topological horseshoes in travelling waves of discretized nonlinear wave equations applying the concept of anti integrable limit to coupled map lattices originated from space time discretized nonlinear wave equations we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions in particular the coupled map lattices display spatio temporal chaos on the horseshoes

complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

multiple nontrivial solutions for semilinear elliptic problems abstract we study the existence of multiple constant sign and nodal sign changing solutions for semilinear elliptic equations our approach is based on a combination of variational techniques using critical point theory of the method of upper lower solutions and of morse theory critical groups we prove several multiplicity results our analytical framework incorporates both coercive and superlinear parametric problems

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

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

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

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

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

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

nanopteron solutions of diatomic fermi pasta ulam tsingou lattices with small mass ratio abstract consider an infinite chain of masses each connected to its nearest neighbors by a nonlinear spring this is a fermi pasta ulam tsingou lattice we prove the existence of traveling waves in the setting where the masses alternate in size in particular we address the limit where the mass ratio tends to zero the problem is inherently singular and we find that the traveling waves are not true solitary waves but rather nanopterons which is to say waves which are asymptotic at spatial infinity to very small amplitude periodic waves moreover we can only find solutions when the mass ratio lies in a certain open set the difficulties in the problem all revolve around understanding jost solutions of a nonlocal schrodinger operator in its semi classical limit

orbits homoclinic to exponentially small periodic orbits for a class of reversible systems application to water waves in this paper a class of reversible analytic vector fields is investigated near an equilibrium for these vector fields the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues

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

critical points and positive solutions of singular elliptic boundary value problems usually we do not think there is variational structure for singular elliptic boundary value problems so it cannot be considered by using critical points theory in this paper we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems especially use the ordinary differential equation theory of banach spaces to obtain new results on the existence of multiple positive solutions the method is useful for other singular problems

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

branches of radial solutions for semipositone problems abstract we consider the radially symmetric solutions to the equation u x u x for x u x 0 for x where denotes the unit ball in r n n 1 centered at the origin and 0 here r r is assumed to be semipositone 0 0 there exists at most one positive solution u such that u belongs to the unbounded branch of positive solutions further when t tp k k 0 and 1

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

two dimensional travelling waves over a moving bottom abstract two dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied the bottom and the wave travel with a same speed the exact euler equations are formulated as a spatial dynamic system by using the stream function a manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions a homoclinic solution to the normal form of this reduced system persists when higher order terms are added which gives a generalized solitary wave the homoclinic solution connecting a periodic solution

the generation of radiating waves in a singularly perturbed korteweg de vries equation abstract we consider a fifth order kdv equation where the fifth order derivative term is multiplied by a small parameter it has been conjectured that this equation admits a non local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core we prove that this solution cannot be exactly steady and instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail the decay rate is calculated in the limit as the parameter tends to zero in order to verify the analytical results we have developed a high precision spectral method for numerical integration of this equation the analytical and numerical result show good agreement

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

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

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

exact theory of secondary supercritical solutions for steady surface waves over a bump abstract two dimensional steady surface waves on an ideal fluid over a small and symmetric bump are studied when the froude number f of the uniform flow far upstream is greater than unity recent numerical results for a symmetric bump indicate that corresponding to f 1 there may exist two solutions for the problem one approaches the uniform flow and the other a solitary wave as the bump size goes to zero in this paper we use a two parameter asymptotic expansion to derive an explicit approximate expression for the later solution and show that the expansion is indeed an asymptotic approximation to the solution of the exact nonlinear equations

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

fucik spectrum sign changing and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity in this paper we use fucik spectrum ordinary differential equation theory of banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity especially with resonance at infinity and obtain new results on the existence of multiple solutions and sign changing solutions with a weakening of the p s condition in one case we get up to seven nontrivial solutions the techniques have independent interest

a shallow water approximation to the full water wave problem we demonstrate that the system of the green naghdi equations as a two directional nonlinearly dispersive wave model is a close approximation to the two dimensional full water wave problem based on the energy estimates and the proof of the well posedness for the green naghdi equations and the water wave problem we compare solutions of the two systems showing that without restrictions on the wave amplitude any two solutions of the two systems remain close at least in some finite time within the shallow water regime provided that their initial data are close in the banach space hs hs 1 for some s as a consequence we show that if the depth of the water compared with the wavelength is sufficiently small the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper 2006 wiley periodicals inc

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

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

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

sign changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems with a reaction term nonzero at zero abstract in this paper we use the ordinary differential equation theory of banach spaces and minimax theory in particular the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero and get new multiple solutions and sign changing solutions theorems moreover the sign changing solutions change sign exactly once under some conditions

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator