a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

stabilized approximations of strongly continuous semigroups this paper introduces stabilization techniques for intrinsically unstable high accuracy rational approximation methods for strongly continuous semigroup the methods not only stabilize the approximations but improve their speed of convergence by a magnitude of up to 1 2

a note on space approximation of parabolic evolution equations our point of interest is the numerical approximation of the abstract cauchy email protected t au t u 0 u email protected x where a generates an analytic semigroup in x x being a separable banach space such equations include e g parabolic partial differential equations of arbitrary order defined on a bounded domain we apply semigroup methods and interpolation theory to get the order of convergence for spatially approximation in arbitrary separable banach spaces

approximation of semigroups and related operator functions by resolvent series we consider the approximation of semigroups e tau a and of the functions varphi j tau a that appear in exponential integrators by resolvent series the interesting fact is that the resolvent series expresses the operator functions e tau a and varphi j tau a respectively in efficiently computable terms this is important for semigroups where the new approximation is different from well known approximations by rational functions as well as for the application of exponential integrators which are currently of high interest and which are usually studied in a semigroup setting on banach spaces the approximation of the operator functions varphi j tau a in a general strongly continuous semigroup setting has not been discussed in the literature so far but this is crucial for an application of these integrators with unbounded operators or bounded operators like discretization matrices with large norm and eigenvalues somewhere in the left half plane

regularity of traveling periodic stratified water waves with vorticity abstract we prove real analyticity of all the streamlines including the free surface of a steady stratified flow of water over a flat bed in the absence of stagnation points with a holder continuous bernoulli function and a holder continuously differentiable density function furthermore we show that if the bernoulli function and the density function possess some gevrey regularity of index s then the stream function admits the same gevrey regularity throughout the fluid domain in particular if the gevrey index s equals to 1 then we obtain analyticity of the stream function the regularity results hold for three distinct physical regimes capillary capillary gravity and gravity water waves

steady periodic waves and formal stability for fixed depth rotational equatorial flows abstract in this paper we apply the bifurcation theory to study the steady two dimensional periodic equatorial water waves for the f plane approximation we consider the waves with vorticity which propagate with a specified fixed depth between the thermocline and the upper boundary of the centre layer moreover we present a functional j and its first variation corresponds to the exact equations in the transformed variables using the second variation of j we prove a formal stability result for the bifurcation inducing the laminar solution

on the convergence of rational approximations of semigroups on intermediate spaces we generalize a result by brenner and thomee on the rate of convergence of rational approximation schemes for semigroups using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the banach space x and the domain of a certain power of the generator of the semigroup the sharpness of the results is also discussed

on the pressure transfer function for solitary water waves with vorticity in this paper we analyse the role which the pressure function on the sea bed plays in determining solitary waves with vorticity we prove that the pressure function on the flat bed determines a unique real analytic solitary wave solution to the governing equations given a real analytic vorticity distribution in particular the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave

deformations in g2 manifolds here we study the deformations of associative submanifolds inside a g2 manifold m7 with a calibration 3 form f a choice of 2 plane field on m which always exists splits the tangent bundle of m as a direct sum of a 3 dimensional associate bundle and a complex 4 plane bundle tm ev and this helps us to relate the deformations to seiberg witten type equations here all the surveyed results as well as the new ones about g2 manifolds are proved by using only the cross product operation equivalently f we feel that mixing various different local identifications of the rich g2 geometry e g cross product representation theory and the algebra of octonions makes the study of g2 manifolds look harder then it is e g the proof of mclean s theorem r c mclean deformations of calibrated submanifolds comm anal geom 6 1998 705 747 we believe the approach here makes things easier and keeps the presentation elementary

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

small internal waves in two fluid systems this paper treats travelling waves in a heterogeneous inviscid non diffusive fluid bounded between two horizontal boundaries the fluid has two incompressible components of different but constant density and is acted on by gravity the flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed the small amplitude solutions are analyzed using a dynamical systems approach a center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all small solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude the result is a characterization of all small waves irrotational in each fluid near the critical speed for the system they are solitary waves surges connecting distinct conjugate flows at extreme ends of the channel conjugate flows and periodic waves

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

nonautonomous analysis of steady korteweg de vries waves under nonlocalised forcing abstract recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the korteweg de vries kdv equation arbitrary but small nonlocalised bottom topographies are amenable to this method two classes of free surface solutions hyperbolic and homoclinic solutions of the associated augmented dynamical system are characterised the first of these corresponds to near uniform free surface flows for which explicit formulae are developed for a range of topographies the second corresponds to solitary waves on the free surface and a method for determining their number is developed formulae for the shape of these solitary waves are also obtained theoretical free surface profiles are verified using numerical kdv solutions and excellent agreement is obtained

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

particle paths in small amplitude solitary waves with negative vorticity abstract we investigate the particle trajectories in solitary waves with vorticity where the vorticity is assumed to be negative and decrease with depth we show that the individual particle moves in a similar way as that in the irrotational case if the underlying laminar flow is favorable that is the flow is moving in the same direction as of the wave propagation throughout the fluid and show that if the underlying current is not favorable some particles in a sufficiently small solitary wave move in the opposite direction to that of wave propagation along a path with a single loop or hump

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

trimodal steady water waves we construct three dimensional families of small amplitude gravity driven rotational steady water waves of finite depth the solutions contain counter currents and multiple crests in each minimal period each such wave is generically a combination of three different fourier modes giving rise to a rich and complex variety of wave patterns the bifurcation argument is based on a blow up technique taking advantage of three parameters associated with the vorticity distribution the strength of the background stream and the period of the wave

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

on convergence rates in approximation theory for operator semigroups abstract we create a new functional calculus approach to approximation formulas for c 0 semigroups on banach spaces restricted to the domains of fractional powers of their generators this approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense in the case of analytic semigroups we improve our general results obtaining better convergence rates which are optimal in that case too the setting of analytic semigroups includes also the case of convergence on the whole space as an illustration of our approach we deduce optimal convergence rates in classical approximation formulas for c 0 semigroups restricted to the domains of fractional powers of their generators

existence of a generalized solitary wave solution for water with positive bond number less than 13 abstract the objective of this paper is to give a rigorous answer to the question whether there exists a generalized solitary wave of elevation on water when the nondimensional surface tension coefficient the so called bond number is positive but less than 1 3 and the froude number is near 1 but greater than 1 the main difficulties to overcome are the small amplitude oscillations at infinity and the choice of appropriate banach spaces for the solution of the problem it is shown that an asymptotic expansion developed for this problem indeed yields an approximate solitary wave solution to the exact equations

finite wavelength stability of capillary gravity solitary waves we consider the euler equations describing nonlinear waves on the free surface of a two dimensional inviscid irrotational fluid layer of finite depth for large surface tension bond number larger than 1 3 and froude number close to 1 the system possesses a one parameter family of small amplitude traveling solitary wave solutions we show that these solitary waves are spectrally stable with respect to perturbations of finite wave number in particular we exclude possible unstable eigenvalues of the linearization at the soliton in the long wavelength regime corresponding to small frequency and unstable eigenvalues with finite but bounded frequency arising from non adiabatic interaction of the infinite wavelength soliton with finite wavelength perturbations

existence and symmetry of ground states to the boussinesq abcd systems we consider a four parameter family of boussinesq systems derived by bona et al j nonlinear sci 12 283 318 2002 we establish the existence of the ground states which are solitary waves minimizing the action functional of the systems we further show that in the presence of large surface tension the ground states are even up to translation

long term stability of variable stepsize approximations of semigroups this paper is concerned with the stability of rational one step approximations of c0 semigroups particular emphasis is laid on long term stability bounds the analysis is based on a general banach space framework and allows variable stepsize sequences under reasonable assumptions on the stepsize sequence asymptotic stability bounds for general co semigroups are derived the bounds are typical in the sense that they contain in general a factor that grows with the number of steps under additional hypotheses on the approximation more favorable stability bounds are obtained for the subclass of holomorphic semigroups

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for gravity capillary solitary waves where the first term in the series is the well known sech2 solution of the kdv equation the asymptotic series is used with nine terms included to investigate the effects of surface tension on the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for surface tension below a critical value the solitary wave with the maximum energy is obtained for large surface tension the series is also used to study the energy related to the solitary waves of depression energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom comparisons are also made with recent experiments

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

on the existence of extreme waves and the stokes conjecture with vorticity this is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity we show that for a certain class of vorticity functions a sequence of regular waves converges to an extreme wave with stagnation points at its crests we also show that for any vorticity function the profile of an extreme wave must have either a corner of 120 or a horizontal tangent at any stagnation point about which it is supposed symmetric moreover the profile necessarily has a corner of 120 if the vorticity is nonnegative near the free surface

on positivity shape and norm bound preservation of time stepping methods for semigroups abstract we use functional calculus methods to investigate qualitative properties of c 0 semigroups that are preserved by time discretization methods preservation of positivity concavity and other qualitative shape properties which can be described via positivity are treated in a banach lattice framework preservation of contractivity or norm bound of the semigroup is investigated in the banach space setting the use of the hille phillips h p functional calculus instead of the dunford taylor functional calculus allows us to extend fundamental qualitative results concerning time discretization methods and simplify their proofs including results on multi step schemes and variable step sizes since the h p functional calculus is used throughout the paper we present an elementary introduction to it based on the riemann stieltjes integral

particle trajectories beneath small amplitude shallow water waves in constant vorticity flows abstract we investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water s free surface within the framework of small amplitude waves we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case and we describe the possible particle trajectories depending on the relation between the initial data and the constant vorticity some particle trajectories are undulating curves to the right or to the left others are loops with forward drift or with backward drift others can follow some peculiar shapes

fully localised solitary wave solutions of the three dimensional gravity capillary water wave problem a model equation derived by kadomtsev petviashvili sov phys dokl 15 539 541 1970 suggests that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction this prediction is rigorously confirmed for the full water wave problem in the present paper the theory is variational in nature a simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties the reduced functional is related to the functional associated with the kadomtsev petviashvili equation and a nontrivial critical point is detected using the direct methods of the calculus of variations

bernoulli free boundary problems in strip like domains and a property of permanent waves on water of finite depth we study weak solutions for a class of free boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth we show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions we also prove that in very general situations the free boundary is necessarily the graph of a function

a flow force reformulation for steady irrotational water waves abstract the classical irrotational water wave problem described by euler equations with a nonlinear free surface boundary condition and influenced by gravity over a flat bed is considered exploiting the monotonicity property of the flow force lines with depth the unknown boundary problem is transformed into a problem with fixed domain a flow force function formulation of the irrotational water wave problem is developed variational approach is used to prove the existence of small amplitude irrotational traveling wave solutions local bifurcation results have been proved relying on crandall rabinowitz bifurcation theorem

existence of capillary gravity water waves with piecewise constant vorticity abstract in this paper we construct small amplitude periodic capillary gravity water waves with a piecewise constant vorticity distribution they describe water waves traveling on superposed linearly sheared currents that have different vorticities this is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump the solutions of the diffraction problem found by using local bifurcation theory are the desired solutions of the hydrodynamical problem

constant vorticity water flows in the equatorial β plane approximation with centripetal forces abstract in this paper we consider three dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the plane approximation with centripetal forces as a striking upshot we show that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity owing to the presence of centripetal forces for the case exhibiting the surface tension we prove that there are no equatorial flows exhibiting a constant vorticity

on the existence of small amplitude solitary waves with strong surface tension an existence theory for small amplitude solitary waves with surface tension effects included is developed for large values of the surface tension parameter 13 using ideas of beale the nash moser implicit function theorem is applied to justify the well known approximation of korteweg and devries some of the recent results of amick and kirchgassner are thereby achieved more directly and additional insight obtained for the open case 13

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for two and three dimensional gravity capillary solitary waves in two dimensions the first term in the asymptotic series is the well known sech 2 solution of the korteweg de vries equation in three dimensions the first term is the rational lump solution of the kadomtsev petviashvili equation i the two dimensional series is used with nine terms included to investigate how small surface tension affects the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for small surface tension the solitary wave with the maximum energy is obtained for large surface tension the two dimensional series is also used to study the energy of depression solitary waves energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom in three dimensions analytic solutions for the high order perturbation terms are computed numerically and the resulting asymptotic series to three terms is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension finally the above asymptotic method is applied to the benney luke bl equation and the resulting asymptotic series to three terms is verified to agree with the solitary wave solution of the bl equation

exact solitary water waves with vorticity the solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field the main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity each solution has a supercritical parameter value and decays exponentially at infinity the proof is based on a generalized implicit function theorem of the nash moser type the first approximation to the surface profile is given by the kdv equation with a supercritical value of the surface tension coefficient a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

on some properties of velocity field for two dimensional rotational steady water waves abstract we investigate the properties of velocity field for steady periodic water waves over a flat bed for a generalized class of c 1 vorticity functions results are proved by exploiting the maximum principles and are based on dubreil jacotin transformation of the fluid domain some properties of velocities interior to the fluid domain and the locations of maximum minimum horizontal fluid velocities are investigated and proved for rotational water waves without any restriction to amplitude for u c and a monotonically varying positive vorticity u y v x 0 the location of maximal horizontal velocity has been proven to be the crest generalizing some of the recently proved results

surface waves on steady perfect fluid flows with vorticity this is a theory of two dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem the volume of fluid per period the circulation per period on the free stream line and the rearrangement class equivalently the distribution function of the vorticity field a minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function this decreasing function can be thought of as an infinite dimensional lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady to illustrate the idea with a minimum of technical difficulties the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved the theory applies equally to irrotational flows and to flows with locally square integrable vorticity 2011 wiley periodicals inc

a uniqueness result for periodic traveling waves in water of finite depth it is shown that in water of finite depth the surface profile eta of a periodic traveling wave uniquely determines the corresponding flow in the body of the fluid this holds for rotational flow as long as the vorticity function gamma psi satisfies the condition gamma psi max xis an element ofr eta 2 x pi 2 this condition is also shown to be sharp

the second variation of the stream function energy of water waves with vorticity abstract we compute the second variation of the stream function energy of two dimensional steady free surface gravity water waves with vorticity in the stream function formulation we prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point the particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy

a new formulation of the water wave problem for stokes waves of constant vorticity we consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs the problem is transformed into one concerning the vertical velocity a representation formula for the stream function in terms of the surface and the vorticity is presented and we show that the surface can be determined from the vertical velocity

mirror duality in a joyce manifold abstract previously the two of the authors defined a notion of dual calabi yau manifolds in a g 2 manifold and described a process to obtain them here we apply this process to a compact g 2 manifold constructed by joyce and as a result we obtain a pair of borcea voisin calabi yau manifolds which are known to be mirror duals of each other

modulational instability in the whitham equation with surface tension and vorticity abstract we study modulational stability and instability in the whitham equation combining the dispersion relation of water waves and a nonlinearity of the shallow water equations and modified to permit the effects of surface tension and constant vorticity when the surface tension coefficient is large we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value and stable otherwise similarly to the benjamin feir instability of gravity waves in the case of weak surface tension we find intervals of stable and unstable wave numbers whose boundaries are associated with the extremum of the group velocity the resonance between the first and second harmonics the resonance between long and short waves and a resonance between dispersion and the nonlinearity for each constant vorticity we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value and stable otherwise moreover it can be made stable for a sufficiently large vorticity the results agree with those based upon numerical computations or formal multiple scale expansions to the physical problem

runge kutta time semidiscretizations of semilinear pdes with non smooth data we study semilinear evolution equations frac mathrm du mathrm dt au b u dudt au b u posed on a hilbert space mathcal y y where a is normal and generates a strongly continuous semigroup b is a smooth nonlinearity from mathcal y ell d a ell yl d al to itself and ell in i subseteq 0 l l i 0 l l ge 0 l 0 0 l in i 0 l i in particular the one dimensional semilinear wave equation and nonlinear schrodinger equation with periodic neumann and dirichlet boundary conditions fit into this framework we discretize the evolution equation with an a stable runge kutta method in time retaining continuous space and prove convergence of order o h p ell p 1 o hpl p 1 for non smooth initial data u 0 in mathcal y ell u0 yl where ell le p 1 l p 1 for a method of classical order p extending a result by brenner and thomee for linear systems our approach is to project the semiflow and numerical method to spectral galerkin approximations and to balance the projection error with the error of the time discretization of the projected system numerical experiments suggest that our estimates are sharp

capillary gravity solitary waves with damped oscillations abstract capillary gravity solitary waves with damped oscillations are studied analytically the analysis follows the work of iooss and kirchgassner who proved that these waves exist for all values of the froude number smaller than one the water wave problem is reduced to a system of ordinary differential equations by using the center manifold theorem the normal form of this reduced system can be obtained and a good approximation to these waves for small amplitude is constructed the limit as the water depth becomes infinite is considered as a special case a comparison with existing numerical results is made for small amplitude waves

dispersion relations for fixed mean depth flows with two discontinuities in vorticity abstract we derive here the dispersion relation for small amplitude steady two dimensional periodic water waves that propagate over a flat bed with a specified and fixed mean depth the water flow has a discontinuous vorticity distribution reflecting an isolated rotational layer of fluid we then use the dispersion relation to obtain necessary and sufficient conditions for local bifurcation to occur moreover a stability analysis of the laminar flows is presented

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

singularities of steady axisymmetric free surface flows with gravity we consider a steady axisymmetric solution of the euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface we analyze stagnation points as well as points on the axis of symmetry at points on the axis of symmetry that are not stagnation points constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation this suggests the presence of downward pointing cusps at those points at stagnation points on the axis of symmetry the unique blowup profile consistent with the invariant scaling of the equation is the garabedian pointed bubble solution with water above air thus at stagnation points on the axis of symmetry with no water above the stagnation point the invariant scaling of the equation cannot be the right scaling a finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve the velocity field scales almost like and is asymptotically given by with a nonzero constant c0 the last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric euler equations by delort while the concentration compactness result alone does not lead to strong convergence in general we prove the convergence to be strong in our application 2014 wiley periodicals inc

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

capillary gravity water waves modified flow force formulation abstract the classical irrotational capillary gravity water wave problem described by the euler equations with a nonlinear free surface boundary condition over a flat bed is considered a modified flow force has been defined and a new formulation of capillary gravity waves in the framework of the modified flow force function has been developed using bifurcation theory the local existence of waves of small amplitude is proved

higher order grunwald approximations of fractional derivatives and fractional powers of operators we give stability and consistency results for higher order grunwaldtype formulae used in the approximation of solutions to fractional in space partial differential equations we use a new carlson type inequality for periodic fourier multipliers to gain regularity and stability results we then generalise the theory to the point where the first derivative operator is replaced by the generator of a bounded group on an arbitrary banach space

small amplitude stokes and solitary gravity water waves with an arbitrary distribution of vorticity this paper presents an existence theory for small amplitude stokes and solitary wave solutions to the classical water wave problem in the absence of surface tension and with an arbitrary distribution of vorticity the hydrodynamic problem is formulated as an infinite dimensional hamiltonian system in which the horizontal spatial coordinate is the time like variable a centre manifold technique is used to reduce the system to a locally equivalent hamiltonian system with one degree of freedom for values of a dimensionless parameter a near its critical value alpha the phase portrait of the reduced system contains a homoclinic orbit for alpha alpha the corresponding solutions of the water wave problem are respectively a solitary wave of elevation and a family of stokes waves c 2008 elsevier b v all rights reserved less

a free boundary approach to two dimensional steady capillary gravity water waves we prove that suitable weak solutions of the two dimensional steady capillary gravity water wave problem even in the presence of vorticity are smooth on a technical level solutions are closely related to critical points of the mumford shah functional so that our main task is to exclude cusps pointing into the water phase we prove that cusps do not exist and that the free surface is smooth

upper bound on the slope of steady water waves with small adverse vorticity abstract we consider the angle of inclination with respect to the horizontal of the profile of a steady 2d inviscid symmetric periodic or solitary water wave subject to gravity there is an upper bound of 31 15 in the irrotational case 1 and an upper bound of 45 in the case of favorable vorticity 13 on the other hand if the vorticity is adverse the profile can become vertical we prove here that if the adverse vorticity is sufficiently small then the angle still has an upper bound which is slightly larger than 45

new solutions for solving problem of particle trajectories in linear deep water waves via lie group method the nonlinear equations of the two dimensional inviscid incompressible fluid in a constant gravitational field describing the wave propagation on the water surface are considered the lie group method has been applied for determining symmetry reductions of the system of partial differential equations lie group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant the determining equations are a set of linear differential equations the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables after the group has been determined a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system effects of the wavelength l and time t on the particle path have been studied and the results are plotted

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

new integrable hierarchy its parametric solutions cuspons one peak solitons and m w shape peak solitons in this paper we propose a new completely integrable hierarchy particularly in the hierarchy we draw two new soliton equations 1 mt 12 1 m2 xxx 12 1 m2 x 2 mt mx u2 ux2 2m2ux 0 m u uxx the first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy both equations can be derived from the two dimensional euler equation by using the approximation procedure all equations in the hierarchy are proven to have bi hamiltonian operators and lax pairs through solving a crucial matrix equation moreover we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints one is for all negative members of the hierarchy on a symplectic submanifold and the other is for all positive members in the standard symplectic space the most interesting things are both equations possess new type of peaked solitons continuous and piecewise smooth w m shape peak soliton solutions in addition we find new cusp solit

a penalization method for calculating the flow beneath traveling water waves of large amplitude we suggest an algorithm for the calculation of large amplitude traveling water waves in irrotational flows and in flows with constant and nonconstant vorticity respectively the algorithm is based on a quite general partial differential equation constrained optimization formulation maximizing the wave amplitude subject to the pde constraint that the euler equations hold numerical examples illustrate the performance of the algorithm for irrotational flows and in flows with constant vorticity where the results are compared with the literature as a proof of principle we also consider a test case with a nonconstant vorticity

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

three dimensional gravity capillary waves on water small surface tension case abstract this paper considers three dimensional gravity capillary waves on water of finite depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction the exact euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time like variable the existence of the solutions of the system is determined by two non dimensional constants the bond number b and the froude number f which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow assume that f 2 c 1 is the curve in the b plane on which the first two eigenvalues for three dimensional waves collide at the imaginary axis and the intersection point of c 1 with 1 is b 1 0 in this paper the case for 0 b b 1 and b near c 1 is considered a center manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations using the existence of a homoclinic orbit connecting to a two dimensional periodic called generalized solitary wave thereafter solution for the reduced system it is shown that such a generalized solitary wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants

periodic traveling gravity water waves with discontinuous vorticity we use elliptic theory to prove the existence of steady two dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity this is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context