connection between the burgers equation with an elastic forcing term and a stochastic process in this paper a complete analytical resolution of the one dimensional burgers equation with the elastic forcing term k2x f t k is an element of r is presented two methods existing for the case k 0 are adapted and generalized using variable and functional transformations valid for all values of space and time the emergence of a fokker planck equation in the method allows the establishment of a connection between the burgers equation and the ornstein uhlenbeck process

premixed flame shapes and polynomials abstract the nonlinear nonlocal michelson sivashinsky equation for isolated crests of unstable flames is studied using pole decompositions as starting point polynomials encoding the numerically computed 2 n flame slope poles and auxiliary ones are found to closely follow a meixner pollaczek recurrence accurate steady crest shapes ensue for n 3 squeezed crests ruled by a discretized burgers equation involve the same polynomials such explicit approximate shapes still lack for finite n pole decomposed periodic flames despite another empirical recurrence

fluid dynamic description of flocking via the povzner boltzmann equation we introduce and discuss the possible dynamics of groups of indistinguishable agents which are interacting according to their relative positions with the aim of deriving hydrodynamic equations these models are developed to mimic the collective motion of groups of species such as bird flocks fish schools herds of quadrupeds or bacteria colonies our starting model for these interactions is the povzner equation 21 which describes a dilute gas in which binary collisions of elastic spheres depend on their relative positions following the cucker and smale model 9 we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance under the assumption of weak dissipation it is shown that the povzner equation is modified through a correction in the form of a nonlinear friction type operator using this correction we formally obtain from the povzner equation in a direct way a fluid dynamic description of a system of agents with weak dissipative interactions with a coefficient of restitution that depends on their relative distance

large scale dynamics of the persistent turning walker model of fish behavior this paper considers a new model of individual displacement based on fish motion the so called persistent turning walker ptw model which involves an ornstein uhlenbeck process on the curvature of the particle trajectory the goal is to show that its large time and space scale dynamics is of diffusive type and to provide an analytic expression of the diffusion coefficient two methods are investigated in the first one we compute the large time asymptotics of the variance of the individual stochastic trajectories the second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories the kinetic model is a fokker planck type equation posed in an extended phase space involving the curvature among the kinetic variables we show that both methods lead to the same value of the diffusion constant we present some numerical simulations to illustrate the theoretical results

dirac lie systems and schwarzian equations abstract a lie system is a system of differential equations admitting a superposition rule i e a function describing its general solution in terms of any generic set of particular solutions and some constants following ideas going back to dirac s description of constrained systems we introduce and analyze a particular class of lie systems on dirac manifolds called dirac lie systems which are associated with dirac lie hamiltonians our results enable us to investigate constants of the motion superposition rules and other general properties of such systems in a more effective way several concepts of the theory of lie systems are adapted to this dirac setting and new applications of dirac geometry in differential equations are presented as an application we analyze solutions of several types of schwarzian equations but our methods can be applied also to other classes of differential equations important for physics

sivashinsky equation for corrugated flames in the large wrinkle limit sivashinsky s acta astron 4 1177 1977 nonlinear integrodifferential equation for the shape of corrugated one dimensional flames is ultimately reducible to a 2n body problem involving the 2n complex poles of the flame slope thual frisch and henon j phys france 46 1485 1985 derived singular linear integral equations for the pole density in the limit of large steady wrinkles n 1 which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise here we solve those analytically for isolated crests next for monocoalesced then bicoalesced periodic flame patterns whatever the large amplitudes involved we compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole decomposition approach good agreement is obtained even for moderately large ns the results are extended to give hints as to the dynamics of supplementary poles open problems are evoked

avoiding collisions in flocks among the many models for flocking systems of interacting particles the one introduced by cucker and smale has attracted attention due to the fact that a convergence to flocking i e to a common velocity could be established depending on conditions on the initial state of the system in this note we extend this model by adding to it a repelling force between particles we show that for this modified model convergence to flocking is established along the same lines while in addition avoidance of collisions i e the respect of a minimal distance between particles is ensured

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

exact solutions for a forced burgers equation with a linear external force abstract we investigate the solutions of the burgers equation t u x t d x 2 u x t x f x t u x t u x t x u x t x t where f x t is an external force and x t represents a forcing term this equation is first analyzed in the absence of the forcing term by taking f x t k 1 t k 2 t x into account for this case the solution obtained extends the usual one present in the ornstein uhlenbeck process and depending on the choice of k 1 t and k 2 t it can present a stationary state or an anomalous spreading afterwards the forcing terms x t 1 t 2 t x and x t 3 x 4 x 3 are incorporated in the previous analysis and exact solutions are obtained for both cases

existence of piecewise weak solutions of a discrete cucker smale s flocking model with a singular communication weight abstract we prove existence of global c 1 piecewise weak solutions for the discrete cucker smale s flocking model with a non lipschitz communication weight s s 0 1 we also discuss the possibility of finite in time alignment of the velocities of the particles

functional separable solutions of nonlinear reaction diffusion equations with variable coefficients abstract the paper presents a number of new functional separable solutions to nonlinear reaction diffusion equations of the form c x u t a x u x x b x u x p x f u where f u is an arbitrary function it is shown that any three of the four variable coefficients a x b x c x p x of such equations can be chosen arbitrarily and the remaining coefficient can be expressed through the others examples of specific equations and their exact solutions are given the results obtained are generalized to more complex multidimensional nonlinear reaction diffusion equations with variable coefficients also some functional separable solutions to nonlinear reaction diffusion equations with delay u t u x x a x f u w w u x t where 0 is the delay time and f u w is an arbitrary function of two arguments are obtained it is important to note that the exact solutions of nonlinear pdes and delay pdes that contain arbitrary functions and therefore have sufficient generality are of the greatest practical interest for testing and evaluating the accuracy of various numerical and approximate analytical methods for solving corresponding initial boundary value problems

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

a new model for self organized dynamics and its flocking behavior we introduce a model for self organized dynamics which we argue addresses several drawbacks of the celebrated cucker smale c s model the proposed model does not only take into account the distance between agents but instead the influence between agents is scaled in term of their relative distance consequently our model does not involve any explicit dependence on the number of agents only their geometry in phase space is taken into account the use of relative distances destroys the symmetry property of the original c s model which was the key for the various recent studies of c s flocking behavior to this end we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems which covers systems with non symmetric influence matrices in particular we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with leaders

riccati ermakov systems and explicit solutions for variable coefficient reaction diffusion equations abstract we present several families of nonlinear reaction diffusion equations with variable coefficients including generalizations of fisher kpp and burgers type equations special exact solutions such as traveling wave rational triangular wave and n wave type solutions are shown by means of similarity transformations the variable coefficients are conditioned to satisfy riccati or ermakov systems of equations when the riccati system is used conditions are established so that finite time singularities might occur we explore solution dynamics across multi parameters in the supplementary material we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions

nondeformability of the complex hyperquadric we prove that the complex hyperquadric of dimension 3 does not allow nontrivial deformation we study the orbit structure of certain vector fields defined on the potential deformation and deduce the nondeformability from the attracting property of these vector fields

asymptotics of the largest zeros of some orthogonal polynomials we study the asymptotics of the largest zeros of the wilson hermite and q laguerre polynomials using two distinct techniques the first is based on the coulomb fluid technique developed in a previous paper where the primary input is the weight function while the second uses the method of chain sequences which supplies inequalities for the largest zeros using the recurrence coefficients we also investigate the asymptotics of the largest zeros of the polynomials orthogonal to the weight functions for c 0 and m a positive even integer

the generalized burgers and zabolotskaya khokhlov equations transformations exact solutions and qualitative properties a point transformation between forms of the generalized burgers equation gbe first given by cates 1989 is investigated applications include generalizations of scott s 1981 classification of long time behaviour for compressive wave solutions of the gbe and the equivalence of the exponential and cylindrical forms of the gbe yielding an exact solution for the exponential gbe applications to nonlinear diffractive acoustics are considered by using a similarity reduction of the dissipative zabolotskaya khokhlov dzk equation describing the evolution of nearly plane waves in a weakly nonlinear medium with allowance for transverse variation effects onto the gbe the result is that waves from parabolic sources may be described by the cylindrical gbe in the case of two dimensions and by the spherical gbe in the three dimensional cylindrically symmetric case furthermore results on the formation of shocks and caustics in the context of the zk equation are presented along with an exact solution to the dzk equation exact solutions with caustic singularities are studied along with a possible mechanism for their control finally results on the evolution of a shock approaching a caustic are given through the identification of a series of parameter regimes dependent on the diffusivity

a cucker smale model with noise and delay a generalization of the cucker smale model for collective animal behavior is investigated the model is formulated as a system of delayed stochastic differential equations it incorporates two additional processes which are present in animal decision making but are often neglected in modeling i stochasticity imperfections of individual behavior and ii delayed responses of individuals to signals in their environment sufficient conditions for flocking for the generalized cucker smale model are derived by using a suitable lyapunov functional as a by product a new result regarding the asymptotic behavior of delayed geometric brownian motion is obtained in the second part of the paper results of systematic numerical simulations are presented they not only illustrate the analytical results but hint at a somehow surprising behavior of the system namely that the introduction of an intermediate time delay may facilitate flocking

the pole dynamics of rational solutions of the viscous burgers equation rational solutions of the viscous burgers equation are examined using the dynamics of their poles in the complex x plane the dynamical system for the motion of these poles is finite dimensional and not hamiltonian nevertheless we show that this finite dimensional system is completely integrable by explicit construction of a sufficient number of conserved quantities the dynamical system has a class of non equilibrium similarity solutions for which all poles have equal real part for t sufficiently large within the context of the finite dimensional dynamical system these solutions are shown to be asymptotically stable

collective behavior models with vision geometrical constraints truncated noises and propagation of chaos abstract we consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains we rigorously derive a vlasov fokker planck type of kinetic mean field equation from the corresponding stochastic integral inclusion system more specifically we construct a global in time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1 wasserstein distance

fast slow dynamics of planar particle models for flocking and swarming abstract we discuss fast slow dynamics of planar particle models appearing in the modeling of flocking and swarming phenomena we employ artstein kevrekidis slemrod titi s unified approach for the singular perturbation to derive a qualitative description of the dynamics in the singular limit our analysis employs minimal assumptions on the communication weight

parametric motion of complex time singularity toward real collapse abstract motion of complex time singularities for solutions of a system of three interacting point vortices is studied by varying initial conditions until collapse occurs when collapse occurs the complex time singularities converge to a point on the real axis a heuristic scaling law between the initial displacement of one vortex and the position of the nearest complex time singularity to the origin is tested numerically the exponent of the scaling law is related to one of the eigenvalues of the linearized stability problem for the motion of the vortices in a regularized blown up frame

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

a kinetic flocking model with diffusion we study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the cucker smale flocking in presence of diffusion whose strength depends on the density this kinetic equation describes the collective behavior of an ensemble of organisms animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration our analysis takes advantage both from the fact that the global equilibrium is a maxwellian distribution function and on the contrary to what happens in the cucker smale model ieee trans autom control 52 852 862 2007 the interaction potential is an integrable function precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found

the long time asymptotics of solutions to the generalized burgers equation the long time i e range following a wavefront behaviour of finite amplitude sound waves in ducts horns of variable cross section is considered the wave evolution being governed by a generalized form of burgers equation the type of initial data far behind and far ahead of the wavefront is restricted to a transition from one constant state to another it is proved that the form of the wave for large range is dependent on the limiting value of a certain function of the duct parameters we have termed the various cases that arise supercylindrical cylindrical and subcylindrical for reasons that should be clear from the details to follow the analytic form of the limiting profile is determined in all but one case in particular it is shown that the similarity solution for cylindrical waves possesses very strong global stability properties

stochastic flocking dynamics of the cucker smale model with multiplicative white noises we present a strong asymptotic stochastic flocking estimate for the stochastically perturbed cucker smale model we characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states when the strength of multiplicative noises is sufficiently large we show that the strong stochastic flocking occurs even for negative communication weights

a characterization of seven compact kaehler submanifolds by holomorphic pinching from the results of simons 9 and chern do carmo and kobayashi 2 we know that in the class of compact minimal submanifolds of a sphere the totally geodesic submanifolds are isolated and that some simple minimal submanifolds can be characterized by suitable pinching on their curvatures these ideas are extended naturally to compact kaehler submanifolds of a complex projective space the problem of the best pinching for the above submanifolds was studied later by several authors e g yau 11 and ogiue 4 for surfaces and hypersurfaces the problem is completely resolved however in the general case we have only partial results let m be a compact kaehler submanifold of complex dimension n immersed in the complex projective space cptm 1 endowed with the fubini study metric of constant holomorphic sectional curvature 1 let h and k be the holomorphic sectional curvature and the sectional curvature of mn respectively ogiue conjectured the following 1 if h or 2 if n 2 and k or 3 if m n n n and k 0 2 then mn is a linear subvariety of cpm 1 recently using natural arguments at the minimum of the function h defined on the unit tangent bundle of mw the author 7 and verstraelen and the author 8 resolved the conjectures 1 and 2 respectively in this paper we obtain the following complete solution of the pinching problem in the kaehlerian case

symbolic computation of solutions for a forced burgers equation in this paper we give exact solutions for a forced burgers equation we make use of the generalized cole hopf transformation and the traveling wave method

global regularity for the fractional euler alignment system we study a pressureless euler system with a non linear density dependent alignment term originating in the cucker smale swarming models the alignment term is dissipative in the sense that it tends to equilibrate the velocities its density dependence is natural the alignment rate increases in the areas of high density due to species discomfort the diffusive term has the order of a fractional laplacian partial xx alpha 2 alpha in 0 1 the corresponding burgers equation with a linear dissipation of this type develops shocks in a finite time we show that the alignment nonlinearity enhances the dissipation and the solutions are globally regular for all alpha in 0 1 to the best of our knowledge this is the first example of such regularization due to the non local nonlinear modulation of dissipation

flocking with short range interactions we study the large time behavior of continuum alignment dynamics based on cucker smale cs type interactions which involve short range kernels that is communication kernels with support much smaller than the diameter of the crowd we show that if the amplitude of the interactions is larger than a finite threshold then unconditional hydrodynamic flocking follows since we do not impose any regularity nor do we require the kernels to be bounded the result covers both regular and singular interaction kernels moreover we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average over balls at small enough scale but otherwise vacuum is allowed at smaller scales consequently our arguments of hydrodynamic flocking apply mutatis mutandis to the agent based cs model with finitely many dirac masses in particular discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents

nonclassical potential symmetries for the burgers equation abstract in this work classes of symmetries for partial differential equations which can be written in a conserved form are found these nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation some of these symmetries are carried out for the burgers equation these symmetries are realized as the nonclassical symmetries of the associated potential system and yield many new nonlocal generators and new explicit solutions of the burgers equation

special symmetries to standard riccati equations and applications by using symmetries associated to riccati equation in standard form sre we obtain a family which can be integrated by quadratures as a consequence we get a new integrability condition for the generalized riccati equation gre we illustrate the result with some examples and we give some applications in the solitons theory

wrinkled flames and geometrical stretch localized wrinkles of thin premixed flames subject to hydrodynamic instability and geometrical stretch of uniform intensity s are studied a stretch affected nonlinear and nonlocal equation derived from an inhomogeneous michelson sivashinsky equation is used as a starting point and pole decompositions are used as a tool analytical and numerical descriptions of isolated centered or multicrested wrinkles with steady shapes in a frame and various amplitudes are provided their number increases rapidly with 1 s 0 a large constant s 0 weakens or suppresses all localized wrinkles the larger the wrinkles the easier the suppression whereas s 0 strengthens them oscillations of s further restrict their existence domain self similar evolutions of unstable many crested patterns are obtained a link between stretch nonlinearity and instability with the cutoff size of the wrinkles in turbulent flames is suggested open problems are evoked

asymptotic behavior and collision avoidance in the cucker smale model for the long range communicated cucker smale c s model asymptotic flocking can be obtained by taking full use of a key equality about the kinetic energy however such behavior does not exist for the short range communicated c s model with general initial data in this note we first establish a new equality about the second order velocity position moments for general c s models then we use this equality to deduce a new type of asymptotic behavior for the short range communicated c s model such behavior not only reflects an essential feature of the short range communication weight but also has some potential applications in the complex multicluster problem besides collision avoidance is also obtained for the c s model with some singular short range communication weights

complete classification of the asymptotical behavior for singular c s model on the real line abstract in this paper we study the singular cucker smale c s model on the real line for long range case i e 1 we prove the uniqueness of the solution in the sense of definition 2 1 and the unconditional flocking emergence moreover the sufficient and necessary condition for collision and sticking phenomenon will be provided for short range case i e 1 we construct the uniform in time lower bound of the relative distance between particles and provide the sufficient and necessary condition for the emergence of multi cluster formation for critical case i e 1 we show the uniform lower bound of the relative distance and unconditional flocking emergence these results provide a complete classification of the collective behavior for c s model on the real line

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative

deciphering singularities by discrete methods we consider the problem of estimating numerically the parameters of singularities of solutions of differential equations we propose a novel approach which is based on discretizing the governing equation and timestepping in the complex domain some applications to ordinary and partial differential equations are discussed