analytical solutions for a nonlinear diffusion equation with convection and reaction abstract nonlinear diffusion equations with the convection and reaction terms are solved by using a power law ansatz this kind of equations typically appears in nonlinear problems of heat and mass transfer and flows in porous media the solutions that we introduce in this work are analytical at least in the convection case the result recovers its linear form as a special limit in the reaction case we define a class of nonlinearity to discuss the evolution of general solutions we also add the verhulst like dynamics and global regulation we think this method based on this kind of ansatz can also be applied to solve other nonlinear partial differential equations

time periodic solutions to a nonlinear wave equation with periodic or anti periodic boundary conditions this paper is concerned with the existence of time periodic solutions to the nonlinear wave equation with x dependent coefficients u x y tt u x y x x au x y y p 2 y f x t on 0 under the periodic or anti periodic boundary conditions y 0 t y t y x 0 t y x t and the time periodic conditions y x t t y x t y t x t t y t x t such a model arises from the forced vibrations of a non homogeneous string and the propagation of seismic waves in non isotropic media a main concept is the notion weak solution to be given in 2 for t 2 k k we establish the existence of time periodic solutions in the weak sense by investigating some important properties of the wave operator with x dependent coefficients

periodic solutions of some infinite dimensional hamiltonian systems associated with non linear partial difference equations ii this is our second paper devoted to the study of some non linear schrodinger equations with random potential we study the non linear eigenvalue problems corresponding to these equations we exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails contrary to our results in the first paper the results established in the present paper hold for an arbitrary strength of the non linear cubic term in the non linear schrodinger equation

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

on the nature of the torus in the complex lorenz equations the complex lorenz equations are a nonlinear fifth order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus in this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter and between zero and nonzero r2 complexity parameter in so doing we are able to gain insight on the effect of the rotational invariance of the system and on how extra weak dispersion r2 0 affects the chaotic behavior of the real lorenz system which describes a weakly dissipative dispersive instability

solution regularity and smooth dependence for abstract equations and applications to hyperbolic pdes first we present a generalized implicit function theorem for abstract equations of the type f u 0 we suppose that u0 is a solution for 0 and that f is smooth for all but mainly we do not suppose that f u is smooth for all u we state conditions such that for all 0 there exists exactly one solution u u0 that u is smooth in a certain abstract sense and that the data to solution map u is smooth then we apply this to time periodic solutions of first order hyperbolic systems tuj aj x xuj bj t x u 0 and second order hyperbolic equations t2u a x 2 x2u b t x u tu xu 0 here we have to prevent small divisors from coming up moreover we need smooth dependence of bj and b on t to get smooth dependence of the solution on this is completely different to the case of parabolic pdes

quasi periodic solutions for 1d wave equation with the nonlinearity u2p 1 abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 2 p 1 0 p n on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

quasiperiodic solutions for dissipative boussinesq systems in this paper we analyze the behavior of the solution of the dissipative boussinesq systems partial t u partial x v alpha partial xxx v beta partial xxt u partial x uv partial t v partial x u c partial xxx u beta partial xxt v v partial x v where c 0 are parameters those systems model two dimensional small amplitude long wavelength water waves for 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it admits solutions that are quasiperiodic in time the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the boussinesq system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem and studying the birkhoff normal form

bifurcation from rotating waves we consider abstract evolution equations which are invariant under a lie group we discuss bifurcation from trajectories that lie in the orbit under the lie group of a single point into trajectories in the orbit of a closed curve

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

an identification problem for a time periodic nonlinear wave equation in non cylindrical domains abstract the existence of a time periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established the problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion

nonlinear wave and schrodinger equations on compact lie groups and homogeneous spaces we develop linear and nonlinear harmonic analysis on compact lie groups and homogeneous spaces relevant for the theory of evolutionary hamiltonian pdes a basic tool is the theory of the highest weight for irreducible representations of compact lie groups this theory provides an accurate description of the eigenvalues of the laplace beltrami operator as well as the multiplication rules of its eigenfunctions as an application we prove the existence of cantor families of small amplitude time periodic solutions for wave and schrodinger equations with differentiable nonlinearities we apply an abstract nash moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the laplace operator we provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on banach scales of sobolev functions

a kam theorem with applications to partial differential equations of higher dimensions the existence of lower dimensional kam tori is shown for a class of nearly integrable hamiltonian systems of infinite dimensions where the second melnikov s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed as a consequence it is proved that there exist many invariant tori and thus quasi periodic solutions for nonlinear wave equations schrodinger equations and other equations of any spatial dimensions

normal forms resonances and meandering tip motions near relative equilibria of euclidean group actions the equivariant dynamics near relative equilibria to actions of noncompact finite dimensional lie groups g can be described by a skew product flow on a center manifold dot g g textbf a v dot v varphi v with g in g with v in a slice transverse to the group action and a v in the lie algebra of g we present a normal form theory near relative equilibria varphi v 0 0 in this general case for the specific case of the euclidean groups se n the skew product takes the form dot r r textbf r v qquad dot s r textbf s v qquad dot v varphi v with textbf r v in so n textbf s v in mathr n we give a precise meaning to the intuitive idea of tip motion of a meandering spiral it corresponds to the dynamics of s t this clarifies the notion of meander radii and drift resonance in the plane n 2 for illustration we discuss the unbounded tip motions associated with a weak focus in v on the verge of hopf bifurcation in the case of resonant hopf and rotation frequencies of the spiral and study resonant relative hopf bifurcation we also encounter random brownian tip motions for trajectories v t rightarrow gamma which become homoclinic for t rightarrow infty we conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the bogdanov takens bifurcation which turn out to be small beyond any finite order

kam tori of hamiltonian perturbations of 1d linear beam equations abstract in this paper one dimensional 1d nonlinear beam equations u tt u xxxx mu f u with hinged boundary conditions are considered the nonlinearity f is an analytic odd function and f u o u3 it is proved that for all real parameters m 0 but a set of small lebesgue measure the above equation admits small amplitude quasi periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system the proof is based on infinite dimensional kam theory developed by kuksin lecture notes in mathematics vol 1556 springer berlin 1993 wayne commun math phys 127 1990 479 528 poschel ann sc norm sup pisa cl sci 23 1996 119 148

newton s laws of motion in the form of a riccati equation we discuss two applications of a riccati equation to newton s laws of motion the first one is the motion of a particle under the influence of a power law central potential v r kr ensuremath epsilon for zero total energy we show that the equation of motion can be cast in the riccati form we briefly show here an analogy to barotropic friedmann robertson lemaitre cosmology where the expansion of the universe can be also shown to obey a riccati equation a second application in classical mechanics where again the riccati equation appears naturally are problems involving quadratic friction we use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems

a birkhoff lewis type theorem for the nonlinear wave equation we give an extension of the celebrated birkhoff lewis theorem to the nonlinear wave equation accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin which is an elliptic equilibrium of the associated infinite dimensional hamiltonian system

kink solitary solutions to generalized riccati equations with polynomial coefficients abstract inverse and direct balancing methods for the construction of kink solitary solutions to generalized riccati equations with polynomial coefficients are presented in this paper necessary and sufficient conditions for the existence of kink solitary solutions are derived in terms of the equation s parameters and initial conditions the proposed technique enables a straightforward derivation of parameters of solitary solutions computational experiments are used to demonstrate the efficiency of the proposed analytical approach

on lie systems and kummer schwarz equations a lie system is a system of first order differential equations admitting a superposition rule i e a map that expresses its general solution in terms of a generic family of particular solutions and certain constants in this work we use the geometric theory of lie systems to prove that the explicit integration of second and third order kummer schwarz equations is equivalent to obtaining a particular solution of a lie system on sl 2 r this same result can be extended to riccati milne pinney and to the here defined generalised kummer schwarz equations which include several types of kummer schwarz equations as particular cases we demonstrate that all the above mentioned equations related to the same lie system on sl 2 r can be integrated simultaneously which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature as a byproduct we recover various properties of the schwarzian derivative

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

kam for reversible derivative wave equations we prove the existence of cantor families of small amplitude analytic linearly stable quasi periodic solutions of reversible derivative wave equations

quasi periodic solutions of nonlinear schrodinger equations of higher dimension abstract it is shown that there are plenty of quasi periodic solutions of nonlinear schrodinger equations of higher spatial dimension where the dimension of the frequency vectors of the quasi periodic solutions are equal to that of the space

periodic solutions on the sturm liouville boundary value problem for two dimensional wave equation in this paper we study the periodic solutions to two dimensional wave equation ytt y ay g y f x t on 0 0 r under the periodic conditions y x t t y x t yt x t t yt x t and the sturm liouville boundary conditions a11y 0 x2 t b11yx1 0 x2 t 0 a12y x2 t b12yx1 x2 t 0 a21y x1 0 t b21yx2 x1 0 t 0 and a22y x1 t b22yx2 x1 t 0 aij2 bij2 0 for i j 1 2 our main concept is that of the weak solution which will be given in sec ii then by means of spectral analysis we investigate some important properties of the wave operator with sturm liouville boundary conditions finally by using the properties we established and the duality principles in the work of brezis bull am math soc 8 409 1983 the existence of periodic solutions is obtained

kam for quasi linear and fully nonlinear forced perturbations of airy equation we prove the existence of small amplitude quasi periodic solutions for quasi linear and fully nonlinear forced perturbations of the linear airy equation for hamiltonian or reversible nonlinearities we also prove their linear stability the key analysis concerns the reducibility of the linearized operator at an approximate solution which provides a sharp asymptotic expansion of its eigenvalues for quasi linear perturbations this cannot be directly obtained by a kam iteration hence we first perform a regularization procedure which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder these transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo differential operators at this point we implement a nash moser iteration with second order melnikov non resonance conditions which completes the reduction to constant coefficients

stability of some waves in the dissipative boussinesq system abstract in this paper we study analytically a class of waves in the variant of the classical dissipative boussinesq system given by t u x v x x x v x x t u x u v t v x u c x x x u x x t v v x v where c 0 is a small parameter and 0 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it contains solutions that are defined for large values of time and they are very close of order o to a linear torus for long times of order o 1 the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates

time periodic solutions of a nonlinear wave equation abstract the existence of a time periodic solution of an n dimensional nonlinear wave equation is established with n 2 and 3

hopf bifurcation and symmetry travelling and standing waves on the circle in this paper we consider hopf bifurcation in the presence of o 2 symmetry the system of reaction diffusion equations u t d u xx f u provided with periodic boundary conditions may serve as a model problem we prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability with asymptotic phase of these periodic solutions giving explicit formulae finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

hopf bifurcation in the presence of symmetry on etablit une generalisation du theoreme de bifurcation de hopf pour des equations differentielles a symetrie

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

parabolic approximations of the convection diffusion equation we propose an approximation of the convection diffusion operator which consists in the product of two parabolic operators this approximation is much easier to solve than the full convection diffusion equation which is elliptic in space we prove that this approximation is of order three in the viscosity and that the classical parabolic approximation is of order one in the viscosity numerical examples are given to demonstrate the effectiveness of our new approximation

invariant tori for 1d quintic nonlinear wave equation abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that for any integer b 2 there are many b dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation this is an extension of the previous work 6 by the same authors where b 2 the proof is based on infinite dimensional kam theory and partial birkhoff normal form

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

quasiperiodic solutions for the cubic complex ginzburg landau equation the existence of two dimensional kolmogorov arnold moser invariant tori is proved for cubic complex ginzburg landau equation of higher spatial dimension as a consequence the equation possesses a cantorian branch of nontraveling wave quasiperiodic solutions of two dimensional frequency vector

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

a domain decomposition method for generating orthogonal polynomials for a gaussian weight on a finite interval abstract a domain decomposition method has been developed for generating orthogonal polynomials for a gaussian weight on 1 1 the method takes advantage of the underlying asymptotic structure of the orthogonal polynomials and hence it is effective in the sense that it makes maximal use of the analytic properties of the solution to increase accuracy and efficiency these polynomials are necessary for constructing gaussian quadrature formulas that are encountered in large quantum chemistry computational packages and in calculating the compton scattering kernel and its associated angular moments

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

cantor families of periodic solutions for wave equations via a variational principle abstract we prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a cantor set of asymptotically full measure via a variational principle a lyapunov schmidt decomposition reduces the problem to a finite dimensional bifurcation equation variational in nature defined on a cantor set of non resonant parameters the cantor gaps are due to small divisors phenomena to solve the bifurcation equation we develop a suitable variational method in particular we do not require the typical arnold non degeneracy condition of the known theory on the nonlinear terms as a consequence our existence results hold for new generic sets of nonlinearities

cantor families of periodic solutions for completely resonant nonlinear wave equations we prove the existence of small amplitude 2 periodic in time solutions of completely resonant nonlinear wave equations with dirichlet boundary conditions for any frequency belonging to a cantor like set of asymptotically full measure and for a new set of nonlinearities the proof relies on a suitable lyapunov schmidt decomposition and a variant of the nash moser implicit function theorem in spite of the complete resonance of the equation we show that we can still reduce the problem to a finitedimensional bifurcation equation moreover a new simple approach for the inversion of the linearized operators required by the nash moser scheme is developed it allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

kam for the derivative nonlinear schrodinger equation with periodic boundary conditions abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f u 2 u x 0 x t r 2 z where f is real analytic in some neighborhood of the origin in c f 0 0 and f 0 0 we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

quasi periodic solutions for the quasi periodically forced cubic complex ginzburg landau equation on td in this paper we discuss the existence of time quasi periodic solutions for quasi periodically forced cubic complex ginzburg landau equation of higher spatial dimension with basic frequency vector 1 2 m by constructing a kam kolmogorov arnold moser theorem for a dissipative system which depends on time in a quasi periodic way we obtain a cantorian branch of m 2 dimensional invariant tori for the equation

periodic solutions to one dimensional wave equation with x dependent coefficients abstract in this paper we study the nonlinear one dimensional periodic wave equation with x dependent coefficients u x y t t u x y x x g x t y f x t on 0 r under the boundary conditions a 1 y 0 t b 1 y x 0 t 0 a 2 y t b 2 y x t 0 a i 2 b i 2 0 for i 1 2 and the periodic conditions y x t t y x t y t x t t y t x t such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media a main concept is the notion weak solution to be given in section 2 for t is the rational multiple of we prove some important properties of the weak solution operator based on these properties the existence and regularity of weak solutions are obtained

abstract forced symmetry breaking and forced frequency locking of modulated waves we consider abstract forced symmetry breaking problems of the typef x y it is supposed that for all the mapsf are equivariant with respect to the action of a compact lie group thatf x0 0 0 and hence thatf x 0 0 for all elementsxof the group orbit o x0 ofx0 we look for solutionsxwhich bifurcate from the solution family o x0 as andymove away from 0and zero respectively especially we describe the number of different solutionsx for fixed control parameters andy their dynamic stability and their asymptotic behavior forytending to zero further generalizations are given to problems of the typef x y x finally our results are applied to a forced frequency locking problem of the typex t f x t y t here it is supposed that the vector fieldsf ares1 equivariant that the unperturbed equationx f x 0 has an orbitally stable modulated wave solution and that the forcingy t is a modulated wave

invariant cantor manifolds of quasi periodic solutions for the derivative nonlinear schrodinger equation abstract this paper is concerned with the derivative nonlinear schrodinger equation with periodic boundary conditions i u t u x x i f x u u x 0 x t r 2 z where f is an analytic function of the form f x u u u 2 u f 4 x u u 0 r and f 4 x u u denotes terms of order at least four in u u we show the above equation possesses cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an infinite dimensional kam theorem for unbounded perturbation vector fields

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

quasi periodic solutions for 1d wave equation with higher order nonlinearity abstract in this paper one dimensional 1d nonlinear wave equation u t t u x x m u u 5 0 on the finite x interval 0 with dirichlet boundary conditions is considered it is proved that there are many 2 dimensional elliptic invariant tori and thus quasi periodic solutions for the above equation the proof is based on infinite dimensional kam theory and partial birkhoff normal form

shape invariance through crum transformation we show in a rigorous way that crum s result regarding the equal eigenvalue spectrum of sturm liouville problems can be obtained iteratively by successive darboux transformations furthermore it can be shown that all neighboring darboux transformed potentials of higher order uk and uk 1 satisfy the condition of shape invariance provided the original potential u does so based on this result we prove that under the condition of shape invariance the nth iteration of the original sturm liouville problem defined solely through the shape invariance is equal to the nth crum transformation

multiplicity of periodic solutions of nonlinear wave equations abstract we prove multiplicity of small amplitude periodic solutions with fixed frequency of completely resonant wave equations with general nonlinearities as 1 the number n of 2 periodic solutions u1 un un tends to the minimal period of the nth solution un is 2 n the proofs are based on the variational lyapunov schmidt reduction comm math phys to appear and minimax arguments 1

hopf bifurcation for semilinear dissipative hyperbolic systems abstract we consider boundary value problems for semilinear hyperbolic systems of the type t u j a j x x u j b j x u 0 x 0 1 j 1 n with smooth coefficient functions a j and b j such that b j x 0 0 for all x 0 1 r and j 1 n we state conditions for hopf bifurcation i e for existence local uniqueness up to phase shifts smoothness and smooth dependence on of time periodic solutions bifurcating from the zero stationary solution furthermore we derive a formula which determines the bifurcation direction the proof is done by means of a liapunov schmidt reduction procedure for this purpose fredholm properties of the linearized system and implicit function theorem techniques are used

iterative solution of elliptic problems by approximate factorization an iterative method for the numerical solution of singularly perturbed second order linear elliptic problems is presented it is a defect correction iteration in which the approximate operator is the product of two first order operators which is readily inverted numerically the approximate operator is generated by formal asymptotic factorization of the original operator hence this is a quasi analytic defect correction iteration quad both its continuous and discrete versions are analyzed in one dimension the scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection diffusion equation numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter

dynamics of breathers in discrete nonlinear schro dinger models abstract we review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic non stationary excitations in discrete nonlinear schrodinger dnls models in two dimensions we show the existence of linearly stable stationary and non stationary localized vortex like solutions we also show that stationary on site localized excitations can have internal breathing modes which are spatially localized and symmetric the excitation of these modes leads to slowly decaying quasiperiodic oscillations finally we show that for some generalizations of the dnls equation where bistability occurs a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value

iterative and non iterative full and approximate factorization methods for multidimensional reaction diffusion equations a variety of second order accurate in both space and time full and approximate factorization methods for the numerical solution of two dimensional reaction diffusion equations is presented these methods may use time linearization and yield linearly implicit techniques and one dimensional operators in each direction it is shown that if the factorization errors are neglected linearly implicit approximate factorization methods provide uncoupled equations whereas if these errors are considered the equations are coupled and must be solved iteratively it is also shown that the allocation of the reaction and diffusion terms to the one dimensional operators plays a paramount role in determining the accuracy of approximate factorization methods and preserving the symmetry of the original differential problem iterative full and approximate factorization methods that do require iterations are also presented and for the problem considered here these methods are shown to converge in about two iterations and provide solutions in agreement with those obtained with linearly implicit full and approximate factorization techniques

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

invariant tori of nonlinear schrodinger equation abstract in this paper we consider one dimensional nonlinear schrodinger equation i u t u x x v x u f u 2 u 0 on 0 r under the boundary conditions a 1 u t 0 b 1 u x t 0 0 a 2 u t b 2 u x t 0 a i 2 b i 2 0 for i 1 2 it is proved that for a prescribed and analytic positive potential v x the above equation admits small amplitude quasi periodic solutions corresponding to d dimensional invariant tori of the associated infinite dimensional dynamical system

conservation of resonant periodic solutions for the one dimensional nonlinear schrodinger equation we consider the one dimensional nonlinear schrodinger equation with dirichlet boundary conditions in the fully resonant case absence of the mass term we investigate conservation of small amplitude periodic solutions for a large measure set of frequencies in particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes provided the corresponding frequencies are large enough say greater than a certain threshold value depending on the number of resonant modes if the frequencies of the latter are close enough to such a threshold then they can not be too distant from each other hence we can interpret such solutions as perturbations of wave packets with large wave number

quasi periodic self excited travelling waves for damped beam equations abstract a two dimensional beam equation is considered it is shown that for a dense set of model parameters the presence of linear and nonlinear damping terms of opposite sign produce small amplitude quasi periodic travelling waves which are continuations of two branches of rotating waves lyapunov schmidt reduction and only the classical implicit function theorem are used

quasi periodic solutions of a non autonomous wave equations with quasi periodic forcing abstract in this paper we prove existence of small amplitude quasi periodic solutions for a non autonomous quasi periodically forced nonlinear wave equations with periodic spatial boundary conditions via kam theory

standing waves on an infinitely deep perfect fluid under gravity the existence of two dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established when formulated as a second order equation for a real valued function w on the 2 torus and a positive parameter the problem is fully nonlinear the highest order x derivative appears in the nonlinear term but not in the linearization at 0 and completely resonant there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter moreover for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial using a nash moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions the existence question can be reduced to one of estimating the inverses of linearized operators at non zero points after changing coordinates these operators become first order and non local in space and second order in time after further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing or quasi one dimensional in the sense of 22 the operator can then be inverted for two reasons first the explicit formula for the approximate solution means that restricted to the infinite dimensional kernel of the linearization at zero the inverse exists and can be estimated second the small divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their diophantine properties a parameter dependent version of the nash moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth corresponding to a set of values of the parameter 1 which is dense at 1 unimodal means that the term of smallest order in the amplitude is cos x cos t which is one of many eigenfunctions of the completely resonant linearized problem

invariant tori for a derivative nonlinear schrodinger equation with quasi periodic forcing this paper is concerned with a one dimensional derivative nonlinear schrodinger equation with quasi periodic forcing under periodic boundary conditions iut uxx ig t f u2 u x 0 x t r 2 z where g t is real analytic and quasi periodic on t with frequency vector 1 2 m f is real analytic in some neighborhood of the origin in f 0 0 and f 0 0 we show that the above equation admits cantor families of smooth quasi periodic solutions of small amplitude the proof is based on an abstract infinite dimensional kolmogorov arnold moser theorem for unbounded perturbation vector fields and partial birkhoff normal form

time periodic solutions to the one dimensional nonlinear wave equation this paper is concerned with the time periodic solutions to the one dimensional nonlinear wave equation with either variable or constant coefficients by adjusting the basis of l2 function space we can circumvent the difficulties caused by u 0 and obtain the existence of a weak periodic solution which was posed as an open problem by baubu and pavel in trans am math soc 349 2035 2048 1997 finally an application to the forced sine gordon equation is presented to illustrate the utility of this technique

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

bifurcation of solutions of the laser equations abstract we prove the existence and stability of bifurcating periodic and quasiperiodic solutions of the equations describing a ring laser

an iterative method for systems of nonlinear hyperbolic equations abstract an iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented parallelism is evident at several levels in the formation of the iteration the equations are decoupled thereby providing large grain parallelism parallelism may also be exploited within the solves for each equation convergence of the iteration is established via a bounding function argument experimental results in two dimensions are presented

stability of some solutions for elliptic equations on a cylindrical domain we study analytically a class of solutions for the elliptic equation formula 1 where 0 and e is a small parameter this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every 0 it contains solutions that are defined for large values of time and they are very close of order o e to a linear torus for long times of order o e 1 the proof uses the fact that the equation leaves invariant a smooth centre manifold and for the restriction of the system to the centre manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates 1 embed graphic 1 gif

quasi periodic solutions of derivative nonlinear schrodinger equations with a given potential abstract it is proved that for a prescribed potential v x there are many quasi periodic solutions of derivative nonlinear schrodinger equation i u t u x x v x u u x 2 u subject to dirichlet boundary condition by means of a kam theorem to a reversible system

non hermitian hamiltonian versus e 0 localized states we analyze the zero energy solutions of a two dimensional system which undergoes a non radial symmetric complex potential v r by virtue of the coherent states concept the localized states are constructed and the consequences of the imaginary part of the potential are found both analytically and schematically

periodic solutions for the schrodinger equation with nonlocal smoothing nonlinearities in higher dimension abstract we consider the nonlinear schrodinger equation in higher dimension with dirichlet boundary conditions and with a nonlocal smoothing nonlinearity we prove the existence of small amplitude periodic solutions in the fully resonant case we find solutions which at leading order are wave packets in the sense that they continue linear solutions with an arbitrarily large number of resonant modes the main difficulty in the proof consists in a small divisor problem which we solve by using a renormalisation group approach

geometry of riccati equations over normed division algebras abstract this work introduces and studies riccati equations over finite dimensional normed division algebras we prove that a riccati equation over a finite dimensional normed division algebra a is a particular case of conformal riccati equation on a euclidean space and it can be considered as a curve in a lie algebra of vector fields v so dim a 1 1 previous results on known types of riccati equations are recovered from a new viewpoint a new type of riccati equations the octonionic riccati equations are extended to the octonionic projective line o p 1 as a new physical application quaternionic riccati equations are applied to study quaternionic schrodinger equations on 1 1 dimensions

periodic solutions for one dimensional wave equation with bounded nonlinearity abstract this paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients the constant coefficient model corresponds to the classical wave equation while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media for finding the periodic solutions of variable coefficient wave equation it is usually required that the coefficient u x satisfies ess inf u x 0 with u x 1 2 u u 1 4 u u 2 which actually excludes the classical constant coefficient model for the case u x 0 it is indicated to remain an open problem by barbu and pavel 1997 6 in this work for the periods having the form t 2 p 1 q p q are positive integers and some types of boundary value conditions we find some fundamental properties for the wave operator with either constant or variable coefficients based on these properties we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded such nonlinearity may cross multiple eigenvalues of the corresponding wave operator in particular we do not require the condition ess inf u x 0