ground waves in atomic chains with bi monomial double well potential abstract ground waves in atomic chains are traveling waves that corresponds to minimal non trivial critical values of the underlying action functional in this paper we study fpu type chains with bi monomial double well potential and prove the existence of both periodic and solitary ground waves to this end we minimize the action on the nehari manifold and show that periodic ground waves converge to solitary ones finally we compute ground waves numerically by a suitable discretization of a constrained gradient flow

existence of infinitely many homoclinic orbits for fourth order difference systems containing both advance and retardation abstract by using the symmetric mountain pass theorem we establish some existence criteria to guarantee the fourth order difference system 4u n 2 q n u n f n u n 1 u n u n 1 have infinitely many homoclinic orbits where n z u r n q z r n n and f z r 3 n r are no periodic in n

two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

minimizing the discrete logarithmic energy on the sphere the role of random polynomials we prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly well suited with respect to the minimal logarithmic energy on the sphere that is roots of random polynomials provide a fairly good approximation to elliptic fekete points

riemann hilbert problems for the shapes formed by bodies dissolving melting and eroding in fluid flows the classical stefan problem involves the motion of boundaries during phase transition but this process can be greatly complicated by the presence of a fluid flow here we consider a body undergoing material loss due to either dissolution from molecular diffusion melting from thermodynamic phase change or erosion from fluid mechanical stresses in a fast flowing fluid in each case the task of finding the shape formed by the shrinking body can be posed as a singular riemann hilbert problem a class of exact solutions captures the rounded surfaces formed during dissolution melting as well as the angular features formed during erosion thus unifying these different physical processes under a common framework this study which merges boundary layer theory separated flow theory and riemann hilbert analysis represents a rare instance of an exactly solvable model for high speed fluid flows with free boundaries 2017 wiley periodicals inc

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

a general framework for solving riemann hilbert problems numerically a new numerical framework for the approximation of solutions to matrix valued riemann hilbert problems is developed based on a recent method for the homogeneous painleve ii riemann hilbert problem we demonstrate its effectiveness by computing solutions to other painleve transcendents an implementation in mathematica is made available online

periodic and homoclinic travelling waves in infinite lattices abstract consider an infinite chain of particles subjected to a potential f where nearest neighbours are connected by nonlinear oscillators the nonlinear coupling between particles is given by a potential v the dynamics of the system is described by the infinite system of second order differential equations q j f q j v q j 1 q j v q j q j 1 j z we investigate the existence of travelling wave solutions two kinds of such solutions are studied periodic and homoclinic ones on one hand we prove under some growth conditions on f and v the existence of non constant periodic solutions of any given period t 0 and speed c c 0 where the constant c 0 depends on f 0 and v 0 on the other hand under very similar conditions we establish the existence of non trivial homoclinic solutions of any given speed c c 0 emanating from the origin moreover we prove that these homoclinics decay exponentially at infinity each homoclinic is obtained as a limit of periodic solutions when the period goes to infinity

method of riemann surfaces in the study of supercavitating flow around two hydrofoils in a channel abstract in the framework of the tulin model of supercavitating flow the problem of reconstructing the free surface of a channel and the shapes of the cavities behind two hydrofoils placed in an ideal fluid is solved in closed form the conformal map that transforms a parametric plane with three cuts along the real axis into the triple connected flow domain is found by quadratures the use of the theory of riemann surfaces the schottky doubles enables the non linear model problem to be reduced to two separate riemann hilbert problems on a hyperelliptic surface of genus two the solution to the first problem is a rational function with certain zeros and poles on a riemann surface the second problem is solved in terms of singular integrals with the weierstrass kernel the essential singularities of the solution at the infinite points of the surface due to a pole of the kernel are removed by solving a real analogue of the jacobi inversion problem on the surface the unknown parameters of the conformal map are recovered from a system of certain algebraic and transcendental equations

motion of a yawed supercavitating wedge beneath a free surface the problem that is studied concerns a yawed wedge under a free surface moving at a uniform speed the model involves a trailing cavity whose boundary is a dividing streamline through the vertex of the wedge the cavity closure mechanism is described according to the tulin terent ev single spiral vortex model a closed form solution to the governing nonlinear boundary value problem is found by the method of conformal mappings the doubly connected flow domain is treated as the image by this map of the exterior of two slits in a parametric plane the mapping function is constructed through the solution to two boundary value problems of the theory of analytic functions the hilbert problem for two slits in a plane and the riemann hilbert problem on an elliptic surface numerical results for the shape of the cavity and the free surface the yaw angle the drag and lift coefficients and the circulation are reported

single spiral vortex model for a cavitating elastic curvilinear foil a two dimensional nonlinear inverse fluid structure interaction problem for a curvilinear elastic hydrofoil is considered a cavity formed behind the foil is modeled according to the single spiral vortex model by tulin the fluid structure problem is decoupled by the method of successive approximations for the cavitation problem the foil is modeled as a polygon the method of conformal mappings and the riemann hilbert problem are employed at this stage the classical detachment mechanism for a smooth arc is satisfied for the polygon approximately the deformation of the smooth foil is described by the governing equations of the thin shell theory with the clamped clamped boundary conditions the loading acting on the middle surface of the foil is prescribed as the difference between the fluid and vapor pressure computed in the fluid problem numerical results include those for the cavity profile the drag coefficient the pressure distribution the speed and the displacements of the elastic foil

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

a generalized traction integral equation for stokes flow with applications to near wall particle mobility and viscous erosion abstract a double layer integral equation for the surface tractions on a body moving in a viscous fluid is derived which allows for the incorporation of a background flow and or the presence of a plane wall the lorentz reciprocal theorem is used to link the surface tractions on the body to integrals involving the background velocity and stress fields on an imaginary bounding sphere or hemisphere for wall bounded flows the derivation requires the velocity and stress fields associated with numerous fundamental singularity solutions which we provide for free space and wall bounded domains two sample applications of the method are discussed we study the tractions on an ellipsoid moving near a plane wall which provides a more detailed understanding of the well studied glancing and reversing trajectories in the context of particle sedimentation and the erosion of bodies by a viscous flow in which the surface is ablated at a rate proportional to the local viscous shear stress simulations and analytical estimates suggest that a spherical body in a uniform flow first reduces nearly but not exactly to the drag minimizing profile and then vanishes in finite time the shape dynamics of an eroding body in a shear flow and near a wall are also investigated stagnation points on the body surface lead generically to the formation of cusps whose number depends on the flow configuration and or the presence of nearby boundaries

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

multiple periodic solutions for lattice dynamical systems with superquadratic potentials abstract in this paper we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction the autonomous dynamical system is described by the following infinite system of second order differential equations q i i 1 q i 1 q i i q i q i 1 i z where i denotes the interaction potential between two neighboring particles and q i t is the state of the i th particle supposing i is superquadratic at infinity for all t 0 we obtain a nonzero t periodic solution of finite energy which may be nonconstant in some range of period if in addition i x is even in x we also obtain infinitely many geometrically distinct solutions for any period t 0 in particular a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period since the functional associated to the above system is invariant under the actions of the non compact group z and the continuous compact group s 1 under our assumptions in order to prove our results we need to extend the abstract critical point theorem about strongly indefinite functional developed by bartsch and ding math nachr 279 2006 1267 1288 to a more general class of symmetry

a fast chebyshev method for simulating flexible wing propulsion abstract we develop a highly efficient numerical method to simulate small amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid we allow the wing s elastic modulus and mass density to vary arbitrarily with an eye towards optimizing these distributions for propulsive performance the method to determine the wing kinematics is based on chebyshev collocation of the 1d beam equation as coupled to the surrounding 2d fluid flow through small amplitude analysis of the euler equations with trailing edge vortex shedding the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1d wing kinematics a class of semi analytical solutions permits fast evaluation of this operator with o n log n operations where n is the number of collocation points on the wing this is in contrast to the minimum o n 2 cost of a direct 2d fluid solver the coupled wing fluid problem is thus recast as a pde with nonlocal operator which we solve using a preconditioned iterative method these techniques yield a solver of near optimal complexity o n log n allowing one to rapidly search the infinite dimensional parameter space of all possible material distributions and even perform optimization over this space

minimal riesz energy point configurations for rectifiable d dimensional manifolds we investigate the energy of arrangements of n points on a rectifiable d dimensional manifold a r d 0 that interact through the power law riesz potential v 1 r s where s 0 and r is euclidean distance in r d 0 with es a n denoting the minimal energy for such n point configurations we determine the asymptotic behavior as n 1 of es a n for each fixed s d moreover if a has positive d dimensional hausdor measure we show that n point configurations on a that minimize the s energy are asymptotically uniformly distributed with respect to d dimensional hausdor measure on a when s d even for the unit sphere s d r d 1 these results are new

nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems abstract in this paper we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q i t i 1 t q i 1 t q i t i t q i t q i 1 t i z where i t x i 2 x 2 v i t x is t periodic in t for t 0 and satisfies i n i for some n n q i t is the state of the i th particle assume that i 0 for some i z and v i t x denoting the derivative of v i respect to x is asymptotically linear with x both at origin and at infinity we would like to point out that this system is resonant both at origin and at infinity and not studied up to now based on some new results concerning the precise computations of the critical groups for a given m z we obtain the existence of nontrivial periodic solutions satisfying q i m n t t q i t for all t r and i z under some additional conditions

travelling wave solutions for the discrete sine gordon equation with nonlinear pair interaction the focus of study is the nonlinear discrete sine gordon equation where the nonlinearity refers to a nonlinear interaction of neighbouring atoms the existence of travelling heteroclinic homoclinic and periodic waves is shown the asymptotic states are chosen such that the action functional is finite the proofs employ variational methods in particular a suitable concentration compactness lemma combined with direct minimisation and mountain pass arguments

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

finding elliptic fekete points sets two numerical solution approaches abstract the objective of this work is to provide a methodology for approximating globally optimal fekete point configurations this problem is of interest in numerical mathematics and scientific modeling following a brief discussion of the analytical background lipschitz global optimization lgo is applied to determine i e to numerically approximate fekete point configurations next to this optimization approach an alternative strategy by formulating a set of differential algebraic equations daes of index 2 will be considered the steady states of the daes coincide with the optima of the function to be minimized illustrative numerical results with configurations of up to 150 fekete points are presented to show the viability of both approaches

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

estimation of fekete points we aim here at presenting a new procedure to numerically estimate the fekete points of a wide variety of compact sets in r 3 we understand the fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of n particles the compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves in order to determine a good initial configuration to start the search of the fekete points of these objects we construct a sequence of approximating regular surfaces our algorithm is based on the concept of disequilibrium degree which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration moreover the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy the numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than n 3

existence of homoclinic orbits for 2nth order nonlinear difference equations containing both many advances and retardations abstract by establishing a proper variational framework and using the critical point theory we establish some new existence criteria to guarantee the 2nth order nonlinear difference equation containing both many advances and retardations n r t n n u t n q t u t f t u t n u t u t n n z 3 t z has at least one or infinitely many homoclinic orbits where r q f are nonperiodic in t our conditions on the potential are rather relaxed and some existing results in the literature are improved

nonlinear steepest descent and numerical solution of riemann hilbert problems the effective and efficient numerical solution of riemann hilbert problems has been demonstrated in recent work with the aid of ideas from the method of nonlinear steepest descent for riemann hilbert problems the resulting numerical methods have been shown in practice to retain accuracy as values of certain parameters become arbitrarily large remarkably this numerical approach does not require knowledge of local parametrices rather the deformed contour is scaled near stationary points at a specific rate the primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved to do so we first construct a general theoretical framework for the numerical solution of riemann hilbert problems second we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes in particular we prove sufficient conditions for numerical methods to retain accuracy finally we compute solutions to the homogeneous painleve ii equation and the modified korteweg de vries equation to explicitly demonstrate the practical validity of the theory 2014 wiley periodicals inc

on the impact of fpu lattice discreteness upon solitary wave interactions abstract this work is concerned with the motions of an infinite one dimensional lattice with nearest neighbour interactions governed by a generic potential the hamiltonian of such a system may be written h i 1 2 p i 2 v q i 1 q i in terms of the momenta p i and the displacements q i of the lattice sites all sites are assumed to be of equal mass certain generic conditions are placed on the potential v the particular results of this paper concern the way in which lattice discreteness impacts upon the solitary wave interaction process which is known to be well approximated in the long wave continuum limit by kdv soliton interaction an evolution equation for discreteness effects on the lattice is proposed and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion acoustic waves it is proved by standard techniques to have a unique classical solution whose profile is explicitly obtained by numerical simulation the most important outcome is the observation that lattice discreteness imposes an effective repulsion between post collision solitary waves relative to their unperturbed pure kdv positions

double cavity flow past a wedge a mathematical model of supercavitating flow past a wedge with sides of arbitrary length is proposed the flow branches at a point on the lower side of the wedge at the vertex of the wedge and at the ends of the wedge the flow breaks away forming a nose bubble and a trailing cavity the closure mechanism is described by the tulin single spiral vortex model the flow domain is mapped into a parametric plane cut along a unit segment the conformal mapping function is reconstructed through the exact solution of two riemann hilbert problems on a genus zero riemann surface to complete the solution one needs to determine five real parameters from a certain system of transcendental equations numerical results are presented for the case when a wedge can rotate about the vertex in the flow domain in this case the flow branches at the vertex and the number of the parameters to be determined is 3

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

a mountain pass algorithm with projector in this paper we present an enhanced version of the minimax algorithm of chen ni and zhou that offers the additional guarantee that the solution found is a fix point of a projector on a cone positivity negativity and monotonicity can be expressed in this way the convergence of the algorithm is proved by means of a computational deformation lemma instead of the usual deformation lemma used in the calculus of variations

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

ideal free streamline flow over a curved obstacle in the classical two dimensional model the description of helmholtz s kirchhoff flow is a problem of complex analysis which can be solved analytically only for a few simple bodies or polygonal contours using the schwarz christoffel map this paper presents a practical method for computing flows over arbitrary obstacles whose boundaries may be piecewise smooth curves while the impinging flow may be an unbounded flow a jet or a semi infinite stream i e the ocean

computational cost of the fekete problem i the forces method on the 2 sphere here we study the computational complexity of the fekete point problem namely we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2 sphere and we characterize the probability distribution of the cost of the different minima in particular we show that a local minimum can be found with an average cost of about o n 2 8

periodic and subharmonic solutions for fourth order nonlinear difference equations abstract by using the critical point theory some new criteria are obtained for the existence and multiplicity of periodic and subharmonic solutions to fourth order nonlinear difference equations the main approach used in our paper is a variational technique and the linking theorem our results generalize and improve the existing ones

homoclinic orbits for nonlinear difference equations containing both advance and retardation in this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions moreover if the nonlinearity is an odd function the existence of an unbounded sequence of homoclinic orbits is obtained

quasi monte carlo rules for numerical integration over the unit sphere mathbb s 2 we study numerical integration on the unit sphere mathbb s 2 subseteq mathbb r 3 using equal weight quadrature rules where the weights are such that constant functions are integrated exactly the quadrature points are constructed by lifting a 0 m 2 net given in the unit square 0 1 2 to the sphere mathbb s 2 by means of an area preserving map a similar approach has previously been suggested by cui and freeden siam j sci comput 18 2 595 609 1997 we prove three results the first one is that the construction is almost optimal with respect to discrepancies based on spherical rectangles further we prove that the point set is asymptotically uniformly distributed on mathbb s 2 and finally we prove an upper bound on the spherical cap l 2 discrepancy of order n 1 2 log n 1 2 where n denotes the number of points this improves upon the bound on the spherical cap l 2 discrepancy of the construction by lubotzky phillips and sarnak commun pure appl math 39 s suppl s149 s186 1986 by a factor of sqrt log n numerical results suggest that the 0 m 2 nets lifted to the sphere mathbb s 2 have spherical cap l 2 discrepancy converging with the optimal order of n 3 4