the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

painleve property and constants of the motion of the complex lorenz model the results of the painleve analysis and of a search for constants of the motion of the complex lorenz model are presented

integrability of two dimensional homogeneous potentials combining the extended painleve conjecture with yoshida s singularity and stability analyses 1986 it is shown that for two dimensional homogeneous potentials of degree 2m integrability restricts kowalevskaya exponents and integrability coefficients to discrete sets of values this result is made use of in the analysis of integrability of symmetric potentials with m 2 3 and 4 direct construction of additional first integrals is successful only in special cases which can be transformed to known integrable ones

new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

atlas of two dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral this paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant quadratic in velocities other than the hamiltonian the configuration space is in general a two dimensional riemannian or pseudo riemannian manifold and the determination of its geometry is a part of the process of solution forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential associated with terms linear in velocities in the lagrangian function of the system the last caused time irreversibility of the system we constructed 41 multiparameter integrable systems of the type described in the title mostly on riemannian manifolds they are mostly new and cover all previously known systems as special cases corresponding to special values of the parameters those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body in the last field we introduce a n

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

new multiple soliton like solutions to the generalized 2 1 dimensional kp equation a new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations pdes in a unified way the main idea of this method is to take full advantage of the riccati equation which has more new solutions more new multiple soliton like solutions are obtained for the the generalized 2 1 dimensional kp equation

proof of integrability for five wave interactions in a case with unequal coupling constants complete integrability is proved for a system of ode describing the double three wave interaction between five waves when the coupling constant in one triplet is twice as large as in the other known results from the painleve analysis for this case are combined with the yoshida kovalevskaya approach 1983 to gain insight about the degree of the missing first integral this integral of degree six is then obtained via a direct search based on irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure

multigrid acceleration of a relaxation procedure for the reduced navier stokes equations on applique la methode multigrille pour ameliorer la convergence d une methode de relaxation iterative pour la resolution numerique des equations de navier stokes reduites

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

towards classification of dimensional integrable equations integrability conditions i in this paper we attempt to extend the symmetry approach well developed in the case of 1 1 dimensional equations to the 2 1 dimensional case presence of nonlocal terms in symmetries and conservation laws is the main feature of integrable 2 1 dimensional equations we have introduced a concept of quasi local functions to characterize nonlocalities we have found a few first integrability conditions for a class of scalar equations in terms of quasi local functions and have demonstrated that they are suitable for testing integrability

regularity for an obstacle problem of hessian equations on riemannian manifolds abstract in this paper we study the regularity for solutions to an obstacle problem of hessian type fully nonlinear equations on riemannian manifolds as an application the existence of a c 1 1 solution is proved

painleve test of coupled gross pitaevskii equations a painleve p test of the coupled gross pitaevskii equations has been carried out with the result that the coupled equations pass the p test only if a special relation containing system parameters masses and scattering lengths is satisfied computer algebra is applied to evaluate the j 4 compatibility condition for admissible external potentials the appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling the connection with recent experiments related to the stability of two component bose einstein condensates of rb atoms is discussed

hessian measures iii abstract in this paper we continue previous investigations into the theory of hessian measures we extend our weak continuity result to the case of mixed k hessian measures associated with k tuples of k convex functions on domains in euclidean n space k 1 2 n applications are given to capacity quasicontinuity and the dirichlet problem with inhomogeneous terms continuous with respect to capacity or combinations of dirac measures

painleve analysis yoshida s theorems and direct methods in the search for integrable hamiltonians treating as an example two dimensional similarity invariant hamiltonians with polynomial velocity dependent potentials the authors show how painleve analysis yoshida s theorems and direct methods can be combined in the search for integrable hamiltonians new integrable systems are found one of which has a polynomial second invariant of fifth order in the momenta most of the necessary calculations have been performed by applying various reduce packages developed for that purpose

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

on the integrability of nonlinear dirac equations the integrability of nonlinear dirac equations is discussed applying recent results in soliton theory using the lie point transformation groups of the nonlinear dirac equations we reduce these partial differential equations to systems of ordinary differential equations and study whether these systems are integrable we also discuss whether lie backlund vector fields exist we conclude that the nonlinear dirac equations are not integrable

non integrability of homogeneous two dimensional hamiltonians with velocity dependent potential the non integrability of classes of homogeneous two dimensional hamiltonian systems with a polynomial velocity dependent potential is shown on the basis of ziglin s theorem an analytic expression for the trace of the relevant monodromy matrices is presented which fits the numerical data perfectly an application is made to fokker planck hamiltonians with quadratic and with cubic drift terms

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

integrability of restricted multiple three wave interactions ii coupling constants with ratios 1 and 2 the system of ordinary differential equations describing the multiple three wave interactions with a common pump daughter wave is proved to be completely integrable by obtaining the necessary 2n 1 independent first integrals in involution for the case when the coupling constants and the frequency mismatches have ratio 1 and or 2 this case was deemed integrable on the basis of a painleve analysis but a direct proof has been lacking for some time the first integrals are the n 1 quadratic manley rowe relations the cubic hamiltonian n 1 quartic integrals analogous to the ones needed for complete integrability in the case of equal coupling constants and detunings in all wave triads and a new sixth order integral involving all wave quantities the form of this last invariant was deduced from the recent result for the analogous interaction between five waves n 2 and essentially made possible by the proper use of irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

on the nonintegrability of yang mils potentials the nonintegrability of three hamiltonian systems obtained by simplifications of the yang mills classical field theory is proved by application of two theorems of yoshida 1987 1988 it is also shown that for the two four degree of freedom systems there exists a set of three independent integrals of motion in involution

new exact solutions of the complex lorenz equations the author presents two classes of particular exact solutions to the complex lorenz equations these solutions possess the painleve property the parameter range for which the above solutions hold includes as special cases two of the three parameter ranges for which the general solution of the complex lorenz equations has the painleve property

the obstacle problem for hessian equations on riemannian manifolds abstract in this paper we consider the obstacle problem for a class of fully nonlinear equations on riemannian manifolds the c 1 1 regularity of the greatest viscosity solutions is established

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

non integrability of su 2 yang mills and yang mills higgs systems the spherically symmetric time dependent su 2 yang mills equations and yang mills higgs equations are shown to be non integrable by using the weiss tabor and carnevale 1983 method of painleve analysis reduced equations corresponding to these systems are also found to be non integrable

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

the cavalcante tenenblat equation does the equation admits physical significance the main purpose of the given paper is to analyze a less studied third order non linear partial differential equation the so called cavalcante tenenblat equation cte in the following form u t u x 1 2 x x u x 3 2 since general class of solutions are of basic interest a complete characterization of the group properties is given the traveling wave ansatz restricts the solution manifold to special class of solutions and hence a generalize algorithm is necessary we determine the lie point symmetry vector fields and calculate similarity ansatze further we also derive a few non linear transformations and some similarity solutions are obtained explicitly due to the complexity of some similarity solutions a numerical procedure is of advantage moreover the non classical case potential symmetries is studied to the first time and further we show how the cte leads to approximate symmetries and we apply the method to the first time we call the disturbed equation the cte e equation and we show how to derive new class of solutions finally the equation does not pass the painleve test and is therefore not soluble by the inverse scattering transform method ist hence suitable alternative algebraic approaches are necessary to derive class of solutions explicitly

stochasticity in yang mills theory some stochastic properties of a subsystem of yang mills classical mechanics are studied which do not fulfill automatically gauss law a possibility not considered before however significant differences with other subsystems already studied are not found for this system

existence of exponentially unstable periodic solutions and the non integrability of homogeneous hamiltonian systems abstract in hamiltonian systems with polynomial potential of homogeneous degree characteristic multipliers floquet multipliers permit an explicit expression for straight line periodic solutions by use of ziglin s theorem it is shown that the existence of an exponentially unstable straight line periodic solution implies the non integrability of the system

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

the painleve property a modified boussinesq equation and a modified kadomtsev petviashvili equation abstract it is shown that the modified boussinesq equation and the modified kadomtsev petviashvili equation both of which are known to be completely integrable possess the painlev e property for differential equations as defined by weiss tabor and carnevale

symmetry properties and solutions of nonlinear dispersive thin film equations in three dimensions the lie point symmetries are obtained for each one of two nonlinear dispersive thin film equations in 2 1 variables by means of the invariants of the one and two dimensional subgroups all reductions to ordinary differential equations and to partial differential equations in two variables are obtained solutions are provided for a number of the reduced equations it is also ascertained that neither one of the two considered thin film equations is integrable in some cases solitary and periodic wave solutions having curved wavecrests are obtained

integrability of hamiltonians with polynomial potentials in this work we study dynamical systems with polynomial potentials such as those of henon heiles yang mills and various generalizations by means of the nonintegrability theory developed by the authors all these problems have also been investigated by using other theories like those proposed by ziglin yoshida morales or the painleve analysis in the examples considered our method allows us to reproduce with quite less work or even to improve the results obtained by other authors

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

interactive airfoil calculations with higher order viscous flow equations a critical examination of the use of higher order viscous flow equations in the computation of airfoil flowfields with a viscid inviscid interaction scheme is presented comparisons are made between experimental data and interactive solutions obtained using the boundary layer equations the second order boundary layer equations and the navier stokes equations with corresponding approximations to the viscid inviscid matching conditions for flows about symmetric and aft loaded airfoils

solution based grid adaptation through segmented multigrid domain decomposition a solution based grid adaptation method that combines elements of the multigrid method for solution acceleration and the domain decomposition philosophy for grid optimization is described unlike other solution based adaptive gridding schemes wherein the overhead of recomputing the grid and re evaluating the solution on the adapted grid leads to higher computational costs compared to a nonadapted calculation the present methodology reduces the computational time required to obtain the solution the computational effort involved in the present calculation is significantly lower than a nonadapted calculation that utilizes the multigrid method purely as a convergence acceleration tool in addition to convergence acceleration the multigrid framework provides a mechanism of information transfer from regions wherein grid refinement is specified to unrefined coarse grid regions the basis for domain decomposition in the current procedure is the variation in grid refinement requirements for each coordinate direction in different portions of the flow field the method is demonstrated herein on an efficient set of governing equations termed the reduced navier stokes equations applied in conjunction with a set of physical boundary conditions the governing equations are discretized through a pressure based flux splitting procedure that is uniformly applicable from incompressible to supersonic mach numbers

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

chaotic behaviour of a hamiltonian with a quartic potential the hamiltonian h x p 1 2 p12 1 2 p22 1 2 x12x22 is investigated the different aspects singular point analysis stability analysis numerical treatment etc for studying chaotic behaviour are discussed moreover the authors discuss the quantised version of the hamiltonian

on the integrability aspects of the one dimensional classical continuum isotropic biquadratic heisenberg spin chain the integrability aspects of a classical one dimensional continuum isotropic biquadratic heisenberg spin chain in its continuum limit up to order o a4 in the lattice parameter a are studied through a differential geometric approach the dynamical equation for the spin chain is expressed in the form of a higher order generalized nonlinear schrodinger equation gnlse an integrable biquadratic chain that is a deformation of the lower order continuum isotropic spin chain is identified by carrying out a painleve singularity structure analysis on the gnlse also through gauge analysis and its properties are discussed briefly for the nonintegrable chain the perturbed soliton solution is obtained through a multiple scale analysis

n soliton solutions to a dimensional integrable equation we give explicitly n soliton solutions of a 2c 1 dimensional equation xtc xxxz 4cxxzcxxz 2c 1 x zzz 4 d 0 this equation is obtained by unifying two directional generalizations of the potential kdv equation the closed ring with the potential kp equation and the calogero bogoyavlenskij schiff equation this equation is also a reduction of the kp hierarchy we also find the miura transformation which yields the same ring of the corresponding modified equations the study of higher dimensional integrable systems is one of the central themes in integrable systems a typical example of a higher dimensional integrable system is obtained by modifying the lax operators of a basic equation the potential kdv p kdv equation in this paper the lax pair of the p kdv equation have the form l x t d 2 xcx x t 1 t x t d l x t 3 2

first integrals and yoshida analysis of nahm s equations first integrals of nahm s equations for monopole solutions in yang mills theories are explicitly given for various special cases taking into account the scale invariance all kowalewski exponents are determined both the polynomial first integrals and the kowalewski exponents are obtained with the help of a reduce program package this leads to a completely consistent picture in the sense of the analysis recently proposed by yoshida 1983

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

integrable nonlocal hirota equations we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular with local structures either in time space or of rogue wave type we construct several new integrable systems corresponding to nonlocal versions of the hirota equation which is a particular example of higher order nonlinear schrodinger equations the integrability of the new models is established by providing their explicit forms of lax pairs or zero curvature conditions the two compatibility equations arising in this construction are found to be related to each other either by a parity transformation p by a time reversal t or a p t transformation possibly combined with a conjugation we construct explicit multisoliton solutions for these models by employing hirota s direct method as well as darboux crum transformations the nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior the multisoliton solutions are of varied type being for instance nonlocal in space nonlocal in time of time crystal type regular

an exp function method for new n soliton solutions with arbitrary functions of a 2 1 dimensional vcbk system in this paper the exp function method is generalized to construct n soliton solutions of the 2 1 dimensional variable coefficient broer kaup vcbk system as a result new and more general 1 soliton 2 soliton and 3 soliton solutions with arbitrary functions are obtained from which the uniform formulae of n soliton solutions are derived based on one of the obtained 2 soliton solution the fission and fusion phenomena between the peakons and solitary waves are investigated it is shown that the exp function method may provide us with a straightforward effective and alternative mathematical tool for generating n soliton solutions of nonlinear evolution equations in mathematical physics

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

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

non isospectral problem in 2 1 dimensions in this paper the singular manifold method allows us to obtain a non isospectral lax pair darboux transformations and miura transformations for an equation in 2 1 dimensions and its modified version in this way we can iteratively build different kinds of solutions with solitonic behaviour

classical su 2 yang mills higgs system time dependent solutions by similarity method a similarity analysis of the wu yang t hooft julia zee ansatz reduced system of nonlinear differential equations of classical su 2 yang mills higgs theory is presented this yields the similarity group g of the equations considering g and one of its subgroups denoted g1 some previously known time dependent solutions in the prasad sommerfield limit are generated two new time dependent solutions are also reported

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

symmetries and first integrals for dissipative systems the connection between symmetries time independent and time dependent first integrals for autonomous systems of first order ordinary differential equations is discussed several new results which have been obtained by applying a reduce package are reported

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points

numerical solution of incompressible flows by a marching multigrid nonlinear method on decrit un schema iteratif de resolution des equations de navier stokes pour un systeme de coordonnees orthogonales curvilignes l algorithme propose est independant du nombre de reynolds et peut etre applique a la resolution des equations d euler incompressibles

on a coupled system of equations describing pulse propagation in quadratic media in the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic in the governing equations we keep linear terms that account for both second and third order dispersion and nonlinear terms describing both nonlinear dispersion and self steepening of the pulse edge we then perform the painleve singularity structure analysis of the most general system of coupled partial differential equations we derived in a specific case when third order dispersion is negligible by using a hirota like method we found zero and one parameter families of bright fundamental frequency and dark second harmonic solitary waves which travel at a locked group velocity