domain walls in the coupled gross pitaevskii equations a thorough study of domain wall solutions in coupled gross pitaevskii equations on the real line is carried out including existence of these solutions their spectral and nonlinear stability their persistence and stability under a small localized potential the proof of existence is variational and is presented in a general framework we show that the domain wall solutions are energy minimizers within a class of vector valued functions with nontrivial conditions at infinity the admissible energy functionals include those corresponding to coupled gross pitaevskii equations arising in modeling of bose einstein condensates the results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall the methods apply to many systems of interest and integrability is not germane to our analysis finally sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential thus giving rigorous justification to earlier results in the physics literature

a priori bounds nodal equilibria and connecting orbits in indefinite superlinear parabolic problems we consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain posed with homogeneous dirichlet boundary conditions the distinguishing feature of this class is the indefinite superlinear but subcritical growth of the nonlinearity at infinity we present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria in addition we can prove the existence of connecting orbits in many cases one advantage of our approach is that the parabolic semiflow is naturally order preserving in contrast to pseudo gradient flows considered when using variational methods therefore we can obtain much information on nodal properties of equilibria that was not known before

symmetry of some entire solutions to the allen cahn equation in two dimensions abstract in this paper we prove even symmetry and monotonicity of certain solutions of allen cahn equation in a half plane we also show that entire solutions with finite morse index and four ends must be evenly symmetric with respect to two orthogonal axes a classification scheme of general entire solutions with finite morse index is also presented using energy quantization

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

two end solutions to the allen cahn equation in r3 abstract in this paper we study axially symmetric solutions of allen cahn equation in the three dimensional euclidean space using a sophisticated continuation method we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates which may be regarded as the analogue of the family of catenoids and hence called two end solutions nonexistence of two end solution with a small growth rate is also shown which differs from the theory of minimal surfaces

a fixed point theorem for s type operators on banach spaces and its applications to boundary value problems abstract in this article we present a fixed point theorem of operator equations defined in cones of banach spaces where the spectral radius of the operator is involved the operator is an s type operator a meaning introduced here as an application of our main theorem we get the existence of positive solutions to two boundary value problems extending several results from the literature

moduli space theory for the allen cahn equation in the plane in this paper we study entire solutions of the allen cahn equation delta u f u 0 where f is an even bistable function we are particularly interested in the description of the moduli space of solutions which have some special structure at infinity the solutions we are interested in have their zero set asymptotic to 2k k 2 oriented affine half lines at infinity and along each of these affine half lines the solutions are asymptotic to the one dimensional heteroclinic solution such solutions are called multiple end solutions and their set is denoted by m 2k the main result of our paper states that if u in m 2k is nondegenerate then locally near u the set of solutions is a smooth manifold of dimension 2k this paper is part of a program whose aim is to classify all 2k ended solutions of the allen cahn equation in dimension 2 for k 2

a contribution to scattering calculation for small wavelengths the high frequency panel method numerical treatment of electromagnetic scattering in the case of small wavelengths too small for reasonable matrix sizes when using discretization methods and too large for traditional asymptotic methods is considered an approach which takes advantage of the user friendly nature of matrix discretization methods while avoiding the explosive increase in computing effort as the wavelength decreases is suggested it appears that no sensible increase of matrix size is required as the wavelength decreases since this is an untried technique applied in an inaccessible computational domain the preliminary work described is confined to two dimensional conducting circular cylinders where the answers are well known and well understood the overall idea however appears to be quite general

higher energy solutions in the theory of phase transitions a variational approach abstract we establish the existence of a higher energy solution to the vector ginzburg landau equation with a triple well potential on a bounded and smooth domain on the plane this solution is obtained by a linking argument in implementing this variational approach we make several considerations on the dynamics of the negative gradient flow in particular we use the conley index to contruct a suitable one dimensional invariant set this solution has morse index two in the nondegenerate case we discuss its structure in connection with the so called triple junction configurations

hamiltonian identities for elliptic partial differential equations abstract new identities for elliptic partial differential equations are obtained several applications are discussed in particular young s law for the contact angles in triple junction formation is proven rigorously structure of level curves of saddle solutions to allen cahn equation are also carefully analyzed

on variational characterization of four end solutions of the allen cahn equation in the plane abstract in this paper a novel variational method is developed to construct four end solutions in r 2 for the allen cahn equation four end solutions have been constructed by del pino kowalczyk pacard and wei when the angle of the ends is close to 2 or 0 using the lyapunov schmid reduction method and later by kowalczyk liu and pacard using a continuation method for general 0 2 by a special mountain pass argument in a restricted space namely a set of monotone paths of monotone functions a family of solutions in bounded domain is constructed with very good control of their nodal sets which then leads to a four end solution after sending the domains to r 2 in this way not only a family of four end solutions is constructed for any angle 0 2 the morse index of such a four end solution is also shown to be one

a fast integral equation approach to acoustic scattering from three dimensional objects using a natural basis set boundary value integral equations and the moment method solution technique provide a succinct and elegant strategy for the solution of acoustic surface scattering problems this paper explores the application of a subdomain basis set based on the phase of the incident radiation which is used to solve for the unknown normal pressure gradient on the surface of the scatterer using the moment method the advantage of this basis set lies in the fact that it can accurately represent the fast phase character of the pressure gradient on the surface of the scatterer this allows for far fewer discretizations basis elements in the moment method formulation and therefore significantly reduces the density of unknowns that must be numerically computed the main contribution of this paper is to show how to efficiently employ this natural basis and to explore the physical justification for using this approach the method is compared with other techniques including the kirchhoff solution technique high frequency single scattering techniques the nussenzveig fock technique and the parabolic equation approximation unlike these methods no approximation to the physical system need be introduced

scattering at a rough boundary extensions of the kirchhoff approximation a helmholtz integral formula gives the acoustic field reflected from a rough surface in terms of the values of the acoustic field and of its normal derivative at all points on the surface for surface reflection problems in which one or the other of these surface field values are specified we can make use of this formula to derive a boundary surface integral equation on the unspecified field values further a series solution of this equation can be constructed by iteration the zeroth order term of the series being the result predicted by a kirchhoff approximation the nth order iterate of this series requires an n fold integration over the region of the rough surface an integration that does not in general converge absolutely for unboundedly large surfaces for a pressure release surface we consider the first two iterates in detail and using stationary phase approximations replace the required integration by summations in this way the source of the convergence difficulty is made clear and is demo

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

application of the integral equation asymptotic phase method to two dimensional scattering a hybrid procedure called the integral equation asymptotic phase ie ap method is investigated for scattering from perfectly conducting cylinders of arbitrary cross section shape the ie ap approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution to leave a slowly varying residual function that can be represented by a coarse density of unknowns in the present investigation the current density appearing within the combined field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function the resulting equation is discretized by the method of moments using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density even for scatterers with corners

heteroclinic connections for multiple well potentials the anisotropic case we investigate the existence of solutions to systems of n differential equations representing connections between minima of potentials with several equal depths in r using variational techniques and in particular a method introduced in af we first prove such existence for n 2 and two minima dealing next with symmetric potentials corresponding to free bulk energies in crystals we establish existence for n 2 in various cases of more than two minima finally we obtain a sufficient condition establishing existence of connections to not necessarily symmetric potentials for arbitrary n and three minima

boundary value problems for singular second order functional differential equations abstract we consider general linear boundary value problems for equations of the form y f x y x 0 0 x x is continuous and f x y has a singularity at y 0 the results improve and extend earlier results for the case x x due to taliaferro 1979 and gatica 1989

multiphase solutions to the vector allen cahn equation crystalline and other complex symmetric structures we present a systematic study of entire symmetric solutions u mathbb r n rightarrow mathbb r m of the vector allen cahn equation delta u w u u 0 quad text for all quad x in mathbb r n where w mathbb r m rightarrow mathbb r is smooth symmetric nonnegative with a finite number of zeros and where w u partial w partial u 1 dots partial w partial u m top we introduce a general notion of equivariance with respect to a homomorphism f g rightarrow gamma g gamma reflection groups and prove two abstract results concerning the cases of g finite and g discrete for the existence of equivariant solutions our approach is variational and based on a mapping property of the parabolic vector allen cahn equation and on a pointwise estimate for vector minimizers

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

existence of positive solutions for boundary value problem of second order fde abstract we use a fixed point theorem in cones to investigate the existence of positive solutions for boundary value problem of a second order functional differential equation fde with the form y t r t y t 0 0 y t y t t r t 0 y t y t n t 1 t 1 a where y t y t for a we allow that r t has the singularity at the endpoints t 0 and t 1 of 0 1 our results include the situations that f is either superlinear or sublinear

positive periodic solutions of nonlinear functional differential equations we apply the generalized form of leggett williams fixed point theorem to prove that the following nonlinear functional differential equationx t a t x t f t x t has at least two positive t periodic solutions where a t is a t periodic function satisfying exp 0 ta u du 1 f t x t is a nonnegative function defined on rxbc where bc denotes the banach space of bounded continuous functions

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

multiple end solutions to the allen cahn equation in r2 we construct a new class of entire solutions for the allen cahn equation u 1 u2 u 0 in r2 c given k 1 we find a family of solutions whose zero level sets are away from a compact set asymptotic to 2k straight lines which we call the ends these solutions have the property that there exist 0 1 2k 0 2 such that limr u rei 1 j uniformly in on compact subsets of j j 1 for j 0 2k 1