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

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

a non local regularization of first order hamilton jacobi equations abstract in this paper we investigate the regularizing effect of a non local operator on first order hamilton jacobi equations we prove that there exists a unique solution that is c 2 in space and c 1 in time in order to do so we combine viscosity solution techniques and green s function techniques viscosity solution theory provides the existence of a w 1 solution as well as uniqueness and stability results a duhamel s integral representation of the equation involving the green s function permits to prove further regularity we also state the existence of c solutions in space and time under suitable assumptions on the hamiltonian we finally give an error estimate in l norm between the viscosity solution of the pure hamilton jacobi equation and the solution of the integro differential equation with a vanishing non local part

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

rational solutions of the master symmetries of the kdv equation in this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for kdv the result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master symmetry kdv hierarchy are bispectral potentials for the schrodinger operator by bispectral potentials we mean that the corresponding schrodinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable this complements certain results of airault mckean moser 4 duistermaat grunbaum 10 and magri zubelli 40 as a consequence of bispectrality the rational solutions of the master symmetries turn out to be solutions of a generalized string equation

soliton structure versus singularity analysis third order completely intergrable nonlinear differential equations in 1 1 dimensions abstract backlund transformations between all known completely integrable third order differential equations in 1 1 dimensions are established and the corresponding transformations formulas for their hereditary operators and hamiltonian formulations are exhibited some of these backlund transformations are not injective therefore additional non commutative symmetry groups are found for some equations these non commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra a 1 1 new completely integrable third order integro differential equations some depending explicitly on x are given these new equations give rise to nonin equation connections between the singularity equations from the painleve analysis and the nonlinear equations for interacting solitons are established a common approach to singularity analysis and soliton structure is introduced the painleve analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine gordon and nonlinear schrodinger equation a method to recover the painleve series from its constant level term is exhibit the soliton singularity transform is recognized to be connected to the mobius group this gives rise to a darboux like result for the spectral properties of the recursion operator these connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton singularity transform this then leads to the result that only for scaling degrees 1 and 2 the usual painleve analysis can be carried out a new invariance principle connected to kernels of differential operators is discovered this new invariance for example connects the explicit solutions of the liouville equation with the miura transform simple methods are exhibited which allow to compute out of n soliton solutions of the kdv bargman potentials explicit solutions of equations like the harry dym equation certain solutions are plotted

min max formulas for nonlocal elliptic operators on euclidean space abstract an operator satisfies the global comparison property if anytime a function touches another from above at some point then the operator preserves the ordering at the point of contact this is characteristic of degenerate elliptic operators including nonlocal and nonlinear ones in previous work the authors considered such operators in riemannian manifolds and proved they can be represented by a min max formula in terms of levy operators in this note we revisit this theory in the context of euclidean space with the intricacies of the general riemannian setting gone the ideas behind the original proof of the min max representation become clearer moreover we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

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

some free boundary problems recast as nonlocal parabolic equations abstract in this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold the canonical examples would be one phase hele shaw flow as well as its two phase analog we also treat nonlinear versions of both one and two phase problems in the special class of free boundaries that are graphs over r d we give a precise characterization that shows their motion is equivalent to that of a solution of a nonlocal fractional nonlinear parabolic equation for functions on r d our main observation is that the free boundary condition defines a nonlocal operator having what we call the global comparison property a consequence of the connection with nonlocal parabolic equations is that for free boundary problems arising from translation invariant elliptic operators in the positive and negative phases one obtains in a uniform treatment for all of the problems one and two phase a propagation of modulus of continuity for viscosity solutions of the free boundary flow

the action angle transformation for soliton equations abstract for third order completely integrable equations in 1 1 dimensions canonical transformations which map in the multi soliton case gradients of action variables to gradients of angle variables are constructed furthermore this action angle map is shown always to be an infinitesimal symmetry group generator of the interacting soliton equation

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

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

further time regularity for nonlocal fully nonlinear parabolic equations we establish holder estimates for the time derivative of solutions of nonlocal parabolic equations under mild assumptions for the boundary data as a consequence we are able to extend the evans krylov estimate for rough kernels to parabolic equations 2016 wiley periodicals inc

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

regularization by 12 laplacian and vanishing viscosity approximation of hjb equations abstract we investigate the regularizing effect of adding small fractional laplacian with critical fractional exponent 1 2 to a general first order hjb equation our results include some regularity estimates for the viscosity solutions of such perturbations making the solutions classically well defined most importantly we use these regularity estimates to study the vanishing viscosity approximation to first order hjb equations by 1 2 laplacian and derive an explicit rate convergence for the vanishing viscosity limit

holder estimates for non local parabolic equations with critical drift abstract in this paper we extend previous results on the regularity of solutions of integro differential parabolic equations the kernels are non necessarily symmetric which could be interpreted as a non local drift with the same order as the diffusion we provide a growth lemma and a harnack inequality which can be used to prove higher regularity estimates

lipschitz regularity for censored subdiffusive integro differential equations with superfractional gradient terms in this paper we are interested in integro differential elliptic and parabolic equations involving nonlocal operators with order less than one and a gradient term whose coercivity growth makes it the leading term in the equation we obtain lipschitz regularity results for the associated stationary dirichlet problem in the case when the nonlocality of the operator is confined to the domain feature which is known in the literature as censored nonlocality as an application of this result we obtain strong comparison principles which allow us to prove the well posedness of both the stationary and evolution problems and steady ergodic large time behavior for the associated evolution problem

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

fractal hamilton jacobi kpz equations nonlinear and nonlocal evolution equations of the form u t lu u q where c is a pseudodifferential operator representing the infinitesimal generator of a levy stochastic process have been derived as models for growing interfaces in the case when the continuous brownian diffusion surface transport is augmented by a random hopping mechanism the goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the diffusive linear and hyperbolic nonlinear terms posed in the whole space r n and supplemented with nonnegative bounded and sufficiently regular initial conditions

a new regularization mechanism for the boltzmann equation without cut off we apply recent results on regularity for general integro differential equations to derive a priori estimates in holder spaces for the space homogeneous boltzmann equation in the non cut off case we also show an a priori estimate in l infty which applies in the space inhomogeneous case as well provided that the macroscopic quantities remain bounded

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries