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

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid abstract in this paper the systems of carbon nanotubes cnts conveying fluid are considered by the combination of lie symmetry analysis and the dynamical system method all of the geometric vector fields of the three systems are obtained then the symmetry reductions and exact solutions to the systems are investigated by the dynamical system and the power series method especially the convergence of the power series solution to the nonlinear system is shown under the condition which is easily satisfied generally

breakdown of smoothness for the muskat problem in this paper we show that there exists analytic initial data in the stable regime for the muskat problem such that the solution turns to the unstable regime and later breaks down that is no longer belongs to c4

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

non uniqueness of admissible weak solutions to compressible euler systems with source terms abstract we consider admissible weak solutions to the compressible euler system with source terms which include rotating shallow water system and the euler system with damping as special examples in the case of anti symmetric sources such as rotations for general piecewise lipschitz initial densities and some suitably constructed initial momentum we obtain infinitely many global admissible weak solutions furthermore we construct a class of finite states admissible weak solutions to the euler system with anti symmetric sources under the additional smallness assumption on the initial densities we also obtain multiple global in time admissible weak solutions for more general sources including damping the basic framework are based on the convex integration method developed by de lellis and szekelyhidi 13 14 for the euler system one of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term

global ill posedness of the isentropic system of gas dynamics we consider the isentropic compressible euler system in 2 space dimensions with pressure law p 2 and we show the existence of classical riemann data i e pure jump discontinuities across a line for which there are infinitely many admissible bounded weak solutions bounded away from the void we also show that some of these riemann data are generated by a 1 dimensional compression wave our theorem leads therefore to lipschitz initial data for which there are infinitely many global bounded admissible weak solutions c 2015 wiley periodicals inc

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

the sine gordon equations complete and partial integrability the sine gordon equation in one space one time dimension is known to possess the painleve property and to be completely integrable it is shown how the method of singular manifold analysis obtains the backlund transform and the lax pair for this equation a connection with the sequence of higher order kdv equations is found the modified sine gordon equations are defined in terms of the singular manifold these equations are shown to be identically painleve also certain rational solutions are constructed iteratively the double sine gordon equation is shown not to possess the painleve property however if the singular manifold defines an affine minimal surface then the equation has integrable solutions this restriction is termed partial integrability the sine gordon equation in n 1 variables n space 1 time where n is greater than one is shown not to possess the painleve property the condition of partial integrability requires the singular manifold to be an einstein space with null scalar curvature the known integrable solutions satisfy this constraint in a trivial manner finally the coupled kdv or hirota satsuma equations possess the painleve property the associated modified equations are derived and from these the lax pair is found

on admissibility criteria for weak solutions of the euler equations we consider solutions to the cauchy problem for the incompressible euler equations satisfying several additional requirements like the global and local energy inequalities using some techniques introduced in an earlier paper we show that for some bounded compactly supported initial data none of these admissibility criteria singles out a unique weak solution as a byproduct in more than one space dimension we show bounded initial data for which admissible solutions to the p system of isentropic gas dynamics in eulerian coordinates are not unique

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

global existence of quasi stratified solutions for the confined ipm equation in this paper we consider a confined physical scenario to prove the global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media ipm equation the result is proved using the stability of stratified solutions combined with an additional structure of our initial perturbation which allows us to get rid of the boundary terms in the energy estimates

lie symmetry analysis and exact explicit solutions for general burgers equation in this paper the lie symmetry analysis is performed for the general burgers equation the exact solutions and similarity reductions generated from the symmetry transformations are provided furthermore the all exact explicit solutions and similarity reductions based on the lie group method are obtained some new method and techniques are employed simultaneously such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science

on the one and two dimensional toda lattices and the painleve property the toda lattice and the two dimensional toda lattice 2 dtl are shown to possess a type of painleve property that is based on the use of separate singular manifolds for each dependent variable the isospectral problem for the 2 dtl found by both mikhailov and by fordy and gibbons can be simply and logically derived from this analysis some remarks are made about the connection between our work and independent work of kametaka and airhault on the relationship between the toda lattice and the second painleve transcendent

spectral analysis of strongly propagative systems on considere la theorie spectrale et la theorie de diffusion pour des systemes symetriques d ordre 1 du type a x t x t l 1 n b t t x x l 0 ou t x est une fonction a valeurs c m sur r n 1 be 1 l n sont des matrices symetriques m m et a x une fonction sur r m avec des valeurs sur les matrices m m symetriques

high dimensionality and h principle in pde in this note we present an analyst s point of view on the nash kuiper theorem and in particular highlight the very close connection to turbulence a paradigm example of a high dimensional phenomenon our aim is to explain recent applications of nash s ideas in connection with the incompressible euler equations and onsager s famous conjecture on the energy dissipation in 3d turbulence

the algebraic complete integrability of geodesic flow on so n we study for which left invariant diagonal metrics onso n the euler arnold equations dot x x lambda x x x ij in so n lambda ij x ij lambda ij lambda ji can be linearized on an abelian variety i e are solvable by quadratures we show that merely by requiring that the solutions of the differential equations be single valued functions of complex timet suffices to prove that under a non degeneracy assumption on the metric the only such metrics are those which satisfy manakov s conditions ij bi bj ai aj 1 the case of degenerate metrics is also analyzed forn 4 this provides a new and simpler proof of a result of adler and van moerbeke 3

scaling exponents for active scalars we provide bounds for dirichlet quotients and for generalized structure functions for 3d active scalars and navier stokes equations these bounds put constraints on the possible extent of anomalous scaling

renormalization of active scalar equations abstract we consider transport equations with an incompressible transporting vector field whereas smooth solutions of such equations conserve every l p norm simply by the chain rule the question arises how regular a weak solution needs to be to guarantee this conservation property whereas the classical diperna lions theory gives sufficient conditions in terms of the regularity of the coefficients with no regularity requirement on the transported scalar we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar this is motivated by the case of active scalar equations where the transporting vector field has the same regularity as the transported scalar we use commutator estimates similar to those of constantin e titi in the context of onsager s conjecture but we require novel arguments to handle the case of l p norms when p 2

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

on one dimensional forward backward diffusion equations with linear convection and reaction abstract we study the initial neumann boundary value problem for a class of one dimensional forward backward diffusion equations with linear convection and reaction the diffusion flux function is assumed to contain two forward diffusion phases we prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases furthermore we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant the results are new even for the pure forward backward diffusion problem without convection and reaction our primary approach relies on a combination of the convex integration and baire s category methods for a related nonlocal differential inclusion

convex integration for a class of active scalar equations we show that a general class of active scalar equa tions including porous media and certain magnetostrophic tur bulence models admit non unique weak solutions in the class of bounded functions the proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in 2 3

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

painleve test generalized symmetries backlund transformations and exact solutions to the third order burgers equations in this paper the painleve analysis is performed on the physical form of the third order burgers equation the painleve property and integrability c integrable of the equation is verified then the generalized symmetries of the equation are presented and the generalized symmetries of the other equation are given by the symmetry transformation method the backlund transformations of the equations are constructed based on the symmetries respectively furthermore the exact explicit solutions to the equations are investigated in terms of the symmetries backlund transformations and transformations of the equations

spectral and scattering theory for symmetric systems of nonconstant deficit abstract we study the spectral and scattering problems for symmetric systems of nonconstant deficit we prove i discreteness of eigenvalues ii absolute continuity of continuous spectrum iii principle of limiting absorption as an application to the scattering problems the decay of local energy and the completeness of wave operators are obtained the class of systems we consider here includes the linearized equation of magnetohydrodynamics as a typical example

spectral theory for symmetric systems in an exterior domain ii abstract our work is devoted to the investigation of spectral theory for symmetric systems with certain coercive boundary conditions in an exterior domain we verify the discreteness of non zero eigenvalues and the principle of limiting absorption

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

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

lie symmetry analysis and exact solutions for the short pulse equation abstract in this paper the lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation spe the symmetries for this equation are given for the traveling wave solutions the exact parametric representations are investigated to guarantee the existence of the above solutions all parameter conditions are determined furthermore the exact analytic solutions are obtained by using the power series method

weak solutions with decreasing energy of incompressible euler equations weak solution of the euler equations is an l2 vector field u x t satisfying certain integral relations which express incompressibility and the momentum balance our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations as both viscosity and compressibility tend to zero thus we believe that weak solutions describe turbulent flows with very high reynolds numbers

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

on the energy dissipation rate of solutions to the compressible isentropic euler system in this paper we extend and complement the results in chiodaroli et al global ill posedness of the isentropic system of gas dynamics 2014 on the well posedness issue for weak solutions of the compressible isentropic euler system in 2 space dimensions with pressure law p 1 first we show that every riemann problem whose one dimensional self similar solution consists of two shocks admits also infinitely many two dimensional admissible bounded weak solutions not containing vacuum generated by the method of de lellis and szekelyhidi ann math 170 1417 1436 2009 arch ration mech anal 195 225 260 2010 moreover we prove that for some of these riemann problems and for 1 3 such solutions have a greater energy dissipation rate than the self similar solution emanating from the same riemann data we therefore show that the maximal dissipation criterion proposed by dafermos in j diff equ 14 202 212 1973 does not favour the classical self similar solutions

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

h principles for the incompressible euler equations in dissipative euler flows and onsager s conjecture arxiv 1205 3626 preprint 2012 de lellis and szekelyhidi construct holder continuous dissipative weak solutions to the incompressible euler equations in the torus mathbb t 3 the construction consists of adding fast oscillations to the trivial solution we extend this result by establishing optimal h principles in two and three space dimensions specifically we identify all subsolutions defined in a suitable sense which can be approximated in the h 1 norm by exact solutions furthermore we prove that the flows thus constructed on mathbb t 3 are genuinely three dimensional and are not trivially obtained from solutions on mathbb t 2

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

convex integration with linear constraints and its applications abstract we study solutions of the first order partial differential inclusions of the form u k where u r n r m and k is a set of m n real matrices and derive a companion version to the result of muller and sverak 16 concerning a general linear constraint on the components of u we then consider two applications the vectorial eikonal equation and a t 4 configuration both under linear constraints

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

cauchy problem for dissipative holder solutions to the incompressible euler equations we consider solutions to the cauchy problem for the incompressible euler equations on the 3 dimensional torus which are continuous or holder continuous for any exponent theta frac 1 16 using the techniques introduced in de lellis and szekelyhidi inventiones mathematicae 9 377 407 2013 dissipative euler flows and onsager s conjecture 2012 we prove the existence of infinitely many holder continuous initial vector fields starting from which there exist infinitely many holder continuous solutions with preassigned total kinetic energy

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

lack of uniqueness for weak solutions of the incompressible porous media equation in this work we consider weak solutions of the incompressible two dimensional porous media ipm equation by using the approach of de lellis szekelyhidi we prove non uniqueness for solutions in l in space and time

integrability characteristics of two dimensional generalizations of nls type equations a recent procedure based on truncated painleve expansions is used to derive lax pairs darboux transformations and various soliton solutions for integrable 2 1 generalizations of nls type equations in particular diverse classes of solutions are found analogous to the dromion instanton lump and ring soliton solutions derived recently for 2 1 korteweg de vries type equations the nizhnik novikov veselov equation and the 2 1 broer kaup system

separation of variables for the constrained flows the separation of variables for some constrained flows of soliton hierarchies are shown the lax matrix and r matrix either constant or dynamical type are used to construct the separation variables and separation equation for the constrained flows

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure