non linear pdes for gap probabilities in random matrices and kp theory abstract airy and pearcey like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials possibly multiplied with a rational function in this work it is shown that such kernels are intimately related to wave functions for polynomial gelfand dickey reductions or rational reductions of the kp hierarchy their fredholm determinant also satisfies linear pdes virasoro constraints yielding in a systematic way non linear pdes for the fredholm determinant of such kernels examples include fredholm determinants giving the gap probability of some infinite dimensional diffusions like the airy process with or without outliers and the pearcey process with or without inliers

wronskian solutions of the constrained kadomtsev petviashvili hierarchy the integrable kadomtsev petviashvili kp hierarchy is compatible with generalized k constraints of the type lk i qi 1xri a large class of solutions among them solitons can be represented by wronskian determinants of functions satisfying a set of linear equations in this paper we shall obtain additional conditions for these functions imposed by the constraints

an analytic description of the vector constrained kp hierarchy in this paper we give a geometric description in terms of the grassmann manifold of segal and wilson of the reduction of the kp hierarchy known as the vector k constrained kp hierarchy we also show in a geometric way that these hierarchies are equivalent to krichever s general rational reductions of the kp hierarchy

on the constrained b type kadomtsev petviashvili hierarchy hirota bilinear equations and virasoro symmetry we derive the bilinear equations of the constrained bkp hierarchy from the calculus of pseudodifferential operators the full hierarchy equations can be expressed in hirota s bilinear form characterized by the functions and besides we also give a modification of the original orlov schulman additional symmetry to preserve the constrained form of the lax operator for this hierarchy the vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless virasoro algebra

a mixed variational formulation based on exact intrinsic equations for dynamics of moving beams abstract a nonlinear intrinsic formulation for the dynamics of initially curved and twisted beams in a moving frame is presented the equations are written in a compact matrix form without any approximations to the geometry of the deformed beam reference line or to the orientation of the intrinsic cross section frame in accordance with previously published work when effects of warping on the dynamics and local constraints on the cross sectional deformation are ignorable the in and out of plane st venant warping displacements which are fully coupled for nonhomogeneous anisotropic beams need only be included explicitly in the determination of a suitable elastic law a two dimensional problem and need not be considered explicitly in the one dimensional equations governing global deformation in this paper it is presupposed that such an elastic law is given as a one dimensional strain energy function thus the present equations which are based on only six generalized strain variables are valid for beams with closed cross sections and for which warping is unrestrained when simplified for various special cases they agree with similar intrinsic equations in the literature although the resulting equations are newtonian in structure closely resembling euler s dynamical equations for a rigid body the formulation adheres to a variational approach throughout thus providing a link between newtonian and energy based methods in particular the present development provides substantial insight into the relationships among variational formulations in which different displacement and rotational variables are used as well as between these formulations and newtonian ones for computational purposes a compact and complete mixed variational formulation is presented that is ideally suited for development of finite element analyses finally a specialized version of the intrinsic equations is developed in which shear deformations is suppressed

on solutions of constrained kp equations we derive solutions of general wronskian form for the vector constrained kp hierarchies as one explicit example we discuss rational solutions in order to introduce our method we give a direct elementary proof of the existence of wronskian solutions for the l modified kp hierarchies l 0 1

method of squared eigenfunction potentials in integrable hierarchies of kp type the method of squared eigenfunction potentials sep is developed systematically to describe and gain new information about the kadomtsev petviashvili kp hierarchy and its reductions interrelation to the function method is discussed in detail the principal result which forms the basis of our sep method is the proof that any eigenfunction of the general kp hierarchy can be represented as a spectral integral over the baker akhiezer ba wave function with a spectral density expressed in terms of sep in fact the spectral representations of the adjoint ba functions can in turn be considered as defining equations for the kp hierarchy the sep method is subsequently used to show how the reduction of the full kp hierarchy to the constrained kp ckp rm hierarchies can be given entirely in terms of linear constraint equations on the pertinent functions the concept of sep turns out to be crucial in providing a description of ckp rm hierarchies in the language of the universal sato grassmannian and finding the non isospectral virasoro symmetry generators acting on the underlying functions the sep method is used to write down generalized binary darboux backlund transformations for constrained kp hierarchies whose orbits are shown to correspond to a new toda model on a square lattice as a result we obtain a series of new determinant solutions for the functions generalizing the known wronskian multi soliton solutions finally applications to random matrix models in condensed matter physics are briefly discussed

constrained kp models as integrable matrix hierarchies we formulate the constrained kp hierarchy denoted by ckp k 1 m as an affine sl m k 1 matrix integrable hierarchy generalizing the drinfeld sokolov hierarchy using an algebraic approach including the graded structure of the generalized drinfeld sokolov hierarchy we are able to find several new universal results valid for the ckp hierarchy in particular our method yields a closed expression for the second bracket obtained through dirac reduction of any untwisted affine kac moody current algebra an explicit example is given for the case sl m k 1 for which a closed expression for the general recursion operator is also obtained we show how isospectral flows are characterized and grouped according to the semisimple non regular element e of sl m k 1 and the content of the center of the kernel of e

regularity of solutions to axisymmetric navier stokes equations with a slightly supercritical condition abstract consider an axisymmetric suitable weak solution of 3d incompressible navier stokes equations with nontrivial swirl v v r e r v e v z e z let z denote the axis of symmetry and r be the distance to the z axis if the solution satisfies a slightly supercritical assumption that is v c ln ln r r for 0 0 028 when r is small then we prove that v is regular this extends the results in 6 16 18 where regularities under critical assumptions such as v c r were proven as a useful tool in the proof of our main result an upper bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper

on a reduction of the multi component kp hierarchy we show that the constrained kp hierarchies and their generalizations are natural reductions of the multi component kp hierarchy and that particular solutions of these hierarchies are obtained in a straightforward way from that of the multi component kp hierarchy

a strong regularity result for parabolic equations we consider a parabolic equation with a drift term u b u ut 0 under the condition divb 0 we prove that solutions possess dramatically better regularity than those provided by standard theory for example we prove continuity of solutions when not even boundedness is expected

solutions for the vector k constrained kp hierarchy a class of 1 1 dimensional integrable systems which are constrained from the kadomtsev petviashvili kp hierarchy is studied the bilinear equations of these integrable systems are shown first and then by using the free fermion operators their rational and soliton solutions which are closely related to the functions of the kp hierarchy are obtained the result implies that a large family of solutions of the 2 1 dimensional kp hierarchy can be obtained by solving these 1 1 dimensional integrable systems in particular the n soliton solutions of the kp hierarchy can be deduced from the soliton solutions of these 1 1 dimensional integrable systems

resonant overlap phenomena in stochastic layers of non linear hamiltonian systems with periodical excitations an accurate standard map approach instead of the approximate standard map approach is developed for the analytical prediction of the onset of a new resonant overlap forming a new resonant separatrix web in the stochastic layer this accurate approach applies to the twin well duffing oscillator for illustration of the resonant overlap phenomena in stochastic layers the excitation strengths are obtained analytically for the onset of the specified resonant separatrix webs in the stochastic layer this accurate approach gives a very good prediction compared to numerical results for the lower order resonant overlap when excitation strength is very weak the accurate and approximate standard map methods are in good agreement and the accurate one is applicable to non linear systems with strong excitations however the further improvement should be completed for a more accurate analytical prediction of the onset of a new resonant overlap in stochastic layers of non linear hamiltonian systems with periodical excitations

chaotic motion in the resonant separatrix bands of a mathieu duffing oscillator with a twin well potential the m 1 resonant bands in the left and right potential wells are skew symmetric and the 2m 1 resonant bands of the large orbit motion are symmetric the analytical conditions for the onset and destruction of a resonant band are developed through the incremental energy approach the numerical predictions of such onset and destruction are also completed by the energy increment spectrum method the sub resonance interaction occurs for strong excitations which needs to be further investigated these results are applicable to the small and large orbit motions of post buckled structure under a parametric excitation

analytic solution of the system of two coupled differential equations with the fifth order non linearity abstract in this paper an approximate method for solving a system of two coupled second order non linear differential equation with strong and weak non linearity is developed the strong non linearities are of the third and of the fifth order the suggested new method is partially based on the elliptic krylov bogolubov method and on the power series method developed for solving the weak non linear differential equations the one frequency solution is considered for the case of certain relations between the equation parameters p and m and initial conditions as a special case the oscillatory motion of the physical system described with a fourth order strong non linear differential equation is considered a wide analysis of the corresponding equation is done the results of the suggested analytic procedure are compared with numerical ones they show a good agreement

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

a fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three dimensional stress effects abstract a new approach is used to develop a geometrically exact nonlinear beam model for naturally curved and twisted solid composite rotor blades undergoing large vibrations in three dimensional space a combination of the new concepts of local displacements and local engineering stresses and strains a new interpretation and manipulation of the virtual local rotations an exact coordinate transformation and the extended hamilton principle is used to derive six fully nonlinear equations of motion describing one extension two bending one torsion and two shearing vibrations of composite beams the formulation is based on an energy approach but the derivation is fully correlated with the newtonian approach and provides a straightforward explanation of all nonlinear structural terms without using complex tensor operations or asymptotic expansions the theory accounts for in plane and out of plane warpings due to bending extensional shearing and torsional loadings elastic couplings among warpings and three dimensional stress effects by using the results of a two dimensional static sectional finite element analysis also the theory fully accounts for extensionality initial curvatures and geometric nonlinearities the equations display linear elastic couplings due to structural anisotropy and initial curvatures and nonlinear geometric couplings the theory contains most of the existing beam theories as special cases and the final equations of motion are put in compact matrix form

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

on the regularity of the axisymmetric solutions of the navier stokes equations we obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional navier stokes equations with nonzero swirl in particular we prove that the integrability of single component of vorticity or velocity fields in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions to obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the navier stokes equations

symmetry reductions of the bkp hierarchy a general symmetry of the bilinear bkp hierarchy is studied in terms of tau functions we use this symmetry to define reductions of the bkp hierarchy among which new integrable systems can be found the reductions are connected to constraints on the lax operator as well as on the bilinear formulation a class of solutions for the reduced equations is derived

a liouville theorem for the axially symmetric navier stokes equations let v x t vrer v e vzez be a solution to the three dimensional incompressible axially symmetric navier stokes equations denote by b vrer vzez the radial axial vector field under a general scaling invariant condition on b we prove that the quantity rv is holder continuous at r 0 t 0 as an application we prove that the ancient weak solutions of axi symmetric navier stokes equations must be zero which was raised by koch nadirashvili seregin and sverak 2009 in 15 and seregin and sverak 2009 in 26 as a conjecture under the condition that b l 0 t bmo 1 as another application we prove that if b l 0 t bmo 1 then v is regular

analytical predictions of chaos in a non linear rod the resonant and chaotic conditions for non dampened non linear planar rods are developed through the chirikov criterion and the subharmonic bifurcation conditions for weakly dampened non linear planar rods are also presented through the melnikov method the analytical conditions are based on a simply supported geometrically non linear planar rod model with a specific single mode response but these conditions are applicable to geometrically non linear planar rods with different supports chaos and transient motion from chaos to periodic motion in the non linear rod are simulated through the approximate rod model and they are illustrated through the poincare mapping section

a priori bound on the velocity in axially symmetric navier stokes equations let v be the velocity of leray hopf solutions to the axially symmetric three dimensional navier stokes equations under suitable conditions for initial values we prove the following a priori bound v x t le frac c ln r 1 2 r 2 qquad 0 r le 1 2 where r is the distance from x to the z axis and c is a constant depending only on the initial value this provides a pointwise upper bound worst case scenario for possible singularities while the recent papers chiun chuan et al commun pde 34 1 3 203 232 2009 koch et al acta math 203 1 83 105 2009 gave a lower bound the gap is polynomial order 1 modulo a half log term

affine lie algebraic origin of constrained kp hierarchies an affine sl n 1 algebraic construction of the basic constrained kp hierarchy is presented this hierarchy is analyzed using two approaches namely linear matrix eigenvalue problem on hermitian symmetric space and constrained kp lax formulation and it is shown that these approaches are equivalent the model is recognized to be the generalized non linear schrodinger gnls hierarchy and it is used as a building block for a new class of constrained kp hierarchies these constrained kp hierarchies are connected via similarity backlund transformations and interpolate between gnls and multi boson kp toda hierarchies our construction uncovers the origin of the toda lattice structure behind the latter hierarchy

a geometrically exact rod model incorporating shear and torsion warping deformation abstract a fully nonlinear three dimensional rod model is developed that incorporates transverse shear and torsion warping deformation the geometric setting is that of a constrained body model with configuration space modeled on r3 so 3 r a differentiable manifold the proposed model incorporates the classical notion of bi moment and bi shear in a fully nonlinear geometrically exact context explicit properly invariant constitutive equations that generalize those of the linearized theory are developed the underlying variational formulation of the model is discussed and computational procedures employing a galerkin projection are addressed numerical examples are presented that illustrate the performance of the formulation

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

solutions of the wdvv equations and integrable hierarchies of kp type we show that reductions of kp hierarchies related to the loop algebra of sl n with homogeneous gradation give solutions of the darboux egoroff system of pde s using explicit dressing matrices of the riemann hilbert problem generalized to include a set of commuting additional symmetries we construct solutions of the witten dijkgraaf e verlinde h verlinde equations

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

modifying the kp the n th constrained kp hierarchies and their hamiltonian structures the kadomtsev petviashvili kp hierarchy has infinitely many hamiltonian pairs then th pair of them is associated withl n wherel is the pseudodifferential operator pdo 3 4 in this paper by the factorizationl n l n l 1 withl j j 1 n being the independent pdos we construct the miura transformation for the kp which leads to a decomposition of the second hamiltonian structure in then th pair to a direct sum each term in the sum is the second structure in the initial pair associated withl j when we impose a constraint 1 9 i e a new type of reduction to the kp hierarchy we obtain the similar results for the constrained kp hierarchy in particular the second hamiltonian structure for this hierarchy is transformed to a vastly simpler one

decay and vanishing of some axially symmetric d solutions of the navier stokes equations abstract we study axially symmetric d solutions of the 3 dimensional navier stokes equations the first result is an a priori decay estimate of the velocity for general domains the second is an a priori decay estimate of the vorticity in r 3 which improves the corresponding results in the literature in addition we prove a similar decay of full 3d solutions except for a small set of angles next we turn to d solutions which are periodic in the third variable and prove vanishing result under a reasonable condition as a corollary we prove that axially symmetric d solutions in the slab r 2 i with suitable boundary condition is 0 here i is any finite interval to the best of our knowledge this seems to be the first vanishing result on a 3 dimensional d solution without extra integral or decay or smallness assumption on the solution the tools used include brezis gallouet inequality dimension reduction scaling green s function bound and liouville theorems for navier stokes equations

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

constraints of the kp hierarchy and multilinear forms we consider the trilinear form of the kaup broer system which gives rise to solutions in wronskian form the kaup broer system is connected with akns system through a gauge transformation the akns hierarchy can be understood as a generalized 1 constraint of the kp hierarchy imposing that constraint on sato s equation we obtain the basic trilinear form and moreover a hierarchy of trilinear equations governing the akns flows similary hierarchies of multilinear forms are derived in the case of generalized k constraints

an algebraic description of generalized k constraints the generalized k constrained kp hierarchy is shown to correspond to a so called pseudo reduction of the two dimensional toda lattice hierarchy described in a free fermion approach which is adapted to the case of two singularities in the spectral parameter range wronskian solutions are discussed and in particular soliton solutions are recovered through a reduction of the toda solitons

dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl in this paper we study the dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl to this purpose we propose a new one dimensional model that approximates the navier stokes equations along the symmetry axis an important property of this one dimensional model is that one can construct from its solutions a family of exact solutions of the three dimensionafinal navier stokes equations the nonlinear structure of the one dimensional model has some very interesting properties on one hand it can lead to tremendous dynamic growth of the solution within a short time on the other hand it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time by exploiting this special nonlinear structure we prove the global regularity of the three dimensional navier stokes equations for a family of initial data whose solutions can lead to large dynamic growth but yet have global smooth solutions