three component nonlinear dynamical system generated by the new third order discrete spectral problem we propose a nonlinear model on a regular infinite one dimensional lattice it describes the three component dynamical system with modulated on site masses and is shown to admit a zero curvature representation the associated auxiliary spectral problem is basically of third order and gives rise to fairly complicated subdivision into domains of regularity of jost functions in the plane of complex spectral parameter as a result both the direct and the inverse scattering problems turn out to be substantially nontrivial the caudrey version of the direct and inverse scattering technique for the needs of model integration is adapted the simplest soliton solution is found

on the global existence of solutions to an aggregation model abstract in this paper we consider a reaction diffusion chemotaxis aggregation model of keller segel type with a nonlinear degenerate diffusion assuming that the diffusion function f n takes values sufficiently large i e takes values greater than the values of a power function with sufficiently high power f n n p for all n 0 where 0 is a constant we prove global in time existence of weak solutions since one of the main features of keller segel type models is the possibility of blow up of solutions in finite time we will derive the uniform in time boundedness which prevents the explosion of solutions the uniqueness of solutions is proved provided that some higher regularity condition on solutions is known a priori finally computational simulation results showing the effect of three different types of diffusion function are presented

pointwise tube formulas for fractal sprays and self similar tilings with arbitrary generators abstract in a previous paper by the first two authors a tube formula for fractal sprays was obtained which also applies to a certain class of self similar fractals the proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling specifically the inner tube formula for each generator of the fractal spray is required to be polynomial now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators and also by giving a proof which holds pointwise rather than distributionally hence our results for fractal sprays extend to higher dimensions the pointwise tube formula for 1 dimensional fractal strings obtained earlier by lapidus and van frankenhuijsen our pointwise tube formulas are expressed as a sum of the residues of the tubular zeta function of the fractal spray in r d this sum ranges over the complex dimensions of the spray that is over the poles of the geometric zeta function of the underlying fractal string and the integers 0 1 d the resulting fractal tube formulas are applied to the important special case of self similar tilings but are also illustrated in other geometrically natural situations our tube formulas may also be seen as fractal analogues of the classical steiner formula

solitons in parametrically driven discrete nonlinear schrodinger systems with the exploding range of intersite interactions we present the sequence of parametrically driven discrete nonlinear schrodinger systems with the progressively extending range of intersite couplings in the case of time independent coupling parameters the sequence is reduced to the ablowitz ladik hierarchy which is known to be integrable by the inverse scattering transform however the models with the time dependent intersite interactions are shown to be integrable too irrespective of a particular form of time dependencies of coupling parameters any of such parametrically driven systems might exhibit rather complex soliton dynamics and is described by the unconserved hamiltonian function we reveal an important subclass of parametrically driven systems demonstrating the parametrical localization of soliton dynamics on a confined domain of space meanwhile an appropriate choice of time dependencies in intersite interactions allow us to transform the original parametrically driven system into another one but subjected to the linear external potential as a result the latter system can be readily integrated as well in particular the peculiarities of bloch oscillations in the systems with time independent long range intersite interactions and linear external potential of constant strength are analyzed in general regulating the range of intersite couplings the strengths and time dependencies of coupling parameters we are able to model a number of physically important quasi one dimensional systems we develop an alternative approach to solve the marchenko equations permitting one to obtain the multisoliton solutions in the most simple and natural way finally we point out how to reformulate any model in row in terms of corrected amplitudes with the standard poisson brackets

enigma of probability amplitudes in hamiltonian formulation of integrable semidiscrete nonlinear schrodinger systems an attempt to find a probability amplitude based hamiltonian representation of the symmetrical version of integrable semidiscrete multicomponent nonlinear schrodinger systems is made thus the on cell locality of the general point transformation in combination with the model multicomponentness is shown to contradict the concept of canonical hamiltonian representation in terms of probability amplitudes nevertheless the above concept can be realized in a slightly adjusted semidiscrete multicomponent nonlinear schrodinger system that preserves some physically valuable solutions of the original either symmetric or asymmetric integrable model the advantages of the adjusted hamiltonian model for the analysis of real physical systems are formulated examples of longitudinal and lateral soliton dynamics on multichain tubular lattices subjected to uniform electric and magnetic fields are given

singular gradient flow of the distance function and homotopy equivalence let m be a riemannian manifold and let varomega be a bounded open subset of m it is well known that significant information about the geometry of varomega is encoded into the properties of the distance d partial varomega from the boundary of varomega here we show that the generalized gradient flow associated with the distance preserves singularities that is if x 0 is a singular point of d partial varomega then the generalized characteristic starting at x 0 stays singular for all times as an application we deduce that the singular set of d partial varomega has the same homotopy type as varomega

nonlinear integrable model of frenkel like excitations on a ribbon of triangular lattice following the considerable progress in nanoribbon technology we propose to model the nonlinear frenkel like excitations on a triangular lattice ribbon by the integrable nonlinear ladder system with the background controlled intersite resonant coupling the system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy the most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach the obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy using the recovered second densities the hamiltonian formulation of integrable nonlinear ladder system with background controlled intersite resonant coupling is presented in doing so the relevant poisson structure turns out to be essentially nontrivial the darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of backlund transformation in justification of its self consistency is pointed out the spectral properties of darboux matrix allow to restore the whole darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system the power of darboux dressing method is explicitly demonstrated in generating the multicomponent one soliton solution to the integrable nonlinear ladder system with background controlled intersite resonant coupling

integrable nonlinear ladder system with background controlled intersite resonant coupling a new spectral problem on one dimensional lattices is found allowing consistently to support the zero curvature representation for a wide class of integrable nonlinear ladder systems the modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived the eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on schrodinger type fields and finite background condition on a concomitant field are solved and the domains of analyticity of jost functions are presented both analytically and graphically this particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct jost functions that have to be involved in the procedure of inverse scattering transform moreover there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of jost functions this crossover should inevitably lead to qualitative rearrangements in the structure of model solutions thus already in the limit of low amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of schrodinger subsystem just above the critical background value of practically unexcited concomitant field whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self site coupling

on the stability of the cut locus abstract in r d we consider a riemannian metric g and an open bounded subset we study the stability of the cut locus associated with and g w r t perturbations both of the set and of the metric g in order to have the stability of the cut locus we assume c 2 regularity of the data the metrics and the sets in the case of sets with c 1 1 boundaries the cut locus may be unstable we prove that to c 2 perturbations both of the set and of the metric correspond small changes of the cut locus w r t the hausdorff distance i e the cut locus is stable in the c 2 category we give some examples showing that c 1 perturbations may lead to large variations of the cut locus

on the extension of the solutions of hamilton jacobi equations abstract we consider the viscosity solution of a homogeneous dirichlet problem for the eikonal equation in a bounded set we suppose that the hamiltonian h x p a x p p 1 is strictly convex w r t the variables p and of class c 1 1 w r t the variables x then the solution of the dirichlet problem admits an extension to a neighbourhood of u such that u is still a viscosity solution of the eikonal equation if and only if satisfies an exterior sphere condition the above result in particular provides a characterization of the boundary singularities and a regularity theorem up to the boundary for the solution of the eikonal equation

propagation of singularities for solutions of nonlinear first order partial differential equations few results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of f x u du 0 this paper provides new techniques to analyse such a set when u is semiconcave and f is a nonlinear convex function with respect to p the main objective achieved here is a classification of the singularities of u that propagate along lipschitz arcs such a propagation phenomenon is also described by means of a generalized characteristics inclusion

integrable semi discretization of the coupled modified kdv equations the discrete version of the inverse scattering method proposed by ablowitz and ladik is extended to solve multi component systems the extension enables one to solve the initial value problem which proves directly the complete integrability of a semi discrete version of the coupled modified korteweg de vries kdv equations and their hierarchy it also provides a procedure to obtain conservation laws and multi soliton solutions of the hierarchy

global existence of classical solutions to an acid mediated invasion model for tumor stromal interactions this paper is purported to investigate an acid mediated invasion model for tumor stromal interactions under no flux boundary condition by applying the contraction mapping principle the parabolic schauder estimates parabolic l p estimates and sobolev embedding theorems we prove that there exists a unique global classical solution of this model

half space theorems for minimal surfaces with bounded curvature first we prove a version of the strong half space theorem for minimal surfaces with bounded curvature in r 3 with the techniques developed in our proof we give criteria for deciding if a complete minimal surface is proper we prove a mixed version of the strong half space theorem turning to 3 dimensional manifolds of bounded geometry and positive ricci curvature we show that complete injectively immersed minimal surfaces with bounded curvature are proper and as a corollary we have a half space theorem in this setting finally we show an application of the maximum principle for nonproper minimal immersions in r 3

lax pairs for integrable lattice systems this paper studies the structure of lax pairs associated with integrable lattice systems where space is a one dimensional lattice and time is continuous it describes a procedure for generating examples of such systems and emphasizes the features that are needed to obtain equations which are local on the spatial lattice

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

symmetry broken canonizations of the semi discrete integrable nonlinear schrodinger system with background controlled inter site coupling the integrable system of nonlinear schrodinger type consisting of two mutually symmetric basic subsystems and one concomitant subsystem given on a ribbon of triangular lattice is characterized by the highly nonstandard form of poisson structure the fact of system criticality against the positive semi definite background parameter provides us the clue how to organize the nonlinear point transformation ensuring the canonization of field variables precisely it allows to introduce two subsets of mutually symmetric intermediate auxiliary field variables and to involve one or another of them into a rational procedure of canonization realizable already in terms of two mutually asymmetric subsets of field amplitudes there are two variants of system asymmetric standardization namely the minus asymmetric and plus asymmetric ones emanated from the two leg structure of underlying space lattice of the primary noncanonical nonlinear system either of the two asymmetrically standardized nonlinear systems consi

the regularity of the distance function propagates along minimizing geodesics abstract we consider the distance function from the boundary of an open bounded set r n associated to a riemannian metric with c 1 1 coefficients we show that the c 1 1 regularity propagates towards the boundary along the distance minimizing geodesics hence we show that the cut locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as

on the local semiconcavity of the solutions of the eikonal equation abstract we show that if the hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave furthermore in the 1d case we show that every viscosity solution of the eikonal equation is semiconcave if and only if the hamiltonian is lipschitz continuous with respect to the state variable

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

minkowski content and fractal curvatures of self similar tilings and generator formulas for self similar sets abstract we study minkowski contents and fractal curvatures of arbitrary self similar tilings constructed on a feasible open set of an ifs and the general relations to the corresponding functionals for self similar sets in particular we characterize the situation when these functionals coincide in this case the minkowski content and the fractal curvatures of a self similar set can be expressed completely in terms of the volume function or curvature data respectively of the generator of the tiling in special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self conformal sets our approach based on the classical renewal theorem is simpler and works for a much larger class of self similar sets and tilings in fact generator type formulas are obtained for essentially all self similar sets when suitable volume functions and curvature functions respectively related to the generator are used we also strengthen known results on the minkowski measurability of self similar sets in particular on the question of non measurability in the lattice case

on the cut locus of closed sets abstract let c be a closed subset of a smooth manifold of dimension n m and let m be endowed with a riemannian metric of class c 2 we study the cut locus of c cut c first we show that cut c is a set of measure zero then we assume that c is the boundary of an open bounded set m in particular this assumption implies that cut c 0 we deduce that cut c is invariant w r t the generalized gradient flow associated with the distance function from the set c as a consequence of the invariance we have that cut c has the same homotopy type as the set furthermore if m is a compact manifold then cut c has the same homotopy type as m c finally we show that the closure of the cut locus stays away from c if and only if c is a manifold of class c 1 1