infinite dimensional counter parts have recently been discovered which are affine kac moody symmetric spaces in this paper we have algebraically explicitly computed the affine kac moody symmetric spaces associated with affine kac moody algebras a1 1 a2 1 a2 2 we hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems

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

admissible positive systems of affine non twisted kac moody lie algebras abstract we introduce admissible positive systems and hermitian real forms of affine non twisted kac moody lie algebras and show that a real form has admissible positive system if and only if it is hermitian we use the vogan diagrams to classify the hermitian real forms and show that their symmetric spaces carry complex structures

well posedness of hyperbolic systems with multiplicities and smooth coefficients we study hyperbolic systems with multiplicities and smooth coefficients in the case of non analytic smooth coefficients we prove well posedness in any gevrey class and when the coefficients are analytic we prove c infty well posedness the proof is based on a transformation to block sylvester form introduced by d ancona and spagnolo boll umi 8 1b 169 185 1998 which increases the system size but does not change the eigenvalues this reduction introduces lower order terms for which appropriate levi type conditions are found these translate then into conditions on the original coefficient matrix this paper can be considered as a generalisation of garetto and ruzhansky math ann 357 2 401 440 2013 where weakly hyperbolic higher order equations with lower order terms were considered

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

equivalence classes of vogan diagrams abstract a vogan diagram is a dynkin diagram with an involution and the vertices fixed by the involution may be painted they represent real simple lie algebras and two diagrams are said to be equivalent if they represent the same lie algebra in this article we classify the equivalence classes of all vogan diagrams in doing so we find that the underlying dynkin diagrams have certain properties in graph painting we show that this combinatorial property provides an easy classification for most of the simply laced dynkin diagrams

non uniqueness in hyperbolic cauchy problems where t e r and x e ri we say that it possesses the uniqueness property at a point to xo if for any solution u e c v of 1 v being a neighborhood of t0o xo such that u 0 on v nf t to it results that u 0 on a neighborhood of to xo the first uniqueness results for equation 1 were supplied by the classical theorems of holmgren 12 carleman 6 and calderon 5 more refined versions of these theorems are the following ones

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

globally analytic triangularization of a matrix function abstract a well known fundamental theorem of schur guarantees that every constant matrix a is unitarily similar to a triangular matrix if a t is a matrix valued function analytic on a b with real eigenvalues it is shown in this paper via a constructive algorithm that there exists a unitary matrix u t analytic on a b such that u t a t u t is triangular in the process of doing so we also show that the gram schmidt orthogonalization process preserves global analyticity a similar algorithm which preserves periodicity works also for a periodic analytic matrix

riccati solutions of discrete painleve equations with weyl group symmetry of type e8 1 we present special solutions of the discrete painleve equations associated with a0 1 a0 1 and a0 1 surfaces these solutions can be expressed by solutions of linear difference equations here the a0 1 surface discrete painleve equation is the most generic difference equation as all discrete painleve equations can be obtained by its degeneration limit these special solutions exist when the parameters of the discrete painleve equation satisfy a particular constraint we consider that these special functions belong to the hypergeometric family although they seem to go beyond the known discrete and q discrete hypergeometric functions we also discuss the degeneration scheme of these solutions

general theory of the matrix formulation of the automorphisms of affine kac moody algebras the determination of conjugacy classes within the group of all automorphisms of an affine kac moody algebra is discussed in detail and the limited role of cartan preserving automorphisms is emphasized a comprehensive method of dealing with all the automorphisms of an untwisted affine kac moody algebra is developed this is based on a matrix formulation of the untwisted affine kac moody algebras four different types of automorphism called type 1a type 1b type 2a and type 2b are identified and analysed within this matrix formulation the development of the detailed properties of automorphisms in the matrix formulation includes formulae for the products and inverses of automorphisms together with the conjugacy and involutive conditions

vogan diagrams of real forms of affine kac moody lie algebras abstract a vogan diagram is actually a dynkin diagram with some additional structure this paper develops theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras here the equivalence classes of vogan diagrams are in one one correspondence with the isomorphism classes of almost compact real forms

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

admissible positive systems of affine kac moody lie algebras the twisted cases abstract we introduce admissible positive systems and hermitian real forms of affine twisted kac moody lie algebras and show that a real form has admissible positive system if and only if it is hermitian we use the vogan diagrams to classify the hermitian real forms and show that their symmetric spaces carry complex structures the affine non twisted kac moody lie algebras have been treated in an earlier work and this article deals with the twisted cases

