hyperasymptotics for nonlinear odes i a riccati equation we illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations the example is a riccati equation the main tools that we need are transseries expansions and the riemann sheet structure of the borel transform of the divergent asymptotic expansions hyperasymptotic expansions determine the solutions uniquely a numerical illustration is included

group classification of variable coefficient generalized kawahara equations an exhaustive group classification of variable coefficient generalized kawahara equations is carried out as a result we derive new variable coefficient nonlinear models admitting lie symmetry extensions all inequivalent lie reductions of these equations to ordinary differential equations are performed we also present some examples on the construction of exact and numerical solutions

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

computation of homoclinic solutions to periodic orbits in a reduced water wave problem abstract this paper concerns homoclinic solutions to periodic orbits in a fourth order hamiltonian system arising from a reduction of the classical water wave problem in the presence of surface tension these solutions correspond to travelling solitary waves which converge to non decaying ripples at infinity an analytical result of amick and toland 1992 showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit is extended numerically also a related result by amick and mcleod 1991 showing the non existence of homoclinic solutions to zero is motivated geometrically a general boundary value method is constructed for continuation of homoclinic orbits to periodic orbits in hamiltonian and reversible systems numerical results are presented using the path following software auto showing that the amick toland solutions persist well away from the singular limit and for large amplitude periodic orbits special account is taken of the phase shift between the two periodic solutions in the asymptotic limits furthermore new multi modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis which is verified a posteriori by explicit computation continuation of these new solutions reveals limit points with respect to the singular parameter

analyticity and nonexistence of classical steady hele shaw fingers this paper concerns analyticity of a classical steadily translating nger in a hele shaw cell and nonexistence of solutions when the relative nger width is smaller than 1 it is proven that any classical solution to the nger problem if it exists for sufciently small but nonzero surface tension is close to some saffman taylor zero surface tension solution and satises some algebraic decay conditions at 1 must belong to the analytic function space a0 as dened in section 1 and chosen in a previous study 34 of the existence of nger solutions further it is proven that for any x ed 2 0 1 there can be no classical steadynger solution when surface tension is sufciently small which contradicts a previous conclusion based on numerical simulation c 2002 wiley periodicals inc

structural stability of the korteweg de vries solitons under a singular perturbation abstract we investigate the stability of a solitary wave solution of the korteweg de vries equation when a fifth order spatial derivative term is added we show that the solution ceases to be strictly localized but develops an infinite oscillating tail and we compute the amplitude of the latter

exponential asymptotics and boundary value problems keeping both sides happy at all orders we introduce templates for exponential asymptotic expansions that in contrast to matched asymptotic approaches enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter graphic 1 and have a single boundary layer at one end of the interval for linear equations the template is a transseries that takes the form of a sliding ladder of exponential scales for nonlinear equations the transseries template is a two dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed an exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of terms in the series beyond all orders in the nonlinear case we also demonstrate how these transseries can be resummed to generate multiple scales type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion 1 embed inline graphic 1 gif

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

asymptotics beyond all orders and analytic properties of inverse laplace transforms of solutions a number of modern mathematical and physical problems require the study of delicate asymptotic properties lying beyond the power series asymptotics in this paper we suggest a link between these asymptotic problems and some analytic properties of inverse laplace transforms of the corresponding solutions the main result claims that these inverse transforms are holomorphic in an appropriately cut complex plane a direct consequence of this is the nonexistence of solutions to the class of asymptotics beyond all orders problems such as regular shocks of the kuramoto sivashinsky equaiton gr needle crystal solutions of the simple geometrical model of crystal growth ks solitary wave solutions to a class of the fifth order kortveg de vries equations ko sect 8 gj homoclinic orbits of some singularly perturbed mappings ec hm and others

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

on the oscillatory tails with arbitrary phase shift for solutions of the perturbed korteweg de vries equation a singularly perturbed korteweg de vries kdv equation e 2 eta xxxx eta xx eta eta 2 0 for infty 0 is a small parameter it is known that the equation has homoclinic solutions connected to small periodic orbits at infinity the relationship between amplitude and phase shift of the periodic orbits was established for noncritical cases to study the critical cases an asymptotic method is introduced here to formally derive the homoclinic solutions to periodic orbits with arbitrary phase shift the exact asymptotic formula relating the amplitude of oscillations to the phase shift is obtained the amplitude of oscillations is determined by the phase shift and the range of the amplitude can be from o e n with any n ge 1 up to o e 2 exp pi e for different phase shift of the oscillations

analytic methods for obstruction to integrability in discrete dynamical systems a unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized borel summability this continuation allows for painleve property methods to be extended to difference equations it is shown that the painleve property pp induces under relatively general assumptions a dichotomy within first order difference equations all equations with pp can be solved in closed form on the contrary absence of pp implies under some further assumptions that the local conserved quantities are strictly local in the sense that they develop singularity barriers on the boundary of some compact set the technique produces analytic formulas to describe fractal sets originating in polynomial iterations 2004 wiley periodicals inc

radiating solitary waves of a model evolution equation in fluids of finite depth abstract the evolution equation proposed by kubota et al j hydronautics 12 1978 157 165 and joseph j phys a 10 1977 225 227 for long waves in stratified fluids of finite depth is considered under a small third order derivative dispersive perturbation solitary wave solutions to this model equation become weakly nonlocal they develop oscillatory tails of infinite extent and exponentially small amplitude here these tails are calculated asymptotically following a perturbation procedure in the wavenumber domain

soliton fusion and fission in a generalized variable coefficient fifth order korteweg de vries equation in fluids under investigation in this paper is a generalized variable coefficient fifth order korteweg de vries equation which describes the interaction between a water wave and a floating ice cover or the gravity capillary waves via the hirota method bell polynomial approach and symbolic computation bilinear forms n soliton solutions backlund transformation and lax pair are derived infinitely many conservation laws are obtained based on the bell polynomial typed backlund transformation soliton fusion and fission and influence of the variable coefficients from the equation are analyzed both variable coefficients c t and n t are in direct proportion to the soliton velocities but have no effect on the amplitudes while another constant coefficient can affect the types of the interactions in the sense of the elastic or inelastic elastic inelastic interactions among the three solitons are presented as well

on asymptotically equivalent shallow water wave equations abstract the integrable third order korteweg de vries kdv equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves however at quadratic order this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other under a group of nonlinear non local normal form transformations introduced by kodama in combination with the application of the helmholtz operator these kodama helmholtz kh transformations are used to present connections between shallow water waves the integrable fifth order kdv equation and a generalization of the camassa holm ch equation that contains an additional integrable case the dispersion relation of the full water wave problem and any equation in this family agree to fifth order the travelling wave solutions of the ch equation are shown to agree to fifth order with the exact solution

weakly non local solutions for capillary gravity waves fifth degree korteweg de vries equation hunter and scheurle have shown that capillary gravity water waves in the vicinity of bond number bo l 3 are consistently modelled by the korteweg de vries equation with the addition of a fifth derivative term this wave equation does not have strict soliton solutions for bo l 3 because the near solitons have oscillatory wings that extend indefinitely from the core of the wave however these solutions are arbitrarily small perturbations of solitary waves because the amplitude of the wings is exponentially small in the amplitude 6 of the core pomeau ramani and grammaticos have calculated the amplitude of the wings by applying matched asymptotics in the complex plane in the limit e 0 in this article we describe a mixed chebyshev radiation function pseudospectral method which is able to calculate the weakly non local solitons for all e we show that for fixed phase speed the solitons form a three parameter family because the linearized wave equation has three eigensolutions we also show that one may repeat the soliton with even spacing to create a three parameter of periodic solutions which we also compute because the amplitude of the wings is exponentially small these non local capillary gravity solitons are as interesting as the classical localized solitons that solve the korteweg de vries equation

from the taylor series of analytic functions to their global analysis abstract we analyze the conditions on the taylor coefficients of an analytic function to admit global analytic continuation complementing a recent paper of breuer and simon on general conditions for natural boundaries to form a new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals the integral representations yield global information such as analytic continuability position of singularities asymptotics for large values of the variable and asymptotic location of zeros

existence and uniqueness for a class of nonlinear higher order partial differential equations in the complex plane we prove existence and uniqueness results for nonlinear third order partial differential equations of the form

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

on exponential asymptotics for nonseparable wave equations i complex geometrical optics and connection kelley s geometrical optics method opens an efficient approach to the study of waves when the variables in the wave equation cannot be separated the trajectories of the hamilton jacobi equation are then needed for access to the wave structure when several disjoint potential wells are present account of the interaction between them requires consideration of trajectories and tori in complex phase space and of the unfamiliar connectivity of such complex toroids it is shown how connection theory across wells and barriers can be reformulated in a manner that is simple and flexible and assures the rigorous reliability needed for exponentially precise approximation to functionals of the hamiltonian operator in several dimensions to make this difficult and unfamiliar subject more accessible its discussion is preceded by a simple example of a double well on a strip domain in two dimensions parts ii and iii will apply the results to the quantitative resolution of degeneracy of discrete and continuous spectra

automorphisms of some banach algebras of analytic functions abstract in this paper we factorize an arbitrary automorphism of the algebra considered by peters in j peters j functional anal 59 1984 498 534 and its weak version and use it to identify an isomorphism invariant which is the quotient group of the full automorphism group over the maximal ideal fixing automorphisms subgroup

a topological study of gravity free surface waves generated by bluff bodies using the method of steepest descents the standard analytical approach for studying steady gravity free surface waves generated by a moving body often relies upon a linearization of the physical geometry where the body is considered asymptotically small in one or several of its dimensions in this paper a methodology that avoids any such geometrical simplification is presented for the case of steady state flows at low speeds the approach is made possible through a reduction of the water wave equations to a complex valued integral equation that can be studied using the method of steepest descents the main result is a theory that establishes a correspondence between different bluff bodied free surface flow configurations with the topology of the riemann surface formed by the steepest descent paths then when a geometrical feature of the body is modified a corresponding change to the riemann surface is observed and the resultant effects to the water waves can be derived this visual procedure is demonstrated for the case of two dimensional free surface flow past a surface piercing ship and over an angled step in a channel

optical uniform estimates and rigorous asymptotics beyond all orders for a class of ordinary differential equations for first order differential equations of the form y9 p p x y p and second order homogeneous linear differential equations y a x y9 b x y 0 with locally integrable coefficients having asymptotic possibly divergent power series when x oo on a ray arg x const under some further assumptions it is shown that on the given ray there is a one to one correspondence between true solutions and complete formal solutions the correspondence is based on asymptotic inequalities which are required to be uniform in x and optimal with respect to certain weights

time asymptotics of the schrodinger wave function in time periodic potentials we study the transition to the continuum of an initially bound quantum particle in d d 1 2 3 subjected for t 0 to a time periodic forcing of arbitrary magnitude the analysis is carried out for compactly supported potentials satisfying certain auxiliary conditions it provides complete analytic information on the time laplace transform of the wave function from this comprehensive time asymptotic properties borel summable transseries follow we obtain in particular a criterion for whether the wave function gets fully delocalized complete ionization this criterion shows that complete ionization is generic and provides a convenient test for particular cases when satisfied it implies absence of discrete spectrum and resonances of the associated floquet operator as an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property

resonance theory for schrodinger operators resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods a general theory is developed with new and weaker conditions allowing for perturbations of threshold eigenvalues and relaxed fermi golden rule the exponential decay rate of resonances is addressed its uniqueness in the time dependent picture is shown in certain cases the relation to the existence of meromorphic continuation of the properly weighted green s function to time dependent resonance is further elucidated by giving an equivalent time dependent asymptotic expansion of the solutions of the schrodinger equation

some point derivations for semicrossed products abstract we consider point derivations of semicrossed product algebras in particular we focus on point derivations into the upper triangular n n matrices and show that such point derivations are inner finally we consider point derivations corresponding to periodic points where the range is a standard semicrossed product considered in previous work on point derivations for graph algebras again we conclude that all such point derivations are inner

exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave abstract a recently discovered phenomenon of capillary gravity waves on water is a solitary wave of elevation plus small amplitude ripples at infinity it has been conjectured by beale comm pure appl math 44 1991 211 257 that the amplitude of the ripples should be exponentially small beyond all orders in this paper we prove this conjecture

exponential asymptotics for thin film rupture the formation of singularities in models of many physical systems can be described using self similar solutions one particular example is the finite time rupture of a thin film of viscous fluid which coats a solid substrate previous studies have suggested the existence of a discrete countably infinite number of distinct solutions of the nonlinear differential equation which describes the self similar behavior however no analytical mechanism for determining these solutions was identified in this paper we use techniques in exponential asymptotics to construct the analytical selection condition for the infinite sequence of similarity solutions confirming the conjectures of earlier numerical studies

annihilator preserving maps multipliers and derivations abstract for a commutative subspace lattice l in a von neumann algebra n and a bounded linear map f n alg l b h we show that if af b c 0 for all a b c n alg l satisfying ab bc 0 then f is a generalized derivation for a unital c algebra a a unital banach a bimodule m and a bounded linear map f a m we prove that if f a b 0 for all a b a with ab 0 then f is a left multiplier as a consequence every bounded local derivation from a c algebra to a banach a bimodule is a derivation we also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier

exponential asymptotics and stokes lines in nonlinear ordinary differential equations a technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented the approach is based on the application of a wkbj type ansatz to the late terms in the naive asymptotic expansion and the identification of stokes lines and is closely related to the well known stokes line smoothing phenomenon in linear ordinary differential equations the method is illustrated by application to examples

solitary waves with damped oscillatory tails an analysis of the fifth order korteweg de vries equation abstract we construct oscillatory solitary wave solutions of a fifth order korteweg de vries equation where the oscillations decay at infinity these waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide our approach is a wave packet analysis about this wavenumber which leads in the first instance to a higher order nonlinear schrodinger equation from which we then obtain the steady solitary wave solution we then describe a complementary normal form analysis which leads to the same result in addition we derive the nonlinear schrodinger equation for all wavenumbers and list all the various anomalous cases

nonexistence of classical steady hele shaw bubble this paper concerns nonexistence of steadily translating bubbles in a hele shaw cell for small but nonzero surface tension it is proved that for any fixed bubble velocity u relative to the fluid velocity at infinity in the interval 2 there can be no classical steady bubble solution

rigorous bounds of stokes constants for some nonlinear ordinary differential equations at rank one irregular singularities a rigorous way of obtaining sharp bounds for stokes constants is introduced and illustrated on a concrete problem arising in applications

investigation on a nonisospectral fifth order korteweg de vries equation generalized from fluids in this paper a nonisospectral fifth order korteweg de vries equation generalized from fluids is investigated with symbolic computation such equation is transformed into its bilinear form through a proposed dependent variable transformation with one more parameter than those in the existing literature n soliton solutions backlund transformation and lax pair in the explicit forms are constructed based on the above results the characteristic line method is applied to discuss the features of the solitons for the nonisospectral problem i e the controllable solitonic velocities and widths four types of solitonic structures with the different solitonic velocities widths amplitudes and backgrounds are also illustrated

existence uniqueness analyticity and borel summability for boussinesq equations through borel summation methods we analyze the boussinesq equations for coupled fluid velocity and temperature fields 1 ut u p u u ae2 f t u we prove that an equivalent system of integral equations in the borel variable p r dual to 1 t has a unique solution in a class of exponentially bounded functions implying the existence of a classical solution to 1 in a complex t region that includes a real positive time axis segment for analytic initial data and forcing it is shown that the solution is borel summable implying that formal series in powers of t is gevrey 1 asymptotic and within the time interval of existence the solution remains analytic with the same analyticity strip width as the initial data and forcing we also determine conditions on the integral equation solution that improve the estimate for existence time

pseudospectral delves freeman computations of the radiation coefficient for weakly nonlocal solitary waves of the third order nonlinear schroedinger equation and their relation to hyperasymptotic perturbation theory we revive a strategy of delves to precondition a spectral calculation using an almost diagonal galerkin matrix we also show that hyperasymptotic singular perturbation theory is a specialization of the delves freeman iteration to a single step with a diagonal galerkin matrix we calculate the first three orders in the hyperasymptotic expansion of the radiation coefficient of the weakly nonlocal envelope solitary waves of the third order nonlinear schroedinger tnls equation which is important in fiber optics telecommunications and water waves in a long appendix it was necessary to develop some non numerical but numerically essential additions to the theory of the tnls equation good numerics often requires a much deeper understanding of a problem than the generation of a string of power series coefficients

nest representations of directed graph algebras this paper is a comprehensive study of the nest representations for the free semigroupoid algebra mathfrak l g of a countable directed graph g as well as its norm closed counterpart the tensor algebra mathcal t g we prove that the finite dimensional nest representations separate the points in mathfrak l g and a fortiori in mathcal t g the irreducible finite dimensional representations separate the points in mathfrak l g if and only if g is transitive in components which is equivalent to being semisimple also the upper triangular nest representations separate points if and only if for every vertex x in mathcal t g supporting a cycle x also supports at least one loop edge we also study faithful nest representations we prove that mathfrak l g or mathcal t g admits a faithful irreducible representation if and only if g is strongly transitive as a directed graph more generally we obtain a condition on g which is equivalent to the existence of a faithful nest representation we also give a condition that determines the existence of a faithful nest representation for a maximal type mathbb n nest

foundational aspects of singular integrals abstract we investigate integration of classes of real valued continuous functions on 0 1 of course difficulties arise if there is a non l 1 element in the class and the hadamard finite part integral p f does not apply such singular integrals arise naturally in many contexts including pdes and singular odes the lebesgue integral as well as p f starting at zero obey two fundamental conditions i they act as antiderivatives and ii if f g on 0 a then their integrals from 0 to x coincide for any x 0 a we find that integrals from zero with the essential properties of p f plus positivity exist by virtue of the axiom of choice ac on all functions on 0 1 which are l 1 e 1 for all e 0 however this existence proof does not provide a satisfactory construction without some regularity at 0 the existence of general antiderivatives which satisfy only i and ii above on classes with a non l 1 element is independent of zf the usual zfc axioms for mathematics without ac and even of zfdc zf with the axiom of dependent choice moreover we show that there is no mathematical description that can be proved within zfc or even extensions of zfc with large cardinal hypotheses to uniquely define such an antiderivative operator such results are precisely formulated for a variety of sets of functions and proved using methods from mathematical logic descriptive set theory and analysis we also analyze p f on analytic functions in the punctured unit disk and make the connection to singular initial value problems

the generation of radiating waves in a singularly perturbed korteweg de vries equation abstract we consider a fifth order kdv equation where the fifth order derivative term is multiplied by a small parameter it has been conjectured that this equation admits a non local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core we prove that this solution cannot be exactly steady and instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail the decay rate is calculated in the limit as the parameter tends to zero in order to verify the analytical results we have developed a high precision spectral method for numerical integration of this equation the analytical and numerical result show good agreement

the stokes phenomenon as a boundary value problem we show that the stokes phenomenon is related to a boundary value problem in two dimensions for a large class of functions and near the stokes lines the subdominant multiplier satisfies a two dimensional boundary value problem of convection diffusion type with discontinuous dirichlet conditions at the boundary the solution of this problem is approximated by an error function of a certain combination of the polar variables of the plane which measures the distance to the stokes line then we offer a different and very simple explanation of the smoothing of the stokes phenomenon showing the universality of the error function as the smoothing factor

rigorous results in selection of steady needle crystals abstract this paper concerns the existence of solutions to a steady needle crystal growth problem in a one sided model we rigorously prove that for small nonzero anisotropy analytic symmetric needle crystal solutions exist in the limit of surface tension e 2 if only if the stokes constant s for a relatively simple nonlinear differential equation is zero this stokes constant s depends on the parameter 2 9 7 e 8 7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of it is also proven that for 0 there can be no symmetric needle crystal solution in the considered space the methodology consists of two steps first the original problem is reduced to a weak half strip problem for any in a compact set of 0 1 by relaxation of the symmetry condition the weak problem is shown to have a unique solution in the function space considered for any 0 m for some m 0 when a symmetry is invoked the weak problem is shown equivalent to the original needle crystal problem next by considering the behavior of the solution in neighborhood of an appropriate complex turning point for 0 m we extract an exponentially small term in e as e 0 that generally violates the symmetric condition we prove that the symmetry condition is satisfied for small e when the parameter is constrained appropriately

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

normal forms of holomorphic matrix valued functions and corresponding forms for singular differential operators abstract normal forms of a holomorphic matrix valued function a z under different classes of similarity transformations and corresponding forms of the singular differential operator z r d dz a z r n r 1 are investigated the problem of holomorphic similarity of matrix valued functions is given special consideration

the ideal structure of certain nonselfadjoint operator algebras soit x un systeme dynamique localement compact et z x co x la sous algebre close en norme du produit croise zx co x engendre par les puissances non negatives de dans le cas homeomorphisme on determine la structure ideale de ces algebres dans le cas ou agit librement

