double phase integral approximations a systematic simplification technique for wave equations with cutoffs and resonances the time independent wave equation d2 dz2 q2 z 0 where q2 z may have arbitrary order zeros and poles on or close to the real axis is transformed to a simpler wave equation of similar properties model approximate transformations leading from the original wave equation to the model are simply related to froman s higher order phase integrals but are nevertheless well defined at the pertinent zeros and poles of q2 z

an elliptic parabolic free boundary problem continuity of the interface on etablit la continuite de l interface de la solution faible d un probleme elliptique parabolique lie a la theorie des ecoulements partiellement satures dans les milieux poreux on utilise une methode basee sur le principe du maximum pour les equations paraboliques en supposant l ecoulement unidimensionnel

degenerate diffusion and the stefan problem on etudie le probleme i u t u xx f x u dans 0 1 r u 0 t 0 u 1 t 0 dans r u x 0 u 0 x dans 0 1 avec certaines hypotheses sur f et u 0 on etudie le caractere des solutions quand s annule sur un intervalle

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

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

on the regularity of weak solutions of a one dimensional two phase degenerate stefan problem with convection we shall be concerned with the following one dimensional two phase stefan problem let t 0 and q 0 1 x 0 t find a pair of functions u x t s t which are defined on qt and 0 t respectively and which satisfy al u uxx f u u in x t o x s l o t t l la az ub4 u xx in x t s t x l o tdt l lb u s t r 0 o t t l lc i t u s t t u s t t o t t lid u 0 t 1 06t t l le u 1 t 1 obt t l lf u x 0 u x o x l 1 k s 0 so e 0 1 l lh here so is a given number and the ar a f and u are known functions of their arguments problem 1 1 may be viewed as a model of heat conduction in a region involving phase change an example situation is an ice water mix in which circulation of the water is present then the term f u u represents the contribution of the field velocity generated by the water flow in the liquid phase a more detailed discussion on the formulation of 1 1 can be found elsewhere see e g s

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

phase integral approximation for coupled ordinary differential equations of the schrodinger type four generalizations of the phase integral approximation pia to sets of ordinary differential equations of schrodinger type uj x k 1nrjk x uk x 0 j 1 2 n are described the recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix r x rjk x either hermitian or non hermitian for hermitian and negative definite r x matrices a wronskian conserving pia theory is formulated which generalizes fulling s current conserving theory pertinent to positive definite r x matrices the idea of a modification of the pia which is well known for one equation u x r x u x 0 is generalized to sets a simplification of wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections if the pia is generated by a nondegenerate eigenvalue of the r x matrix the eliminated integration is the only one present in that case the simplified theory becomes fully algorit

local behavior of solutions of an elliptic parabolic equation abstract consideration of the nonsteady filtration of an incompressible fluid in a homogeneous partially saturated porous medium leads to a degenerate nonlinear parabolic equation which is of elliptic type in the space variables in the region of degeneracy i e in the saturated region results concerning the existence of weak solutions of such equations have been established here the local properties of weak solutions are studied in particular it is shown that under the realistic assumption that the hydrostatic potential is bounded the solution moisture content can be defined pointwise and is continuous author

the existence of regular orbits for nilpotent groups consider the following situation suppose r a cr where r is an r group and c is a cyclic group of order qe where r and q are distinct primes let k be a finite field of characteristic p and suppose v is a faithful irreducible k cr module this situation has been widely studied when p q it is similar to the situation occurring in the proof of theorem b of hall and higman 9 when p q it is similar to the situation occurring in the corresponding theorem of shult 131 in this paper a generalization of this situation is considered suppose g 4 ag where g is a solvable group a is nilpotent and ia g 1 again we assume v is a faithful irreducible k ag module two related questions arise in the study of this situation

homogenization of richardsʼ equation of van genuchten mualem model abstract this paper is devoted to the homogenization of richards equation of van genuchten mualem model which is a nonlinear degenerate parabolic differential equation it is usually used to model the motion of saturated unsaturated water flow in porous media we firstly apply the kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term then under the real assumption for van genuchten mualem model we obtain the homogenized equation based on the two scale convergence theory some results on the first order corrector are also presented

transmission through cutoffs and resonances in the double phase integral approximation using the double phase integral approximation technique developed earlier for the wave equation d2 dz2 q2 z 0 we derive analytical formulas for the reflection r transmission t and absorption a coefficients they are valid to arbitrary order in the expansion parameter for functions q2 z having either two cutoffs or one cutoff and one resonance for two examples of this type the formulas for r t and a are checked against numerical results using approximations up to fifth order

asymptotic behaviour of solutions to parabolic elliptic variational inequalities in a real hilbert space h where j is a given interval in r 0 co j is the subdifferential of a convex function 4 on h and b is a maximal monotone operator in h the abstract setting reflects the form of a class of parabolic elliptic boundary value problems with time dependent obstacles such problems arise in particular as mathematical models of various processes in electrochemical technology and some flows with saturations in porous media time dependent obstacles may represent geometric controls moving control actions in these problems in particular such controls may be comprehended in variable boundary conditions prescribed over time dependent parts of the boundary of domain or in source terms distributed over time dependent subdomains the former refers to control of flows via variable water table in reservoirs the latter characterizes control of electrochemical process via mechanically driven motion of a toolpiece cathode the most typical control objective consists there in stabilizing the state of the process at a given spatial distribution for example constant water head and desired shape of anode this motivates our interest in the asymptotic behavior of solutions to the corresponding boundary value problems as t a to the knowledge of the authors this question has so far not been studied in literature as it has been shown in kenmochi pawlow 16 171 the class of considered time dependent problems admits a representation in the form of evolution equation e b in 16 171 the corresponding results on existence uniqueness and stability of solutions have been established in the present paper we study the asymptotic properties of solutions to e c b as t 00 provided some hypotheses are imposed on the family t e r and the limit 4 of as t a for the solution u u of e d b on r we prove that

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

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

formal upscaling and numerical validation of unsaturated flow models in fractured porous media abstract in this work we consider a mathematical model for describing flow in an unsaturated porous medium containing a fracture both the flow in the fracture as well as in the matrix blocks are governed by richards equation coupled by natural transmission conditions using formal asymptotics we derive upscaled models as the limit of vanishing e the ratio of the width and length of the fracture our results show that the ratio of porosities and permeabilities in the fracture to matrix determine to the leading order of approximation the appropriate effective model in these models the fracture is a lower dimensional object for which different transversally averaged models are derived depending on the ratio of the porosities and permeabilities of the fracture and respective matrix blocks we obtain a catalogue of effective models which are validated by numerical computations

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

letter to the editor spurious localized highest frequency modes in schrodinger type equations solved by finite difference methods high frequency solutions of one or several schrodinger type equations are well known to differ very little from the plane wave solutions exp ikx that is the potential terms impact the envelope of a high frequency plane wave to only a small extent however when such equations are solved by a finite difference method the highest frequency solutions may under certain conditions turn out to be localized this may puzzle the researcher and suggest that the code may have an error however this is not an error but a numerical artifact and in this note we explain it

mixed finite elements for the richards equation linearization procedure we consider mixed finite element discretization for a class of degenerate parabolic problems including the richards equation after regularization time discretization is achieved by an euler implicit scheme while mixed finite elements are employed for the discretization in space based on the results obtained in radu et al rana preprint 02 06 eindhoven university of technology 2002 this paper considers a simple iterative scheme to solve the emerging nonlinear elliptic problems

efficient integration of the one dimensional time independent wave equation for bound states and for wave propagation the wave equation is solved with the phase integral approximation in the asymptotic regions and integrated numerically elsewhere in this application the phase integral approximation is relatively simple in higher orders and its accuracy can be precisely controlled with the derived error estimates for the bound states the analytical formulas defining the termination error in the wave function are derived for the wave propagation an efficient formalism is given for calculating the reflection transmission and absorption coefficients

periodic solutions of parabolic elliptic obstacle problems this paper deals with parabolic elliptic obstacle problems the problem is formulated as follows let sz be a bounded domain in rn iv 1 and for each i e r let 9 t be a relatively compact domain in sz for a given interval z to tl cc d to t co we then define non cylindrical subsets qjz i 1 2 of ix sz by

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

an elliptic parabolic free boundary problem in which c is a continuous nondecreas rg function defined on r such that c u is strictly increasing when u 0 ui 1 if bi 0 and bi 1 if ai 0 for i 0 and i 1 and fo fi and t are prescribed functions problem i arises in the theory of fluid aow through pargaily saturated media then t6 denotes the potential due to capiitary suction and c the moisture content the dependence of c on ie is found empirically to be as in fig i c being bounded above by the saturation value c 1 thus in regions where the medium is saturated the flow is of potential type and described by an ehiptic equation and in regions where the medium is unsaturated the flow is of diffusive type and described by a parabolic equation at the boundary between these regions the interface one expects u 0 and u continuous

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

nonstationary filtration in partially saturated porous media from the mathematical formulation of a one dimensional flow through a partially saturated porous medium we arrive at a nonlinear free boundary problem the boundary being between the saturated and the unsaturated regions in the medium in particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part

detailed behavior of the phase integral approximations at zeros and singularities of the square of the base function approximate solutions to the one dimensional time independent wave equation called the phase integral approximations are analyzed in the vicinity of characteristic points the approximations are of arbitrary order and are generated from an unspecified base function the general theory is illustrated by examples involving the power and or the exponential behavior of the square of the base function in these cases simple estimates are derived for the integrals which define the accuracy of the phase integral approximation and the optimum approximation order is determined

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

uniqueness of the bounded solution to a strongly degenerate parabolic problem abstract this paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem the equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies by using holmgren s approach we prove that the bounded solution of the associated initial boundary value problem is unique under some essentially necessary condition on the growth of the convection

slow damping of internal waves in a stably stratified fluid we study the damping of internal gravity waves in a stably stratified fluid with constant viscosity in two and three dimensional bounded domains for the linearized navier stokes equations for incompressible flow with no slip boundary conditions that model this fluid we prove there are non oscillatory normal modes with arbitrarily small exponential decay rates the proof is very different from that for a horizontally periodic layer and depends on a structure theorem for compact operators which are self adjoint with respect to an indefinite scalar product in a hilbert space we give a complete proof of this theorem which is closely related to results of pontrjagin

regularity of the interfaces with sign changes of solutions of the one dimensional porous medium equation abstract in previous papers we considered the cauchy problem for the one dimensional evolution p laplacian equation for nonzero bounded and nonnegative initial data having compact support and showed that after a finite time the set of spatial critical points of the nonnegative solution u u x t in u 0 consists of one point the spatial maximum point of u and the curve of the spatial maximum points is continuous with respect to the time variable since the spatial derivative x u satisfies the porous medium equation with sign changes the curve of the spatial maximum points is regarded as an interface with sign changes of x u on the other hand in a paper by m bertsch and d hilhorst 1991 appl anal 41 111 130 the interfaces where the solutions change their sign were studied in detail for the initial boundary value problems of the generalized porous medium equation over two dimensional cylinders but the monotonicity of the initial data is assumed there as is noted in section 4 of our earlier work 1996 j math anal appl 203 78 103 the monotonicity of x u t in some neighborhood of the spatial maximum point of u t cannot be assumed and therefore if this monotonicity for some large t 0 is proved then by the method of bertsch and hilhorst cited above one may get more precise regularity properties of the curve of the spatial maximum points the purpose of the present paper is twofold one is to remove some monotonicity assumption for initial data in bertsch and hilhorst s theorem concerning the regularity of the interfaces with sign changes of solutions of the one dimensional generalized porous medium equation by comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time the other is as a by product of the method to get c 1 regularity of the curves of the spatial maximum points of nonnegative solutions of the cauchy problem for the evolution p laplacian equation for sufficiently large t

nonlinear elliptic parabolic problems we introduce a notion of viscosity solutions for a general class of elliptic parabolic phase transition problems these include the richards equation which is a classical model in filtration theory existence and uniqueness results are proved via the comparison principle in particular we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data these results are new even in the linear case where we also show that viscosity solutions coincide with the regular weak solutions introduced in alt and luckhaus math z 183 311 341 1983

bounded weak solutions of an elliptic parabolic neumann problem abstract in this paper we establish existence and uniqueness for boundedweak solutions of an elliptic parabolic neumann problem we also describethe asymptotic behavior as t oo 1 introduction let c r r be a continuous function which is strictlyincreasing on r and identically equal to a constant on r see figure 1 fora bounded domain fi c rn with smooth boundary dfl we consider the neumann problem 1 1 f c u t au inqt fix 0 t 1 2 i du du f on du x 0 t 3 c u x 0 vo x xeu here u u x t a denotes the laplacian in r and 3 du the outward normal derivative on du for given functions and i n we discuss existence uniqueness and asymptotic behavior of weak solutions of problem i equation 1 1 is elliptic if u 0 and parabolic if u 0 because of this change of type we cannot a prioriexpect solutions to be classical the physical background of 1 1 lies in the theory of partially saturated flowsin porous media in that context u stands for the hydrostatic potential and c u for the moisture content or saturation the part of fi where u is negative is calledthe unsaturated region and that where u is positive the saturated region the set

the regular module problem i in the study of induced representations the following problem arises let h ag be a finite solvable group and k a field with chark a let v be an irreducible faithful primitive k 4g module suppose h contains a normal extraspecial r subgroup r with z r z h and that a acts faith fully on r under what conditions does a have a regular direct summand in vi in this paper we consider this question under the hypotheses that g mr where 1 m is normal abelian in am a is nilpotent a mr m r 1 and r z r is a faithful irreducible am modu e we show that a has at least three regular direct summands in v unless a exp m and r satisfy certain very restrictive conditions let h ag be a finite solvable group where a is nilpotent g is normal and y4 g 1 let k be a field of characteristic not dividing a and v a zc module assume that v is faithful and irreducible the study of induced representations often leads to the following two questions 1 regular orbit problem under what conditions does a have a regular orbit on f

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples

strong solutions to the richards equation in the unsaturated zone abstract we study the unsaturated case of the richards equation in three space dimensions with dirichlet boundary data we first establish an a priori l estimate with its help by means of a fixed point argument we prove global in time existence of a unique weak solution in sobolev spaces finally we are able to improve the regularity of this weak solution in order to gain a strong one