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

real forms and conjugacy classes of involutive automorphisms of the d 1 affine kac moody algebras the real forms of the complex affine kac moody algebras d 1 for 4 are found by determining all the conjugacy classes of the involutive automorphisms of d 1 using the matrix formulation of automorphisms of an affine kac moody algebra

a note on weakly hyperbolic equations with analytic principal part in this note we show how to include low order terms in the c well posedness results for weakly hyperbolic equations with analytic time dependent coefficients this is achieved by doing a different reduction to a system from the previously used one we find the levi conditions such that the c well posedness continues to hold

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

maps of convex sets and invariant regions for finite difference systems of conservation laws general results about maps of convex sets in r n are proved we outline their extensions to an infinite dimensional context such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps here we explore applications only in the finite dimensional context more specifically we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite difference schemes applied to systems of conservation laws in several space variables in particular we establish a final characterization of the invariant regions under the lax friedrichs scheme and also give sufficient conditions for the local invariance further we give sufficient conditions for the global and local invariance of regions under flux splitting finite difference schemes an example of the multi dimensional euler equations for non isentropic gas dynamics is discussed

involutive automorphisms and iwasawa decomposition of affine kac moody superalgebras the involutive automorphisms of affine kac moody superalgebras are computed from satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider a 1 0 1 and b 1 1 1 as representative examples for the purpose of illustration

hyperbolic systems with non diagonalisable principal part and variable multiplicities i well posedness in this paper we analyse the well posedness of the cauchy problem for a rather general class of hyperbolic systems with space time dependent coefficients and with multiple characteristics of variable multiplicity first we establish a well posedness result in anisotropic sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal namely the terms below the diagonal at a distance k to it must be of order k this setting also allows for the jordan block structure in the system second we give conditions for the schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied we give explicit details for the appearing conditions and constructions for 2 times 2 and 3 times 3 systems complemented by several examples

involutive automorphisms of the affine kac moody algebras al 1 for l or 3 by the matrix method the determination of all the conjugacy classes of involutive automorphisms of the affine kac moody algebras al 1 for l or 3 is presented using the matrix formulation of automorphisms of an affine kac moody algebra that was developed in a previous paper ibid vol 25 p 3955 1992

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

new airy type solutions of the ultradiscrete painleve ii equation with parity variables the q difference painleve ii equation admits special solutions written in terms of determinants whose entries are the general solution of the q airy equation an ultradiscrete limit of the special solutions is studied using the procedure of ultradiscretization with parity variables then we obtain new airy type solutions of the ultradiscrete painleve ii equation with parity variables and the solutions have a richer structure than the known solutions

the conjugacy classes of the involutive automorphisms of affine kac moody algebra a2 1 all the conjugacy classes of involutive automorphisms of the affine kac moody algebra a2 1 are obtained using the matrix formulation of automorphisms of an affine kac moody algebra that was developed in a previous paper

on two parameter solutions of simultaneous ultradiscrete painleve ii equation with parity variables we introduce a simultaneous ultradiscrete painleve ii equation with parity variables which is shown to be more suitable for studying two parameter solutions than the single second order ultradiscrete painleve ii equation with parity variables we investigate several types of two parameter solutions and the solutions which are related with the ultradiscrete limit of determinant type solutions of q painleve ii equation we introduce a simultaneous ultradiscrete painleve ii equation with parity variables which is shown to be more suitable for studying two parameter solutions than the single second order ultradiscrete painleve ii equation with parity variables we investigate several types of two parameter solutions and the solutions which are related with the ultradiscrete limit of determinant type solutions of q painleve ii equation

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

hyperbolic equations with non analytic coefficients we prove that the cauchy problem for a hyperbolic homogeneous equation with mathcal c infty coefficients depending on time is well posed in every gevrey class although in general it is not well posed in mathcal c infty provided the characteristic roots satisfy the condition lambda i t 2 lambda j t 2 leq m lambda i t lambda j t 2 quad i neq j

iwasawa decomposition of affine kac moody algebras using satake diagrams we carry out the iwasawa decomposition of the affine kac moody algebras within the general framework of involutive automorphisms which are determined with the help of the satake diagrams corresponding to these algebras the parabolic subalgebras essential for the construction of induced representations are computed from the langlands decomposition in particular we consider the untwisted versions viz a1 1 and a2 1 as representative examples for the purpose of illustration

the ultimate discretisation of the painleve equations abstract we present a systematic way to construct ultra discrete versions of the painleve equations starting from know discrete forms these ultra discrete equations are generalised cellular automata in the sense that the dependent variable takes only integer values the ultra discrete painleve equations have the properties characteristic of the continuous and discrete painleve s namely coalescence cascades particular solutions and auto backlund relations

nonstrictly hyperbolic nonlinear systems on etudie le caractere bien pose du probleme aux valeurs initiales localement en temps pour une classe de systemes hyperboliques non lineaires

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

some forms of kac moody algebras let k be an algebraically closed field of char 0 the unique up to isomorphism three dimensional simple lie algebra s1 2 k plays an important role in the theory of the semisimple lie algebras over k for example chevalley harish chandra and serre s construction of the semisimple lie algebras g over k corresponding to a dynkin diagram 9 can be viewed as glueing together copies of sl 2 k in lf but we can not choose the distinct tds in an arbitrary way we need some compatibility conditions between the tds attached to neighbouring vertices they are described by some scalars sij and for simplicity we restrict our attention in this definition to dynkin diagrams without cycles for both definitions we also make the additional hypothesis that the

on hyperbolic equations and systems with non regular time dependent coefficients in this paper we study higher order weakly hyperbolic equations with time dependent non regular coefficients the non regularity here means less than holder namely bounded coefficients as for second order equations in 14 we prove that such equations admit a very weak solution adapted to the type of solutions that exist for regular coefficients the main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter classical solutions are recovered as limit of very weak solutions finally by using a reduction to block sylvester form we conclude that any first order hyperbolic system with non regular coefficients is solvable in the very weak sense

invariants of real forms of affine kac moody lie algebras abstract the theory of vogan diagrams which are dynkin diagrams with an overlay of certain additional information allows one to give a rapid classification of finite dimensional real semisimple lie algebras and to make use of this classification in practice this paper develops a corresponding theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras in this case also the equivalence classes of vogan diagrams correspond to the isomorphism classes of almost compact real forms although the real forms of such algebras had already been classified the theory of vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

weakly hyperbolic systems with holder continuous coefficients abstract we study the cauchy problem for a hyperbolic system with multiple characteristics and non smooth coefficients depending on time we prove in particular that if the leading coefficients are holder continuous and the system has size m 3 then the problem is well posed in each gevrey class of exponent s m

algebraic conditions for hyperbolicity of systems of partial differential equations partial differential equations or systems of such equations are classified according to type such as elliptic parabolic hyperbolic etc hyperbolic systems with respect to the initial plane t 0 are those for which the initial value problem is well posed in the sense of hadamard see 8 general conditions that are both necessary and sufficient for hyperbolicity are complicated see 10 11 2 3 14 15 16 17 the situation however is rather simple for an m th order linear homogeneous system of equations with constant coefficients p left frac partial partial t frac partial partial x 1 frac partial partial x n right u 0 1 for a vector u u t x 1 x n u t x with n components here p is an n x n square matrix whose elements p ik are m th degree forms in their n 1 arguments we associate with 1 the matrix the symbol of 1 p lambda xi p lambda xi 1 cdot cdot cdot xi n pik lambda xi 1 cdot cdot cdot xi n 2 and the characteristic form q lambda xi q lambda xi 1 cdot cdot cdot xi n det p lambda xi 1 cdot cdot cdot xi n 3 of degree mn in its arguments

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

hyperbolic second order equations with non regular time dependent coefficients in this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients this means assuming that the coefficients are less regular than holder the characteristic roots are also allowed to have multiplicities for such equations we describe the notion of a very weak solution adapted to the type of solutions that exist for regular coefficients the construction is based on considering friedrichs type mollifiers of coefficients and corresponding classical solutions and their quantitative behaviour in the regularising parameter we show that even for distributional coefficients the cauchy problem does have a very weak solution and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist in concrete applications the dependence on the regularising parameter can be traced explicitly

direct ultradiscretization of ai and bi functions and special solutions for the painleve ii equation ultradiscrete ai and bi functions are directly derived through the ultradiscrete limit from q difference analogues of the ai and bi functions respectively an infinite number of identities among the number of restricted partitions are obtained as by products a direct relationship between a class of special solutions for the ultradiscrete painlev ii equation and those of the q painlev ii equation which have a determinantal structure is also established

on the gevrey well posedness of the cauchy problem for weakly hyperbolic equations of higher order abstract we shall consider the cauchy problem for weakly hyperbolic equations of higher order with coefficients depending only on time the regularities of the distinct characteristic roots and the multiple characteristic roots independently influence gevrey well posedness of the cauchy problem

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

ultradiscretization without positivity we present a new ultradiscretization approach which can be applied to discrete systems the solutions of which are not positive definite this was made possible thanks to an ansatz involving the hyperbolic sine function we apply this new procedure to simple mappings for the linear and homographic mappings we obtain ultradiscrete forms and explicitly construct their solutions two discrete painleve ii equations are also analysed and ultradiscretized we show how to construct the ultradiscrete analogues of their rational and airy type solutions