a simple explanation of the stokes phenomenon the stokes phenomenon is known to be a pervasive feature of asymptotics but its explanation in the literature is obscured by intricate and lengthy technicalities this article presents a simpler approach to its understanding and treatment as a natural aspect of a well motivated characterization of functions by approximands of different multivaluedness

thin layer solutions of the helmholtz and related equations this paper concerns a certain class of two dimensional solutions to four generic partial differential equations the helmholtz modified helmholtz and convection diffusion equations and the heat conduction equation in the frequency domain and the connections between these equations for this particular class of solutions specifically we consider thin layer solutions valid in narrow regions across which there is rapid variation in the singularly perturbed limit as the coefficient of the laplacian tends to zero for the well studied helmholtz equation this is the high frequency limit and the solutions in question underpin the conventional ray theory wkb approach in that they provide descriptions valid in some of the regions where these classical techniques fail examples are caustics shadow boundaries whispering gallery and creeping waves and focusing and bouncing ball modes it transpires that virtually all such thin layer models reduce to a class of generalized parabolic wave equations of which the heat conduction equation is a special case moreover in most situations we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the helmholtz equation we also show how reasonably well understood thin layer phenomena associated with any one of the four generic equations may translate into less well known effects associated with the others in addition our considerations also shed some light on the relationship between the methods of matched asymptotic wkb and multiple scales expansions

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

unfolding the high orders of asymptotic expansions with coalescing saddles singularity theory crossover and duality we study the leading behaviour of the late coefficients high orders r of asymptotic expansions in a large parameter k for contour integrals involving a cluster of coalescing saddles and thereby establish the form of the divergence of the expansions the two principal cases are saddle to cluster where the integral is through a simple saddle and its expansion diverges because of a distant cluster and cluster to saddle where the integral is through a cluster and its expansion diverges because of a distant simple saddle in both the large r coefficients are dominated by the factorial divided by power familiar in asymptotics but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters a a 1 a 2 in the integrand the crossover between different forms is described by a series of canonical integrals built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence the arguments of these integrals involve not only the a but also fractional powers of r which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals a by product of the cluster to saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions

exponential asymptotics and stokes lines in a partial differential equation a singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered a steepest descent analysis of the fourier transform solution identifies asymptotic contributions from saddle points end points and poles and the stokes lines across which these may be switched on and off these results are then derived directly from the equation by optimally truncating the naive perturbation expansion and smoothing the stokes discontinuities the analysis reveals two new types of stokes switching a higher order stokes line which is a stokes line in the approximation of the late terms of the asymptotic series and which switches on or off stokes lines themselves and a second generation stokes line in which a subdominant exponential switched on at a primary stokes line is itself responsible for switching on another smaller exponential the new stokes lines discussed by berk et al berk et al 1982 j math phys 23 988 1002 are second generation stokes lines while the vanishing stokes lines discussed by aoki et al aoki et al 1998 in microlocal analysis and complex fourier analysis ed k f t kawai pp 165 176 are switched off by a higher order stokes line

experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix the d o r t method french acronym for decomposition of the time reversal operator is an active remote sensing technique using arrays of antennas for the detection and localization of scatterers prada et al j acoust soc am 99 2067 2076 1996 the analogy between the time reversal operator and the covariance matrix used for classical sources separation in passive remote sensing bienvenu et al ieee trans assp 31 1235 1247 1983 is established then an experiment of subwavelength detection and localization of point like scatterers with a linear array of transducers is presented using classical estimators in reception like maximum likelihood and multiple signal characterization music two point like scatterers separated by 3 and placed at 100 from the array of transducers are resolved in these experiments the role of multiple scattering and the existence of additional eigenvectors associated with dipolar and monopolar radiation of each scatterer is discussed

stokes phenomenon and matched asymptotic expansions this paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting stokes phenomenon in particular the approach highlights the way in which the local structure and the possibility of finding stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

derivations for a class of matrix function algebras abstract we study a class of matrix function algebras here denoted t c n we introduce a notion of point derivations and classify the point derivations for certain finite dimensional representations of t c n we use point derivations and information about n n matrices to show that every t c n valued derivation on t c n is inner

exponential asymptotics and capillary waves recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension in the absence of gravity exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow for example stagnation points and corners meet it the amplitude of these waves is determined and the implications are considered for many quite general flows the asymptotic results are compared to numerical simulations of the full problem for flow over a polygonal plough and for flow round a right angled corner and they show remarkably good agreement even for quite large values of the surface tension parameter

a relationship between time reversal imaging and maximum likelihood scattering estimation time reversal methods have attracted increasing interest recently the so called computational time reversal approach creates an image of the illuminated scene by computing the back propagated field and is useful for detecting and estimating targets in the scene in shi and nehorai ldquomaximum likelihood estimation of point scatterers for computational time reversal imaging rdquo communications in information and systems vol 5 no 2 pp 227 256 2005 we estimated point scatterers by maximum likelihood estimate mle using the born approximated physical model as well as the foldy lax model in this correspondence we further find an explicit relationship between energy based basic time reversal imaging and the mle approach the time reversal imaging function differs by only a scaling factor from the likelihood imaging function using the estimated scattering potential when a single scatterer model is employed furthermore this scaling factor is a function of the imaging position only we show that as a result time reversal imaging has a near far problem that tends to produce a weaker image for areas further away from the imaging arrays whereas the mle based image is more balanced experimental results confirm this conclusion

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

stokes phenomena in discrete painleve i in this study we consider the asymptotic behaviour of the first discrete painleve equation in the limit as the independent variable becomes large using an asymptotic series expansion we identify two types of solutions which are pole free within some sector of the complex plane containing the positive real axis using exponential asymptotic techniques we determine stokes phenomena effects present within these solutions and hence the regions in which the asymptotic series expression is valid from a careful analysis of the switching behaviour across stokes lines we find that the first type of solution is uniquely defined while the second type contains two free parameters and that the region of validity may be extended for appropriate choice of these parameters

weakly nonlocal solitary waves in a singularly perturbed korteweg de vries equation a fifth order korteweg de vries equation is considered where the fifth order derivative term is multiplied by a small parameter it is known that solitary wave solutions of this model equation are nonlocal in that the central core of the wave is accompanied by copropagating trailing oscillations here using the techniques of exponential asymptotics these solutions are reexamined and it is established that they form a one parameter family characterized by the phase shift of the trailing oscillations explicit asymptotic formula relating the oscillation amplitude to the phase shift are obtained

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

analytical and numerical studies of a singularly perturbed boussinesq equation we study the singularly perturbed sixth order boussinesq equation recently introduced by daripa and hua appl math comput 101 1999 159 motivated by their work we formally derive this equation from two dimensional potential flow equations governing the small amplitude long capillary gravity waves on the surface of shallow water for bond number very close to but less than 1 3 on the basis of far field analyses and heuristic arguments we show that the traveling wave solutions of this equation are weakly non local solitary waves characterized by small amplitude fast oscillations in the far field we review various analytical and numerical methods originally devised to obtain this type of weakly non local solitary wave solutions of the singularly perturbed fifth order kdv equation using these methods we obtain weakly non local solitary wave solutions of the singularly perturbed sixth order boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far field

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

multi pulse embedded solitons as bound states of quasi solitons abstract embedded solitons ess is a name given to localised solutions of multi component dispersive wave models that occur in resonance with the linear spectrum they exist at isolated codimension one parameter values within the steady state equations which have linearisation with eigenvalues and i at the same time quasi solitons qss or generalised solitary waves with non decaying radiation tails are known to be endemic we consider the general question of when two qss may be glued together to form a two humped bound state bs es in the limit 0 a generalisation of the method of gorshkov and ostrovsky is used within the framework of general normal forms for hamiltonian reversible systems a simple asymptotic formula is derived that governs the existence symmetry and accumulation rate of the bss this generalises earlier ad hoc calculations for specific examples the formula depends on few simple ingredients the eigenvalues of the equilibrium an asymptotic estimate for the tail amplitude of the qss the birkhoff signature and symmetry properties of the hamiltonian only the tail amplitude estimate is difficult to calculate requiring exponential asymptotics but it is shown that this only affects the third order term in the asymptotic formula the calculation is worked out in detail for the steady states of several example pde systems taken from nonlinear optics and fluid mechanics and excellent agreement with numerical results is found

on the non universality of the error function in the smoothing of stokes discontinuities the non universality of the error function in the smoothing of stokes lines is demonstrated by means of an example with smoothing function e u 2 m g u d u where m is any integer greater than 2 and g is any polynomical of degree less than or equal to 2 m 1

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude