on finite strain micromorphic elastoplasticity abstract in the micromorphic continuum theory of eringen it was proposed that microstructure of materials could be represented in a continuum framework using a micro deformation tensor governing micro element deformation in addition to the deformation gradient governing macro element deformation the paper formulates finite strain micromorphic elastoplasticity based on micromorphic continuum mechanics in the sense of eringen multiplicative decomposition into elastic and plastic parts of the deformation gradient and micro deformation are assumed and the clausius duhem inequality is formulated in the intermediate configuration b to analyze what stresses elastic deformation measures and plastic deformation rates are used defined in the constitutive equations the resulting forms of plastic and internal state variable evolution equations can be viewed as phenomenological at their various scales i e micro continuum and macro continuum the phenomenology of inelastic mechanical material response at the various scales can be different but for demonstration purposes j2 flow plasticity is assumed for each of three levels of plastic evolution equations identified with different stress internal state variables and material parameters all evolution equations and a semi implicit time integration scheme are formulated in the intermediate configuration for future coupled lagrangian finite element implementation a simpler two dimensional model for anti plane shear kinematics is formulated to demonstrate more clearly how such model equations simplify for future finite element implementation

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

isobe kakinuma model for water waves as a higher order shallow water approximation abstract we justify rigorously an isobe kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom it is known that the full water wave equations are approximated by the shallow water equations with an error of order o 2 where is a small nondimensional parameter defined as the ratio of the mean depth to the typical wavelength the green naghdi equations are known as higher order approximate equations to the water wave equations with an error of order o 4 in this paper we show that the isobe kakinuma model is a much higher order approximation to the water wave equations with an error of order o 6

asymptotic stability of solitary waves for the regularized long wave equation we show that a family of solitary waves for the regularized long wave rlw equation is asymptotically stable the large time dynamics of a solution near a solitary wave are studied by decomposing the solution into a modulating solitary wave with speed and phase shift that are functions of t plus a perturbation the strategy of proof follows that used by pego and weinstein 24 who considered the asymptotic stability of solitary waves of korteweg devries kdv type equations for rlw it is necessary to modify the basic ansatz to incorporate a new time scale which must be determined by the scheme different techniques are also required to analyze the spectral theory of the differential operator that arises in the linearized equation for a solitary wave perturbation in particular we use a result of pruss 27 to show that the linearized operator generates a semigroup with exponentially decaying norm on a certain weighted function space and we exploit the formal convergence of rlw to kdv under a certain scaling kdv scaling in order to rule out the existence of nonzero eigenvalues of the linearized operator 1996 john wiley sons inc

linear stability of solitary waves of the green naghdi equations we investigate the eigenvalue problem obtained from linearizing the green naghdi equations about solitary wave solutions unlike weakly nonlinear water wave models the physical system considered here has nonlinearity in its highest derivative term this results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter combining the technique of singular perturbation with the evans function we show that for solitary waves of small amplitude the problem has no eigenvalues of positive real part and the evans function is nonvanishing everywhere except the origin this fact then leads to the linear stability of these solitary waves 2001 john wiley sons inc

3 manifolds with nonnegative ricci curvature for a complete noncompact 3 manifold with nonnegative ricci curvature we prove that either it is diffeomorphic to 3 or the universal cover splits this confirms milnor s conjecture in dimension 3

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity viscoplasticity and damage the construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the helmholtz free energy function of the constitutive material model the combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator within the small strain framework the choice of a quadratic potential w r t the gradient term provides the widely used helmholtz operator whose regularization properties are well known smoothing of discontinuities at interfaces and boundary layers in hardening materials and finite width localization bands in softening materials the objective is to review and propose nonlinear extensions of micromorphic and strain damage gradient models along two lines the first one introducing nonlinear relations between generalized stresses and strains the second one envisaging several classes of finite deformation model formulations the generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach additive split of some lagrangian strain measure or choice of a local objective rotating frame three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient a new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

a mixed solution strategy for the nonlinear analysis of brick masonry walls the paper presents a discrete mechanical model for masonry walls based on a lagrangean description where each brick is described as a rigid body and each mortar joint as an interface element constitutive assumptions characterized by elasticity damage and friction are associated to the joints only a numerical solution strategy based on a mixed path following approach in terms of stresses strains displacements damage and load parameters is proposed for avoiding convergence problems related to the joint softening behaviour some numerical results are also presented showing the robustness and effectiveness of this proposal

constant mean curvature spheres in rm sol _3 based on work of daniel and mira we prove that for each h0 there exists a unique up to ambient isometry immersed constant mean curvature h sphere s h in rm sol 3 where rm sol 3 is the non abelian solvable 3 dimensional lie group equipped with its usual left invariant metric that makes it into a model space for one of the eight thurston geometries see bonahon or scott for a discussion of the thurston geometries that can occur for compact 3 manifolds since this proof also demonstrates that s h has index one the results in daniel mira imply s h is embedded and the left invariant gauss map of s h is a diffeomorphism one key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature h graphs in rm sol 3 which depend only on any fixed positive lower bound for h by the work in daniel mira the existence of the spheres s h for all h0 implies the existence for all h geq 0 of a complex holomorphic only when h 0 quadratic differential q h defined on all constant mean curvature h surfaces m in rm sol 3 and which for h0 vanishes on an open set of m if and only if m is contained in some left translation of s h furthermore when q h is not identically zero on m it has isolated zeros on m and negative index at these zeros

formulation of implicit finite element methods for multiplicative finite deformation plasticity some constitutive and computational aspects of finite deformation plasticity are discussed attention is restricted to multiplicative theories of plasticity in which the deformation gradients are assumed to be decomposable into elastic and plastic terms it is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non symmetric even for associated plastic flow both explicit i e no equilibrium iteration and implicit finite element formulations are considered an assumed strain formulation is used to accommodate the near incompressibility associated with fully developed isochoric plastic flow as an example of explicit integration the rate tangent modulus method is reviewed in some detail an implicit scheme is derived for which the consistent tangents resulting in quadratic convergence of the equilibrium iterations can be written out in closed form for arbitrary material models all the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

on standard and vector finite element analysis of a strict anti plane shear plasticity model with elastic curvature abstract a finite deformation plasticity model with plasticity driven by curvature of lattice elastic deformation due to geometrically necessary dislocations gnds and by scalar statistically averaged lattice elastic deformation due to statistically stored dislocations ssds is formulated for strict anti plane shear kinematics and implemented by standard and vector finite element fe methods the curl of the elastic part of the deformation gradient represents the elastic lattice curvature that a grain or subgrain cell experiences as a result of the presence of gnds at grain or subgrain cell boundaries strict anti plane shear implies that the form of the elastic and plastic parts of the deformation gradient are the same as the total deformation gradient which is the unity tensor plus the dyadic product of the out of plane normal and displacement gradient the plasticity model with elastic curvature leads to a partial differential equation pde to solve for the plastic vector rather than an ordinary differential equation for standard plasticity without curvature along with the linear momentum balance this leads to a coupled system of pdes for which to solve numerically by a coupled fe implementation aside from the out of plane displacements a plastic vector is solved for at the nodes for the standard element and along element edges for the vector element numerical examples verify the standard and vector element implementations and demonstrate the additional hardening due to the elastic curvature generated at the notched crack tip for a mode iii anti plane shear crack simulation

nonlinear microstrain theories a hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes rotation and straining of an underlying microstructure an increase in the number of degrees of freedom represents a refinement of the material description in addition to available nonlinear cosserat and micromorphic theories general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua a microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element in each case balance equations and boundary conditions are derived decompositions of the finite strain measures into elastic and plastic parts are provided the formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials thus complying with requirements of continuum thermodynamics some guidelines for the selection of a suitable higher order model for a given material close the discussion

exact closed form solutions of a fully nonlinear asymptotic two fluid model abstract a fully nonlinear model of choi and camassa 1999 describing one dimensional incompressible dynamics of two non mixing fluids in a horizontal channel under a shallow water approximation is considered an equivalence transformation is presented leading to a special dimensionless form of the system involving a single dimensionless constant physical parameter as opposed to five parameters present in the original model a first order dimensionless ordinary differential equation describing traveling wave solutions is analyzed several multi parameter families of physically meaningful exact closed form solutions of the two fluid model are derived corresponding to periodic solitary and kink type bidirectional traveling waves specific examples are given and properties of the exact solutions are analyzed

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

long time effects of bottom topography in shallow water abstract we present and discuss new shallow water equations that provide an estimate of the long time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity we consider the regime where the froude number is much smaller than the aspect ratio of the shallow domain the new equations are obtained at first order in an asymptotic expansion of the solutions of the euler equations for a shallow fluid by using the small parameter 2 the leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects both sets of equations conserve energy and circulation convect potential vorticity and have a hamiltonian formulation the corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects

hamiltonian formulation of the extended green naghdi equations abstract a novel method is developed for extending the green naghdi gn shallow water model equation to the general system which incorporates the arbitrary higher order dispersive effects as an illustrative example we derive a model equation which is accurate to the fourth power of the shallowness parameter while preserving the full nonlinearity of the gn equation and obtain its solitary wave solutions by means of a singular perturbation analysis we show that the extended gn equations have the same hamiltonian structure as that of the gn equation we also demonstrate that zakharov s hamiltonian formulation of surface gravity waves is equivalent to that of the extended gn system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface