integrability of two dimensional homogeneous potentials combining the extended painleve conjecture with yoshida s singularity and stability analyses 1986 it is shown that for two dimensional homogeneous potentials of degree 2m integrability restricts kowalevskaya exponents and integrability coefficients to discrete sets of values this result is made use of in the analysis of integrability of symmetric potentials with m 2 3 and 4 direct construction of additional first integrals is successful only in special cases which can be transformed to known integrable ones

stability of a strongly stabilizing control for systems with a skew adjoint operator in hilbert space the stabilization problem of systems with a skew adjoint operator in a hilbert space is considered we prove that an one dimensional stabilizing feedback control allows a wide class of perturbations such that the system under consideration is strongly stabilizable with the aim of the perturbed controls

time dependent invariants and quantum mechanics in two dimensions a class of time dependent problems in two space dimensions possessing time dependent invariants bilinear in momenta is considered explicit expression of the potentials and the corresponding invariants are derived quantum mechanics is introduced in these time dependent problems directly through a feynman propagator defined as a path integral involving the classical action the propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators equivalence of the present theory to that of lewis and riesenfeld h r lewis jr and w b riesenfeld j math phys 10 1458 1969 is discussed

sliding mode control in indefinite dimensional systems abstract a mathematical description of motion in a sliding mode is considered for systems with discontinuous controls described by differential equations in banach space applying the regularization principle to such systems the sliding mode equations are obtained for a set of interconnected heat processes the control system with the prescribed dynamics is designed through deliberate introduction of sliding modes

the weak painleve property as a criterion for the integrability of dynamical systems we investigate the validity of the weak painleve property as an integrability criterion we present an example of a time dependent hamiltonian system which possesses a weak painleve type expansion while presenting a chaotic behavior however this system presents also critical fixed singularities the importance of the latter as far as integrability is concerned is discussed here

a new stabilizing technique for boundary integral methods for water waves boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities a previous stability analysis by beale hou and lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability such balance can be preserved by applying suitable numerical filtering at certain places of the discretization while this filtering technique is effective for two dimensional 2 d periodic fluid interfaces it does not apply to nonperiodic fluid interfaces moreover using the filtering technique alone does not seem to be sufficient to stabilize 3 d fluid interfaces here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3 d interfaces a stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order this modified boundary integral method still has the same order of accuracy as the point vortex method a detailed stability analysis is presented for the point vortex method for 2 d water waves the effect of various stabilizing terms is illustrated through careful numerical experiments

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

integrability and the painleve property for low dimensional systems we examine a system described by two first order nonlinear differential equations from the point of view of integrability the singularity analysis in the complex time plane is used to investigate the painleve property which according to the ablowitz ramani segur conjecture is a prerequisite for integrability for infinite dimensional systems we show that for such low dimensional systems the painleve analysis is still a most useful guide but integrable cases also exist which do not possess the painleve property

a new integrable toda related three particle system a study of a three particle system that possesses in addition to the energy function two nonlinear constants of motion is presented herein the system is obtained by modification of the lagrangian of the toda lattice the explicit expression of the two integrals is obtained and it is proven that this new three particle lagrangian is integrable finally the possible generalization to the case of n particles is discussed

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

lax formalism for a family of integrable toda related n particle systems this article can be divided in two parts first a lax representation is obtained for a family of integrable three particle toda related systems the second part contains the generalization to the case of n particles

an efficient surface integral algorithm applied to unsteady gravity waves a computationally fast method for calculating the unsteady motion of a surface on a two dimensional fluid is described cauchy s integral theorem is used iteratively to solve laplace s equation for successive time derivatives of the surface motion and time stepping is performed using truncated taylor series this allows fairly large time steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae this reduces the overall number of required calculations the numerical implementation of the method is found to be accurate and efficient a fairly thorough examination of this implementation is carried out revealing that high accuracies are often achievable using surprisingly few numerical surface points extensive calculations are also performed using modest computing resources some numerical instabilities are identified although these would not usually be significant in practical calculations a model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time stepping should the third steep wave instability become significant it is shown that it can be completely controlled by using high order smoothing techniques at little cost to accuracy using a routine to ensure asymptotic conservation of energy this is confirmed by time stepping a very steep but stable wave over thousands of wave periods 0 1992 academx press inc since all of the interior properties of a body of fluid undergoing inviscid incompressible irrotational motion are fully determined by properties at its boundary it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone in a numerical scheme the entire motion can thus be modelled using only a point discretisation of the surface such an approach was first suggested by svendsen in 1971 as a means of calculating the motion of two dimensional gravity waves on water 13 since then the idea has been implemented using a variety of boundary integral or conformal mapping techniques by longuet higgins and cokelet 2 vinje and brevig 3 baker meiron and orszag 4 roberts s fornberg 6 and based on his original formulation by svendsen and co workers 7 although this paper presents yet another numerical method the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity in numerically representing complicated surface motions ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves it typically becomes necessary to use a large number of surface points if running times tend to increase too rapidly with the number of points then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface the penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques yielding running times per time step that increase like n3 the cube of the number of surface points by using iterative techniques the penalty can be reduced to running times per time step that increase in proportion to n2 while being less severe this still has the effect that increased numbers of surface points become increasingly expensive to calculate the use of a conformal mapping technique 6 can lead to running times per timestep that vary like n log n however this method is severely limited in its ability to resolve steep wave surfaces as are found in breaking waves requiring disproportionately many surface points to achieve a given accuracy it is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required the most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface in this paper a further improvement is introduced by showing that higher time derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep this yields significantly more information about the motion for relatively little extra computation in turn this allows larger time steps to be performed for a given accuracy so that the overall number of calculations can be reduced the method presented here combines these two ideas into producing a practical numerical algorithm the properties of which are examined in the following pages in appendix a the formulae are derived that make it

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

stabilization of uncertain distributed parameter systems the stabilization of uncertain systems governed by an evolution equation in a real hilbert space is studied the uncertain systems under consideration are described by state equations in which both the state and input operators depend upon uncertain parameters exponential stability is achieved by the suitable choice of state feedback control the proofs are based upon perturbation analysis of semigroup and lyapunov theory a number of examples are given to illustrate the approach

robustness of uncertain systems in the absence of matching assumptions abstract we consider a class of uncertain dynamical systems containing uncertain elements these may be due to model parameter uncertainty input disturbance and measurement noise we require no a priori information on the uncertain elements except that their restraint sets are assumed to be known and compact this paper describes a general approach for designing controls to assure practical stability in the absence of matching assumptions which most previous results have needed the result is applicable to non linear systems a specialization to linear systems is then given and a comparison is made with previous work

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

symmetries of hamiltonian systems with two degrees of freedom we classify the lie point symmetry groups for an autonomous hamiltonian system with two degrees of freedom with the exception of the harmonic oscillator or a free particle where the dimension is 15 we obtain all dimensions between 1 and 7 for each system in the classification we examine integrability

integrability of toda lattice by generalized variational symmetry approach a systematic method of finding nontrivial generalized variational symmetries are presented herein and the corresponding first integrals for nonlinear dynamical system with two degrees of freedom are derived then a method is applied to a toda lattice given by the potential v ex y c1e2y c2ey c3e 2x c4 e x where c1 c2 c3 and c4 are parameters the symmetries and constants of motion are presented explicitly

numerical techniques on improving computational efficiency of spectral boundary integral method numerical techniques are suggested in this paper in order to improve the computational efficiency of the spectral boundary integral method initiated by clamond grue d clamond and j grue a fast method for fully nonlinear water wave computations j fluid mech 2001 447 337 355 for simulating nonlinear water waves this method involves dealing with the high order convolutions by using fourier transform or inverse fourier transform and evaluating the integrals with weakly singular integrands a de singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity an anti aliasing technique is developed in this paper to overcome the aliasing problem associated with fourier transform or inverse fourier transform with a limited resolution this paper also presents a technique for determining a critical value of the free surface under which the integrals can be neglected numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques the tests will demonstrate that the improved method can significantly accelerate the computation in particular when waves are strongly nonlinear

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

modeling extreme waves based on equations of potential flow with a free surface a method for numerical investigation of nonlinear wave dynamics based on direct hydrodynamical modeling of 1 d potential periodic surface waves is created the model is a part of an interactive wind wave model using a non stationary conformal mapping the principal equations are rewritten in a surface following coordinate system and reduced to two simple evolutionary equations for the elevation and the velocity potential of the surface fourier expansion is used to approximate these equations high accuracy was confirmed by validation of the non stationary model against known solutions and by comparison between the results obtained with different resolution in the horizontal the method developed is applied to the simulation of waves evolution with different initial conditions numerical experiments with initially monochromatic waves with different steepness show that the model is able to simulate breaking conditions when the surface becomes a multi valued function of the horizontal coordinate an estimate of the critical initial wave height that divides between non breaking and eventually breaking waves is obtained simulations of nonlinear evolution of a wave field is represented initially by two modes with close wave numbers amplitude modulation and a wave field with a phase modulation both runs result in the appearance of large and very steep waves these also break if the initial amplitudes are sufficiently large

explicit integrability for hamiltonian systems and the painleve conjecture we analyze a class of hamiltonian systems in two dimensions for which we proved that a second constant of motion exists it is shown that using the two first integrals the equations of motion can be written in a form which allows their integration by quadratures an analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex time plane it is shown explicitly that the systems belonging to this class possess the weak painleve property i e their solutions in complex time can present singularities of a specific algebraic type

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

nonlinear equations with superposition formulas and the exceptional group g2 i complex and real forms of g2 and their maximal subalgebras in order to study nonlinear ordinary differential equations with superposition principles related to the exceptional simple lie group g2 the complex and real forms of its lie algebra are examined and their maximal subalgebras are summarized in particular the parabolic subalgebras of the noncompact real form gnc2 r are determined explicit matrix realizations of the fundamental representation d 1 0 are used and studied in connection with invariant subspaces in a seven dimensional complex or real vector space the results are collected in three tables of specific interest for the study of nonlinear differential equations which will be developed in paper ii of this series

darboux points and integrability of hamiltonian systems with homogeneous polynomial potential in this paper we study the integrability of natural hamiltonian systems with a homogeneous polynomial potential the strongest necessary conditions for their integrability in the liouville sense have been obtained by a study of the differential galois group of variational equations along straight line solutions these particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom we call them darboux points we analyze in detail the case of two degrees of freedom we show that except for a radial potential the number of darboux points is finite and it is not greater than the degree of the potential moreover we analyze cases when the number of darboux points is smaller than maximal for two degrees of freedom the above mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the hessian of potential calculated at a darboux point we prove that for a given potential these nontrivial eigenvalues calculated for all darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form we use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case for a given degree of the potential there is only a finite number of potentials which satisfy these conditions we also describe the nongeneric cases as an example we give a full list of potentials of degree four satisfying these conditions then investigating the differential galois group of higher order variational equations we prove that except for one discrete family among these potentials only those which are already known to be integrable are integrable we check that a finite number of potentials from the exceptional discrete family are not integrable and we conjecture that all of them are not integrable

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

an integrable three particle system a study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized dynamical or hidden symmetries two lagrangians are considered both obtained by modifying the toda lagrangian first a two particle system is studied and then the results are generalized to a three particle system it is shown that in both cases the lagrangians possess nonlinear constants of motion in involution and thus they are integrable

periodic orbits and nonintegrability of generalized classical yang mills hamiltonian systems the averaging theory of first order is applied to study a generalized yang mills system with two parameters two main results are proved first we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy and these families are given up to first order in a small parameter second we prove that for the nonintegrable classical yang mills hamiltonian systems in the sense of liouville arnold which have the isolated periodic orbits found with averaging theory cannot exist in any second first integral of class c1 this is important because most of the results about integrability deals with analytic or meromorphic integrals of motion

existence of exponentially unstable periodic solutions and the non integrability of homogeneous hamiltonian systems abstract in hamiltonian systems with polynomial potential of homogeneous degree characteristic multipliers floquet multipliers permit an explicit expression for straight line periodic solutions by use of ziglin s theorem it is shown that the existence of an exponentially unstable straight line periodic solution implies the non integrability of the system

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

a numerical study on parasitic capillary waves using unsteady conformal mapping this paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth within the framework of irrotational plane flow as an alternative to the widely used boundary integral method with mixed eulerian lagrangian mel time updating we focus on a numerical method based on unsteady conformal mapping which will be hereafter referred to as the unsteady hodograph transformation uht method in this method we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle the aim of this work is to compare the uht method with the mel method and find a more efficient method to compute parasitic capillary waves from linear stability analysis it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals and the uht method can avoid some numerical instability due to it numerical examples demonstrate that the uht method is more suitable than the mel method for not only parasitic capillary waves but also capillary dominated waves in particular the uht method requires no artificial techniques such as filtering to control numerical errors in these examples in addition another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface depending on the restoring force of waves gravity or surface tension

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

integrability of the toda chain with free boundaries the hamilton function for the toda chain with free boundaries is studied for different interaction and mass parameters for certain systems the connection to simple lie algebras is shown and the integrability is proved by constructing the corresponding lax representations for the four particle system a painleve analysis is presented

a list of all integrable two dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta we searched for integrable two dimensional homogeneous polynomial potentials with a polynomial first integral by using the so called direct method of searching for first integrals we proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than four

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

new bi hamiltonian systems on the plane we discuss several new bi hamiltonian integrable systems on the plane with integrals of motion of third fourth and sixth orders in momenta the corresponding variables of separation separated relations compatible poisson brackets and recursion operators are also presented in the framework of the jacobi method

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

on monoparametric families of orbits sufficient for integrability of planar potentials with linear or quadratic invariants by combining darboux s results 1901 on the direct construction of integrable systems with results from the inverse problem of dynamics the authors prove that if a planar potential admits a monoparametric family of conic sections with constant focal distance or a family of confocal parabolas it is integrable and the second invariant is quadratic in the moments while the existence of a family of straight lines either parallel or intersecting at one point also suffices for integrability the second invariant being linear in the momenta

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

bounded feedback stabilization and global separation principle of distributed parameter systems we show that the infinite dimensional system spl sigma x spl dot t ax t bu t x sub 0 spl epsiv h is globally strongly asymptotically stabilizable by an arbitrarily small smooth feedback here the operator a is the infinitesimal generator of a c sub 0 semigroup of contractions e sup ta on real hilbert space h and b is a bounded linear operator mapping a hilbert space of controls u into h an explicit smooth feedback control law is given further we identify the class of perturbations for which the system is still stabilizable by the same feedback law as for the nominal system based on these results and some differential lyapunov operator equations we then establish a global separation principle for the system spl sigma with a kalman like observer finally these results are illustrated via an example dealing with the wave equation

convergence of a non stiff boundary integral method for interfacial flows with surface tension boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities in addition surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics this makes the spatial discretization even more difficult and at the same time imposes a severe time step constraint for stable explicit time integration methods a proof of the convergence of a reformulated boundary integral method for two density fluid interfaces with surface tension is presented the method is based on a scheme introduced by hou lowengrub and shelley j comp phys 114 1994 pp 312 338 to remove the high order stability constraint or stiffness some numerical filtering is applied carefully at certain places in the discretization to guarantee stability the key of the proof is to identify the most singular terms of the method and to sho through energy estimates that these terms balance one another the analysis is at a time continuous space discrete level but a fully discrete case for a simple hele shaw interface is also studied the time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the cfl constraint depends on the curvature and regularity of the solution the robustness of the method is illustrated with several numerical examples a numerical simulation of an unstably stratified two density interfacial flow shows the roll up of the interface the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed the method remains stable even in the full nonlinear regime of motion another application of the method shows the process of drop formation in a falling single fluid

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified