renormalization methods for higher order differential equations we adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second order equation in particular we study equations involving the derivative of a double well potential such as u u3 or u 2u3 using momentum fourier space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation once length is properly renormalized yielding an approximation to solutions of the higher order equation from the second order the results are confirmed using numerical computations additional numerics confirm that the main role of the higher order derivatives is in rescaling the length

a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

on a shallow water model with the coriolis effect abstract in the present study an asymptotic model for wave propagation in shallow water with the effect of the coriolis force is derived from the governing equation in two dimensional flows the transport equation theory is then applied to investigate the local well posedness and wave breaking phenomena for this model the nonexistence of the camassa holm type peaked solution and classification of various traveling wave solutions to the new system are also established moreover it is shown that all the symmetric waves to this model are traveling waves

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

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

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

on a shallow water approximation to the green naghdi equations with the coriolis effect abstract we consider an asymptotic 1d in space rotation camassa holm r ch model which could be used to describe the propagation of long crested shallow water waves in the equatorial ocean regions with allowance for the weak coriolis effect due to the earth s rotation this model equation has similar wave breaking phenomena as the camassa holm equation it is analogous to the rotation green naghdi r gn equations with the weak earth s rotation effect modeling the propagation of wave allowing large amplitude in shallow water we provide here a rigorous justification showing that solutions of the r gn equations tend to associated solution of the r ch model equation in the camassa holm regime with the small amplitude and the larger wavelength furthermore we demonstrate that the r gn model equations are locally well posed in a sobolev space by the refined energy estimates

differential systems with fuchsian linear part correction and linearization normal forms and multiple orthogonal polynomials abstract differential systems with a fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations such systems are not necessarily analytically equivalent to their linear part they are not linearizable and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable more generally normal forms are found the corrections and the normal forms are found constructively expansions in multiple orthogonal polynomials and their generalization to matrix valued polynomials are instrumental to these constructions

a nonlocal shallow water model with the weak coriolis and equatorial undercurrent effects abstract we derive a quasilinear nonlocal shallow water model equation for moderate amplitude equatorial waves with effects of the weak coriolis force and equatorial undercurrent this mathematical modeling is analogous to the camassa holm approximation of the two dimensional incompressible and rotational euler equations in the equatorial region moreover we investigate the influences and interactions of the weak coriolis force caused by the earth rotation vorticity and nonlocal higher nonlinearities on the wave breaking phenomena in certain cases by applying the method of characteristics and extremal property of the solution to the riccati type differential inequality we demonstrate that wave breaking phenomena occur only depending on a shape of wave initially in the local spacial point

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

a note on the adomian decomposition method for the generalized huxley equation in a recent paper ismail et al adomian decomposition method for burgers huxley and burgers fisher equations appl math comput 159 2004 291 301 employed the adomian decomposition method adm to solve in particular a generalized nonlinear huxley equation the purpose of this note is to correct ismail et al s numerical solutions of the generalized huxley equation in addition a convergence proof of the adm applied to the generalized huxley equation is also given

non uniform continuity of the solution map to the rotation two component camassa holm system abstract studied in this paper is the rotation two component camassa holm system which can be considered as a model for the propagation of long crested shallow water waves in the equatorial ocean regions with the weak coriolis effect due to the earth s rotation it is shown that the data to solution map is not uniformly continuous on both the line and the circle in certain sobolev spaces depending on the effect of the coriolis force and stretching balance parameters the non uniform dependence result is established by using the method of approximate solutions with refined energy estimates

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

correction and linearization of resonant vector fields and diffeomorphisms we extend the classical siegel brjuno russmann linearization theorem to the resonant case by showing that under a d brjuno s diophantine condition any resonant local analytic vector field resp diffeomorphism possesses a well defined correction which 1 depends on the chart but in any given chart is unique 2 consists solely of resonant terms and 3 has the property that when substracted from the vector field resp when factored out of the diffeomorphism the vector field or diffeomorphism thus corrected becomes analytically linearizable with a privileged or canonical linearizing map moreover in spite of the small denominators and contrary to a hitherto prevalent opinion the correction s analyticity can be established by pure combinatorics without any analysis

wave breaking analysis for the periodic rotation two component camassa holm system abstract in this paper we study the blow up phenomenon of the initial boundary value problem for the periodic rotation two component camassa holm r2ch system the results include a precise blow up criterion when 0 and two new blow up statements and the exact blow up rate for strong solutions of the system

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

on the rotation two component camassa holm system modelling the equatorial water waves abstract in this paper a modified two component camassa holm system with the effect of the coriolis force in the rotating fluid is derived which is a model in the equatorial water waves the effects of the coriolis force caused by the earth s rotation and nonlocal nonlinearities on blow up criteria and wave breaking phenomena are then investigated our refined analysis relies on the method of characteristics and conserved quantities and is proceeded with the riccati type differential inequality finally conditions which guarantee the permanent waves are obtained by using a method of the lyapunov function

on the wave breaking phenomena and global existence for the periodic rotation two component camassa holm system abstract considered herein is the periodic rotation two component camassa holm system which can be derived from the f plane governing equations for the geophysical water waves with a constant underlying current the nonlocal nonlinearities on blow up criteria and wave breaking phenomena are established finally a sufficient condition for global solutions is obtained by using a method of the lyapunov function

an explicit and numerical solutions of some fifth order kdv equation by decomposition method in this paper by considering the adomian decomposition method explicit and numerical solutions are calculated for a various fifth order korteweg de vries equations with initial condition the method does not need linearization or weak nonlinearity assumptions perturbation theory the decomposition series explicit solution of the equation is quickly obtained by observing the existence of the self canceling noise terms where sum of components vanishes in the limit

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

on the cauchy problem for the shallow water model with the coriolis effect abstract in this paper we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the coriolis force we first establish the local well posedness in a range of the besov spaces as well as the local well posedness in the critical space then we study the gevrey regularity of the shallow water model by using a generalized cauchy kovalevsky theorem which implies that the shallow water model admits analytical solutions locally in time and globally in space moreover we obtain a precise lower bound of the lifespan and the continuity of the solution finally working with moderate weight functions that are commonly used in time frequency analysis some persistence results to the shallow water model are illustrated

painleve analysis and normal forms theory nonlinear vector fields have two important types of singularities the fixed points in phase space and the time singularities in the complex plane the first singularities are locally analyzed via normal form theory whereas the second ones are studied by the painleve analysis in this paper normal form theory is used to describe the solutions around their complex time singularities to do so a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced regular normal form theory is then used in this new system it is shown that a vector field has the painleve property only if the associated system is locally linearizable around its fixed points a problem analogous to the classical problem of the center moreover the connection between partial and complete integrability and the structure of local series around both types of singularities are established a new proof of the convergence of the local psi series is given and an explicit method to prove the existence of finite time blow up manifold in phase space is presented

the global gevrey regularity of the rotation two component camassa holm system abstract in this paper we study the rotation two component camassa holm system which is a model in the equatorial water waves with the effect of the coriolis force we establish the global gevrey regularity of the rotation two component camassa holm system in gevrey class g with 1 in time our obtained result improves considerably the recent result in 45

isochronous centers of polynomial hamiltonian systems and a conjecture of jarque and villadelprat abstract we study the conjecture of jarque and villadelprat stating that every center of a planar polynomial hamiltonian system of even degree is nonisochronous this conjecture has already been proved for quadratic and quartic systems using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields we are able to describe a very large class of nonisochronous hamiltonian systems of even arbitrarily large degree

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

numerical solutions of coupled burgers equations with time and space fractional derivatives in this paper by introducing the fractional derivative in the sense of caputo the adomian decomposition method is directly extended to study the coupled burgers equations with time and space fractional derivatives as a result the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components the figures show the effectiveness and good accuracy of the proposed method

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

convergence or generic divergence of the birkhoff normal form we prove that the birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent the same result is proved for the normalization mapping and any formal first integral

on the curvature blow up phenomena for the fokas qiao xia li equation abstract we study the blow up phenomena of the strong solutions for the fokas qiao xia li fqxl equation with quadratic and cubic nonlinearities which include the celebrated camassa holm equation and the fokas olver rosenau qiao forq equation as its special case the fqxl equation models the unidirectional propagation of the shallow water waves over a flat bottom and describes the physics more accurately than its celebrated counterpart the gardner equation two kinds of conditions on the initial data including cases where the initial momentum density is not of one sign are formulated to guarantee finite time blow up for the strong solutions for this equation the first condition relies on the sign persistence property of the momentum density m the second condition is based on the full use of the conservation laws of the equation especially the conservation quantity h 2 u 0

a numerical implementation of the decomposition method for the lienard equation in this study a kind of explicit exact and numerical solutions to the lienard equation is obtained and the applications of the results to this equation has been compared with their known theoretical solution this paper is particularly concerned with the adomian decomposition method and the numerical results demonstrate that the new method is relatively accurate and easily implemented