nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis abstract in this paper we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed by a change of variable from the well known keller segel model describing cell bacteria movement toward the concentration gradient of the chemical that is consumed by the cells we prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

exact solution of an n sublattice vertex model the exact diagonalization of the diagonal to diagonal transfer matrix of an n sublattice vertex model with interactions of the vertex arrow type between sublattices is carried out using the generalized nested bethe ansatz method an exact expression for the free energy is obtained involving the solution of an integral equation which is studied numerically

asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity abstract this paper establishes the nonlinear stability of traveling wave solutions to a chemotaxis model with singular or logarithmic sensitivity and its transformed parabolic hyperbolic system depending on the parameter signs we discuss the linear instability of traveling wave solutions using the spectral analysis and nonlinear asymptotic stability of traveling wave solutions with zero end state by the weighted energy estimates where the latter result solves the open question left in a previous work li and wang 2009 7

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

a wave propagation algorithm for viscoelastic fluids with spatially and temporally varying properties an algorithm is presented for simulating the fluid and stress equations that arise in a continuum model of platelet aggregation a l fogelson r d guy platelet wall interactions in continuum models of platelet thrombosis formulation and numerical solution math med biol 21 2004 293 334 the model is equivalent to models of viscoelastic fluids with spatially and temporally varying material parameters a subsystem containing the elastic terms is handled using a wave propagation algorithm two different methods for propagating waves are compared in computational tests one method linearizes the equations about cell edges and the other is based on the propagation of waves in a heterogeneous elastic medium when the viscoelastic fluid is in contact with a newtonian fluid the method based on linearization about the edges gives more accurate results no high weissenberg number instabilities were observed as long as the time step is chosen to obey a cfl condition the algorithm is stable and no unphysical oscillations appear in the solution 2007 elsevier b v all rights reserved

the supersymmetric t j model in one dimension the t j model related to the strong correlation limit of the hubbard model is shown to be soluble in one dimension using the bethe ansatz the solution holds only when the hamiltonian is supersymmetric the ground state in the presence of holes is found to be gapless and to have no magnetization

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

traveling waves and shocks in a viscoelastic generalization of burgers equation we consider traveling wave phenomena for a viscoelastic generalization of burgers equation for asymptotically constant velocity profiles we find three classes of solutions corresponding to smooth traveling waves piecewise smooth waves and piecewise constant shock solutions each solution type is possible for a given pair of asymptotic limits and we characterize the dynamics in terms of the relaxation time and viscosity

new integrable generalization of the one dimensional model a new generalization of the t j model with a nearest neighbour hopping is formulated and solved exactly by the bethe ansatz method in the thermodynamic limit the model describes the dynamics of fermions with different spins and with isotropic and anisotropic interactions

chemotactic traveling waves with compact support abstract a logarithmic model type chemotaxis equation is introduced with porous medium diffusion and a population dependent consumption rate the classical assumption that individual bacterium can sense the chemical gradient is not taken instead the chemotactic term appears by assuming that the migration distance is inversely proportional to the amount of food if food is the reason for migration the existence and uniqueness of a traveling wave solution of the model are obtained in particular solutions have interfaces that divide into constant and non constant regions in particular the profile of the population distribution has compact support numerical simulations are provided and compared with analytic results

exact solution of generalized t j models in one dimension a relation between the anisotropic t j model and a generalized six vertex model has been established on the basis of this relation we obtain the exact bethe ansatz solution for the ground state energy of anisotropic t j model for arbitrary spin and special values of coupling parameters we consider also the anisotropic generalization of t j model with localons

thermodynamics of the quantum perk schultz model the quantum mechanical analogue of the classical perk schultz model is considered which comprises the uimin sutherland model and the integrable t j chain the quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic bethe ansatz only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature the bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions furthermore the eigenvalue of the quantum transfer matrix is expressed in terms of these functions the integral formulation allows for taking the limit of infinite trotter suzuki number analytically the low temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

exponential stability of large amplitude traveling fronts for quasi linear relaxation systems with diffusion a b s t r a c t this paper is concerned with the stability of traveling front solutions for 2 2 quasi linear relaxation systems with small diffusion rate by applying geometric singular perturbation method special evans function estimates detailed spectral analysis and c0 semigroup theories we prove that all the non degenerate waves for semi linear relaxation systems are locally exponentially stable in some exponentially weighted spaces we also obtain the linear exponential stability of the non degenerate waves for quasi linear relaxation systems where the wave strengths can be large

the solvability of some chemotaxis systems abstract in this paper we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations these systems are essentially quasi linear strongly coupled partial differential equations we also study the maximal interval of existence in time of solutions the results are illustrated in application to a number of partial differential equation models arising in biology

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians