on critical behaviour in generalized kadomtsev petviashvili equations an asymptotic description of the formation of dispersive shock waves in solutions to the generalized kadomtsev petviashvili kp equation is conjectured the asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the painleve i hierarchy several examples are discussed numerically to provide strong evidence for the validity of the conjecture the numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the subcritical kp equations while in the supercritical kp equations a blow up occurs after the formation of the dispersive shock waves

hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs we consider time harmonic wave equations in cylindrical wave guides with physical solutions for which the signs of group and phase velocities differ the perfectly matched layer methods select modes with positive phase velocity and hence they yield stable but unphysical solutions for such problems we derive an infinite element method for a physically correct discretization of such wave guide problems which is based on a laplace transform in propagation direction in the laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are hardy spaces of a curved domain the hardy space is constructed such that it contains a simple and convenient riesz basis with small condition numbers in this paper the new method is only discussed for a one dimensional fourth order model problem exponential convergence is shown the method does not use a modal separation and works on an interval of frequencies numerical experiments confirm exponential convergence

convergence of the infinite element methods for the helmholtz equation in separable domains to the best knowledge of the authors this work presents the first convergence analysis for the infinite element method iem for the helmholtz equation in exterior domains the approximation applies to separable geometries only combining an arbitrary finite element fe discretization on the boundary of the domain with a spectral like approximation in the radial direction with shape functions resulting from the separation of variables the principal idea of the presented analysis is based on the spectral decomposition of the problem

nonlinear diffraction and caustic formation this paper reports a new family of symmetries to the zabolotskaya khokhlov dissipative zabolotskaya khokhlov and kadomtsev petviashvili equations it also reports the details of the corresponding set of exact similarity solutions to the zabolotskaya khokhlov equation and the corresponding reduction of the dissipative zabolotskaya khokhlov equation onto the generalized burgers equation and implies that of the kadomtsev petviashvili equation onto a simpler equation the bearing that the symmetries and exact solutions have on other work is discussed the first non trivial smooth global solutions to the zabolotskaya khokhlov equation are presented answering a conjecture as to the existence of such the formation of line caustics is examined using the exact solutions quasi statically giving new results

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

topological galois theory abstract we introduce an abstract topos theoretic framework for building galois type theories in a variety of different mathematical contexts such theories are obtained from representations of certain atomic two valued toposes as toposes of continuous actions of a topological group our framework extends grothendieck s theory of galois categories and allows to build galois type equivalences in new contexts such as for example graph theory and finite group theory

sites whose topoi are the smooth representations of locally prodiscrete monoids abstract we define a class of sites such that the associated topos is equivalent to the category of smooth sets representations of some locally prodiscrete monoids to be defined examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups this is a generalization of the fact that the topos associated with the etale site of a scheme is equivalent to the category of sets with continuous action by the etale fundamental group we then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group

dispersive shock waves in the kadomtsev petviashvili and two dimensional benjamin ono equations abstract dispersive shock waves dsws in the kadomtsev petviashvili kp equation and two dimensional benjamin ono 2dbo equation are considered using step like initial data along a parabolic front employing a parabolic similarity reduction exactly reduces the study of such dsws in two space one time 2 1 dimensions to finding dsw solutions of 1 1 dimensional equations with this ansatz the kp and 2dbo equations can be exactly reduced to the cylindrical korteweg de vries ckdv and cylindrical benjamin ono cbo equations respectively whitham modulation equations which describe dsw evolution in the ckdv and cbo equations are derived and riemann type variables are introduced dsws obtained from the numerical solutions of the corresponding whitham systems and direct numerical simulations of the ckdv and cbo equations are compared with very good agreement obtained in turn dsws obtained from direct numerical simulations of the kp and 2dbo equations are compared with the ckdv and cbo equations again with good agreement it is concluded that the 2 1 dsw behavior along self similar parabolic fronts can be effectively described by the dsw solutions of the reduced 1 1 dimensional equations

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

a tannakian context for galois theory abstract strong similarities have been long observed between the galois and tannaka theories of the representation of groups in this paper we construct an explicit neutral tannakian context for the galois theory of atomic topoi and prove equivalence for the fundamental theorem since the theorem is known for the galois context this yields a proof of the fundamental recognition theorem for a new tannakian context this example is different from the additive cases or their generalization for which the theorem is known to hold and for which the unit of the tensor product is always an object of finite presentation which is not the case in our context

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

whitham modulation theory for 2 1 dimensional equations of kadomtsev petviashvili type whitham modulation theory for certain two dimensional evolution equations of kadomtsev petviashvili kp type is presented three specific examples are considered in detail the kp equation the two dimensional benjamin ono 2dbo equation and a modified kp m2kp equation a unified derivation is also provided in the case of the m2kp equation the corresponding whitham modulation system exhibits features different from the other two the approach presented here does not require integrability of the original evolution equation indeed while the kp equation is known to be a completely integrable equation the 2dbo equation and the m2kp equation are not known to be integrable in each of the cases considered the whitham modulation system obtained consists of five first order quasilinear partial differential equations the riemann problem i e the analogue of the gurevich pitaevskii problem for the one dimensional reduction of the m2kp equation is studied for the m2kp equation the system of modulation equations is used to analyze the linear stability of traveling wave solutions

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

whitham modulation theory for the kadomtsev petviashvili equation the genus 1 kadomtsev petviashvili kp whitham system is derived for both variants of the kp equation namely the kpi and kpii equations the basic properties of the kp whitham system including symmetries exact reductions and its possible complete integrability together with the appropriate generalization of the one dimensional riemann problem for the korteweg de vries equation are discussed finally the kp whitham system is used to study the linear stability properties of the genus 1 solutions of the kpi and kpii equations it is shown that all genus 1 solutions of kpi are linearly unstable while all genus 1 solutions of kpii are linearly stable within the context of whitham theory

the generalized burgers and zabolotskaya khokhlov equations transformations exact solutions and qualitative properties a point transformation between forms of the generalized burgers equation gbe first given by cates 1989 is investigated applications include generalizations of scott s 1981 classification of long time behaviour for compressive wave solutions of the gbe and the equivalence of the exponential and cylindrical forms of the gbe yielding an exact solution for the exponential gbe applications to nonlinear diffractive acoustics are considered by using a similarity reduction of the dissipative zabolotskaya khokhlov dzk equation describing the evolution of nearly plane waves in a weakly nonlinear medium with allowance for transverse variation effects onto the gbe the result is that waves from parabolic sources may be described by the cylindrical gbe in the case of two dimensions and by the spherical gbe in the three dimensional cylindrically symmetric case furthermore results on the formation of shocks and caustics in the context of the zk equation are presented along with an exact solution to the dzk equation exact solutions with caustic singularities are studied along with a possible mechanism for their control finally results on the evolution of a shock approaching a caustic are given through the identification of a series of parameter regimes dependent on the diffusivity

the halfspace matching method a new method to solve scattering problems in infinite media abstract we are interested in acoustic wave propagation in time harmonic regime in a two dimensional medium which is a local perturbation of an infinite isotropic or anisotropic homogeneous medium we investigate the question of finding artificial boundary conditions to reduce the numerical computations to a neighborhood of this perturbation our objective is to derive a method which can extend to the anisotropic elastic problem for which classical approaches fail the idea consists in coupling several semi analytical representations of the solution in halfspaces surrounding the defect with a finite element computation of the solution around the defect as representations of the same function they have to match in the infinite intersections of the halfspaces it leads to a formulation which couples via integral operators the solution in a bounded domain including the defect and its traces on the edge of the halfspaces a stability property is shown for this new formulation

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

pole condition a numerical method for helmholtz type scattering problems with inhomogeneous exterior domain this paper presents a new numerical method for the solution of exterior helmholtz scattering problems which is applicable to inhomogeneous exterior domains and a wide class of geometries the algorithm is based on the pole condition which is a general radiation condition and allows a treatment of exterior helmholtz problems without an explicit knowledge of green s functions or a series representation our algorithm is based on a numerical approximation of the singularities of a laplace transform of the exterior solution numerical examples illustrate the performance of the method

discrete transparent boundary conditions for the numerical solution of fresnel s equation the paper presents a construction scheme of deriving transparent i e reflection free boundary conditions for the numerical solution of fresnel s equation being formally equivalent to schrodinger s equation in contrast to previous suggestions the method advocated here treasts the discrete problem after discretization of the time like variable i e in a rothe method which leads to a sequence of coupled boundary value problems the thus obtained boundary conditions appear to be of a nonlocal cauchy type as it turns out each kind of linear implicit discretization induces its own discrete transparent boundary conditions numerical experiments on technologically relevant examples from integrated optics are included

hardy space infinite elements for scattering and resonance problems this paper introduces a new type of infinite element for scattering and resonance problems that is derived from a variant of the pole condition as radiation condition this condition states that a certain transform of the exterior solution belongs to the hardy space of l 2 boundary values of holomorphic functions on the unit disc if and only if the solution is outgoing we obtain a symmetric variational formulation of the problem in this hardy space our infinite elements correspond to a galerkin discretization with respect to the standard monomial orthogonal basis of this hardy space and lead to simple element matrices hardy space infinite elements are particularly well suited for solving resonance problems since they preserve the eigenvalue structure of the problem we prove superalgebraic convergence for a separated problem numerical experiments exhibit fast convergence over a wide range of wave numbers

three dimensional time harmonic elastodynamic green s functions for anisotropic solids a method based on the radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time harmonic point force the radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations this system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues the resulting simplified system can be solved easily a back transformation to the original coordinate system and a subsequent application of the inverse radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term both terms are integrals over a unit sphere for the regular dynamic term the surface integration can be evaluated numerically without difficulty for the singular static term the surface integral has been reduced to a line integral over half a unit circle reductions to the cases of isotropy and transverse isotropy have been worked out in detail examples illustrate applications of the method

localic galois theory abstract in proposition i of memoire sur les conditions de resolubilite des equations par radicaux galois considers the splitting field a of a polynomial with rational coefficients given any intermediate extension x he proves that the action of the galois group gal a q on the set of morphisms x a is transitive and that x is the fixed field of its galois group gal a x in this article we first state and prove a dual categorical formulation of these statements which turns out to be a theorem about atomic sites with a representable point these abstract developments correspond exactly to classical galois theory we then consider the same situation but with the point no longer representable it determines a proobject and it becomes tautologically prorepresentable mutatis mutandis we state and prove the prorepresentable version of the categorical formulation of galois theorems in this case the classical group of automorphisms has to be replaced by the localic group of automorphisms these developments form the content of a theory that we call localic galois theory from our results it immediately follows the theorem a topos with a point is connected atomic if and only if it is the classifying topos of a localic group and this group can be taken to be the locale of automorphisms of the point the hard implication in this equivalence was first proved in print in joyal and tierney mem amer math soc 151 1984 theorem 1 section 3 chapter viii and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales our proof of this theorem is completely independent of descent theory and of any other result in joyal and tierney 1984 the theorem follows as a straightforward consequence of our direct generalization of the fundamental results of galois we explain and give the necessary definitions to understand this paper without previous knowledge of the theory of locales other than this our principal results theorems 2 2 and 2 3 require only elementary notions of category theory

on the evolution of perturbations to solutions of the kadomtsev petviashvilli equation using the benney luke equation the benney luke equation which arises as a long wave asymptotic approximation of water waves contains the kadomtsev petviashvilli kp equation as a leading order maximal balanced approximation the question analyzed is how the benney luke equation modifies the so called web solutions of the kp equation it is found that the benney luke equation introduces dispersive radiation which breaks each of the symmetric soliton like humps well away from the interaction region of the kp web solution into a tail of multi peaked oscillating profiles behind the main solitary hump computation indicates that the wave structure is modified near the center of the interaction region both analytical and numerical techniques are employed for working with non periodic non decaying solutions on unbounded domains

numerical solution of the small dispersion limit of korteweg de vries and whitham equations the cauchy problem for the korteweg de vries kdv equation with small dispersion of order 2 1 is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order these oscillations are approximately described by the elliptic solution of kdv where the amplitude wave number and frequency are not constant but evolve according to the whitham equations in this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the kdv equation in the small dispersion limit and the corresponding approximate solution for values of between 10 1 and 10 3 the numerical results are compatible with a difference of order close to the center within the interior of the whitham oscillatory zone of order 1 3 at the left boundary outside the whitham zone and of order at the right boundary outside the whitham zone 2007 wiley periodicals inc

mathematical analysis of the junction of two acoustic open waveguides the present paper concerns the scattering of a time harmonic acoustic wave by the junction of two open uniform waveguides where the junction is limited to a bounded region we consider a two dimensional problem for which wave propagation is described by the scalar helmholtz equation the main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave we use here modal radiation conditions which extend the classical conditions used for closed waveguides they are based on the generalized fourier transforms which diagonalize the transverse contributions of the helmholtz operator on both sides of the junction we prove the existence and uniqueness of the solution which seems to be the first result in this context the originality lies in the proof of uniqueness which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized fourier variable

wave collapse and instability of solitary waves of a generalized kadomtsev petviashvili equation abstract the solutions of a family of generalized kadomtsev petviasvili equation in two diemnsions u t u m u x u xxx x 2 u yy are studied with appropriate boundary conditions and constraints on the initial data for m in a dense set m 0 and for 2 1 if m 4 then the solution can blow up in finite time we show that if 1 m 4 a solitary wave so exists which is unstable if m 4 3 in this and related problems th e instability of solitary waves is associated with onset of wave collapse we give numerical evidence to support this

an accurate and efficient acoustic eigensolver based on a fast multipole bem and a contour integral method an accurate numerical solver is developed in this paper for eigenproblems governed by the helmholtz equation and formulated through the boundary element method a contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations in order to accelerate the solution procedure the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared moreover a wideband fast multipole method is implemented with a block idr s solver to reduce the overall solution cost of the boundary element systems of equations with multiple right hand sides the burton miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains the actual effect of the burton miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as i k is confirmed through exterior sphere examples furthermore the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method

numerical study of shock formation in the dispersionless kadomtsev petviashvili equation and dispersive regularizations abstract the formation of singularities in solutions to the dispersionless kadomtsev petviashvili dkp equation is studied numerically for different classes of initial data the asymptotic behavior of the fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity the approach is first tested in detail in 1 1 dimensions for the known case of the hopf equation where it is shown that the break up of the solution can be identified with prescribed accuracy for dissipative regularizations of this shock formation as the burgers equation and for dispersive regularizations as the korteweg de vries equation the fourier coefficients indicate as expected global regularity of the solutions the kadomtsev petviashvili kp equation can be seen as a dispersive regularization of the dkp equation the behavior of kp solutions for small dispersion parameter 1 near a break up of corresponding dkp solutions is studied it is found that the difference between kp and dkp solutions for the same initial data at the critical point scales roughly as 2 7 as for the korteweg de vries equation

a new finite element realization of the perfectly matched layer method for helmholtz scattering problems on polygonal domains in two dimensions in this paper we propose a new finite element realization of the perfectly matched layer method pml method our approach allows to deal with a wide class of polygonal domains and with certain types of inhomogeneous exterior domains among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics we give a detailed insight into implementation aspects numerical examples show exponential convergence behavior to the exact solution with the thickness of the pml sponge layer

analysis of a coupled finite infinite element method for exterior helmholtz problems this analysis of convergence of a coupled fem iem is based on our previous work on the fem and the iem for exterior helmholtz problems the key idea is to represent both the exact and the numerical solution by the dirichlet to neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle the investigation of convergence can then be related to a spectral analysis of these dtn operators we give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere our main goal is to explore the convergence mechanism in this context we show well posedness of both the continuous and the discrete models we further show that the discrete inf sup constants have a positive lower bound that does not depend on the number of dof of the iem the proofs are based on lemmas on the spectra of the continuous and the discrete dtn operators where the spectral characterization of the discrete dtn operator is given as a conjecture from numerical experiments in our convergence analysis we show algebraic in terms of n convergence of arbitrary order and generalize this result to exponential convergence