remarks on kdv type flows on star shaped curves abstract we study the relation between the centro affine geometry of star shaped planar curves and the projective geometry of parametrized maps into r p 1 we show that projectivization induces a map between differential invariants and a bi poisson map between hamiltonian structures we also show that a hamiltonian evolution equation for closed star shaped planar curves discovered by pinkall has the schwarzian kdv equation as its projectivization for both flows the curvature evolves by the kdv equation using algebro geometric methods and the relation of group based moving frames to akns type representations we construct examples of closed solutions of pinkall s flow associated with periodic finite gap kdv potentials

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

velocity model estimation with data derived wavefront attributes kinematic information for constructing velocity models can be extracted in a robust way from seismic prestack data with the common reflection surface crs stack this data driven process results in addition to a simulated zero offset section in a number of wavefront attributes wavefront curvatures and normal ray emergence angles associated with each simulated zero offset sample a tomographic inversion method is presented that uses this kinematic information to determine smooth laterally heterogeneous isotropic subsurface velocity models for depth imaging the input for the inversion consists of wavefront attributes picked at a number of locations in the simulated zero offset section the smooth velocity model is described by b splines an optimum model is found iteratively by minimizing the misfit between the picked data and the corresponding modeled values the required forward modeled quantities are obtained during each iteration by dynamic ray tracing along normal rays pertaining to the input dat

computing true amplitude reflections in a laterally inhomogeneous earth recently bortfeld 1982 gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero offset reflection from the common datum point traveltime measurements of the event to underline the significance and consequences of this method a derivation and discussion of geometrical spreading factors is now given for two and three dimensional earth models with curved reflecting boundaries the spreading factors can be used easily to transform primary reflections in a zero offset seismic section to true amplitude reflections these permit an estimation of interface reflection coefficients either directly or in connection with a true amplitude migration a seismic section with true amplitude reflections can be described by one physical experiment the tuned reflector model hence the application of the wave equation in connection with a migration after stack is justified on such a seismic section also the geometrical spreading factors that are deri

geometric poisson brackets on grassmannians and conformal spheres in this paper we relate the geometric poisson brackets on the 2 grassmannian in r4 and on the 2 2 mobius sphere we show that when written in terms of local moving frames the geometric poisson bracket on the mobius sphere does not restrict to the space of differential invariants of schwarzian type but when the concept of conformal natural frame is trans ported from the conformal sphere into the grassmannian and the poisson bracket is written in terms of the grassmannian natural frame it restricts and results into either a decoupled system or a complexly coupled system of kdv equations depending on the character of the invariants we also show that the bihamiltonian grassmannian geometric brackets are equivalent to the non commutative kdv bihamiltonian structure both integrable systems and hamiltonian structure can be brought back to the conformal sphere

a certain class of starlike log harmonic mappings in this paper we investigate some properties of log harmonic starlike mappings for this aim we use the subordination principle or lindelof principle lewandowski 1961 7

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

bi hamiltonian operators integrable flows of curves using moving frames and geometric map equations moving frames of various kinds are used to derive bi hamiltonian operators and associated hierarchies of multi component soliton equations from group invariant flows of non stretching curves in constant curvature manifolds and lie group manifolds the hierarchy in the constant curvature case consists of a vector mkdv equation coming from a parallel frame a vector potential mkdv equation coming from a covariantly constant frame and higher order counterparts generated by an underlying vector mkdv recursion operator in the lie group case the hierarchy comprises a group invariant analogue of the vector nls equation coming from a left invariant frame along with higher order counterparts generated by a recursion operator that is like a square root of the mkdv one the corresponding respective curve flows are found to be given by geometric nonlinear pdes specifically mkdv and group invariant analogues of schrodinger maps in all cases the hierarchies also contain variants of vector sine gordon equations arising from the kernel of the respective recursion operators the geometric pdes that describe the corresponding curve flows are shown to be wave maps

period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations flows on the moduli space of the algebraic riemann surfaces preserving the periods of the corresponding solutions of the soliton equations are studied we show that these flows are gradient with respect to some indefinite symmetric flat metric arising in the hamiltonian theory of the whitham equations the functions generating these flows are conserved quantities for all the equations simultaneously we show that for 1 1 systems these flows can be imbedded in a larger system of ordinary nonlinear differential equations with a rational right hand side finally these flows are used to give a complete description of the moduli space of algebraic riemann surfaces

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

hamiltonian evolution of curves in classical affine geometries abstract in this paper we study geometric poisson brackets and we show that if m g r n g endowed with an affine geometry in the klein sense and if g is a classical lie group then the geometric poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves except for the case g gl n r this trivial extension does not exist in other nonaffine cases projective conformal etc

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

on completely integrable geometric evolutions of curves of lagrangian planes in this paper we find an explicit moving frame along curves of lagrangian planes invariant under the action of the symplectic group we use the moving frame to find a family of independent and generating differential invariants we then construct geometric hamiltonian structures in the space of differential invariants and prove that if we restrict them to a certain poisson submanifold they become a set of decoupled korteweg de vries kdv first and second hamiltonian structures we find an evolution of curves of lagrangian planes that induces a system of decoupled kdv equations on their differential invariants we call it the lagrangian schwarzian kdv equation we also show that a generalized miura transformation takes this system to a modified matrix kdv equation in the four dimensional case we show that there are no unrestricted compatible geometric pairs

curvatures and inhomogeneities an improved common reflection surface approach abstractmultiparameter stacking is an important tool to obtain a first reliable time image of the subsurface in addition it provides wavefield attributes which form the basis for many important applications the quality of the image and the attribute estimates relies heavily on the accuracy of the traveltime moveout description the commonly used hyperbolic common reflection surface crs operator reduces to the nmo hyperbola in the common midpoint gather its accuracy however depends on the curvature of the reflector under consideration the conventional multifocusing mf operator a time shifted double square root expression leads to good results for high reflector curvatures and moderate inhomogeneities of the overburden we used a new implicit crs formulation that combines the robustness of crs regarding heterogeneities with the high sensitivity to curvature of the mf approach it assumes reflectors to be locally circular and can be applied in an iterative fashion for simple but common acquisi

determination of wavefront attributes by differential evolution in the presence of conflicting dips abstractthe common reflection surface crs method represents a multidimensional stacking approach i e the stacking surface is determined in the midpoint and offset directions in the 2d case three attributes span the stacking surface thus requiring a three parameter search contrary to a one parameter search in the classic common midpoint stack however crs wavefront attributes use data redundancy in the midpoint direction as well which makes them very useful in several seismic applications e g data preconditioning velocity model building and migration contrary to previous works we simultaneously estimate crs attributes using differential evolution in subcubes of the 3d search space differential evolution is a global optimization technique that performs particularly well when the objective function is unknown because we apply de for each subcube we could find local maxima additionally to the global maximum therefore conflicting dips are recognized and can be used for the stack and subs

variational problems for plane curves in centro affine geometry in this paper closed extremals of variational problems defined by quadratic polynomials in the centro affine curvature are considered the closure of the trajectories is discussed and the existence of countably many closed critical curves is proven the geometrical properties of closed trajectories are analyzed by numerical methods

the vortex filament in the minkowski 3 space and generalized bi schrodinger maps by extending the concept of generalized bi schrodinger maps to the case that the target manifold is a para kahler manifold we show that the third order timelike and spacelike correction models of the vortex filament in the minkowski 3 space are equivalent to the generalized bi schrodinger maps from to the hyperbolic 2 space and the de sitter 2 space respectively as a consequence all three typical second to fourth order integrable systems of the akns hierarchy are interpreted in a unified way in terms of generalized bi schrodinger maps based on this exploitation and a general discussion of the generalized bi schodinger maps from into an arbitrary riemannian surface we reveal a property of generalized bi schrodinger maps that is not admitted for schrodinger maps

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

univalent functions in h _ h d etude des fonctions univalentes dans h h d solutions d une equation aux derivees partielles non lineaire elliptique

quaternionic soliton equations from hamiltonian curve flows in bb h bb p n a bi hamiltonian hierarchy of quaternion soliton equations is derived from geometric non stretching flows of curves in the quaternionic projective space the derivation adapts the method and results in recent work by one of us on the hamiltonian structure of non stretching curve flows in riemannian symmetric spaces m g h by viewing as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups as main results scalar vector multi component versions of the sine gordon sg equation and the modified korteweg de vries mkdv equation are obtained along with their bi hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator the corresponding geometric curve flows in are shown to be described by a non stretching wave map and a mkdv analog of a non stretching schr dinger map

integrable systems and invariant curve flows in centro equiaffine symplectic geometry abstract in this paper the theory for curves in centro equiaffine symplectic geometry is established integrable systems satisfied by the curvatures of curves under inextensible motions in centro equiaffine symplectic geometry are identified it is shown that certain non stretching invariant curve flows in centro equiaffine symplectic geometry are closely related to the matrix kdv equations and their extension

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

one pointed univalent logharmonic mappings abstract a complete characterization of univalent logharmonic mappings from the exterior of the unit disku onto c 1 is given

integrable systems and invariant curve flows in symplectic grassmannian space abstract in this paper local geometry of curves in the symplectic grassmannian homogeneous space sp 4 r sp 2 r sp 2 r and its connection with that of the pseudo hyperbolic space h 2 2 are studied the group based serret frenet equations and the associated maurer cartan differential invariants for the grassmannian curves are obtained by using the equivariant moving frame method the grassmannian natural frame is also constructed by a gauge transformation from the serret frenet frame relating to the hyperbolic natural frame by the local lie group isomorphism using the natural frames invariant curve flows in the grassmannian and the hyperbolic spaces are studied it is shown that certain intrinsic curve flows induce the bi hamiltonian integrable matrix mkdv equation on the maurer cartan differential invariants

invariant differential equations and the adler gel fand dikii bracket in this paper we find an explicit formula for the most general vector evolution of curves on rpn 1 invariant under the projective action of sl n r when this formula is applied to the projectivization of solution curves of scalar lax operators with periodic coefficients one obtains a corresponding evolution in the space of such operators we conjecture that the formula we have found gives another alternative definition of the second kdv hamiltonian evolution under appropriate conditions in other words both evolutions are identical provided that the vector differential invariant characterizing the sl n r invariant evolution on the space of projectivized curves is identified with the coefficients of the hamiltonian pseudo differential operator we prove the above facts for n 6 and further simplify both evolutions in appropriate coordinates so that one can attempt to prove the equivalence for arbitrary n

an analytical approach to migration velocity analysis prestack depth migration provides a powerful tool for velocity analysis in complex media both prominent approaches to velocity analysis depth focusing analysis and residual curvature analysis rely on approximate formulas to update velocity generally these formulas are derived under the assumptions of horizontal reflector lateral velocity homogeneity or small offset therefore the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large lateral variations here based on ray theory i find the analytic representation for the derivative of imaged depths with respect to migration velocity this derivative function characterizes a general relationship between residual moveout and residual velocity using the derivative function and the perturbation method i derive a new formula to update velocity from residual moveout in the derivation i impose no limitation on offset dip or velocity distribution consequently i revise the residual curvature a

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

imaging and velocity estimation with depth focusing analysis prestack depth migration uses two imaging conditions zero time and zero offset during downward continuation to form a migrated depth section when the migration velocities are exact the two imaging conditions act in a complementary fashion to yield a focused image when the migration velocities are in error reflected energy collapses to zero offset at depths that are inconsistent with the zero time imaging condition the result is a deteriorated seismic image however by interpreting the nonzero times at which focusing actually occurs the migration velocities can be updated iteratively in a process called depth focusing analysis to produce a well focused seismic image the goal of depth focusing analysis must be the elimination of focusing errors however practical considerations can prevent this goal from being achieved therefore to relax the sensitivity of the migrated image to focusing errors we introduce a nonzero time imaging condition by extracting the data along the interpreted surface o

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

poisson brackets associated to the conformal geometry of curves in this paper we present an invariant moving frame in the group theoretical sense along curves in the mobius sphere this moving frame will describe the relationship between all conforrnal differential invariants for curves that appear in the literature using this frame we first show that the kac moody poisson bracket on lo n 1 1 can be poisson reduced to the space of conformal differential invariants of curves the resulting bracket will be the conformal analogue of the adler gel fand dikii bracket secondly a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow we give the conditions on the invariant flow ensuring that the induced evolution is hamiltonian with respect to the reduced poisson bracket because of a certain parallelism with the euclidean case we study what we call frenet and natural cases we comment on the implications for completely integrable systems and describe conformal analogues of the hasimoto transformation

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed