Persistence of hyperbolic tori in Hamiltonian systems

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system.     

On semi-global invariants for focus-focus singularities

This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a four-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus type.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

Singular periodic waves of an integrable equation from short capillary-gravity waves

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

The modular hierarchy of the Toda lattice

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

Spectral and modulational stability of periodic wavetrains for the nonlinear Klein–Gordon equation

This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein–Gordon equation _utt−_uxx+V^′(u)=0$u_{tt}-u_{xx}+V^{\prime }(u)=0$, where u is a scalar-valued function of x and t  , and the potential V(u)$V(u)$ is of class C²$C^2$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and present new spectral stability results for periodic traveling waves, and to make a solid connection between these results and predictions of the (formal) modulation theory, which has been developed by others but which we review for completeness.     

Moser's theorem for lower dimensional tori

Moser's C^?-version of Kolmogorov's theorem on the persistence of maximal quasi-periodic solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C^? for any ?>6n+5 and the quasi-periodic solutions are proved to be of class C^p for any p with 2<p<p_* for a suitable p_*=p_*(n,?)>2 (which tends to infinity when ?→∞).     

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I - ?Birkhoff Normal Forms

. The aim of this paper (part I and II) is to explore the relationship between the effective (Nekhoroshev) stability for near-integrable Hamiltonian systems and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set Q \Theta defined by a Diophantine condition, we are going to find a family L_w, w ? Q \Lambda_{\omega}, \omega \in \Theta , of KAM invariant tori of H with frequencies w ? Q \omega \in \Theta which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union L \Lambda of the invariant tori which can be viewed as a simultaneous Birkhoff normal form of H around all invariant tori L_w \Lambda_{\omega} . This leads to effective stability of the quasiperiodic motion near L \Lambda . As an application we obtain in part II (semiclassical) quasimodes with exponentially small error terms which are associated with a Gevrey family of KAM tori for its principal symbol H. To do this we construct a quantum Birkhoff normal form of the Schrödinger operator around L \Lambda in suitable Gevrey classes starting from the Birkhoff normal form of H.     

Integrable Hamiltonian systems with positive topological entropy

In this paper we provide a class of integrable Hamiltonian systems on a three-dimensional Riemannian manifold whose flows have a positive topological entropy on almost all compact energy surfaces. As our knowledge, these are the first examples of C^∞ Liouvillian integrable Hamiltonian flows with potential energy on a Riemannian manifold which has a positive topological entropy.     

About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels

We study completely integrable quasi-bi-Hamiltonian systems whose common level surfaces are compact and prove in particular their separability in the sense of Falqui and Pedroni.     

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed.     

Integrability versus topology of configuration manifolds and domains of possible motions

We establish a generic sufficient condition for a compact n-dimensional manifold bearing an integrable geodesic flow to be the n-torus. As a complementary result, we show that in the case of domains of possible motions with boundary, the first Betti number of the domain of possible motions may be arbitrarily large. Received: 28 January 2005; revised: 16 March 2005     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

Small amplitude steady internal waves in stratified fluids

We study the weakly non linear solutions of theDubreil-Jacotin—Long elliptic equation in a strip, which describes two dimensional gravity internal waves propagating steadily in a stratified fluid. In the neighborhood of the first critical value of the Froude number, the center manifold theorem ensures that small solutions are parametrized by two coordinates which verify a system of nonlinear ordinary differential equations. We compute numerically the coefficients of the normal form of this reduced system for a three parameters family of stratifications and show that the quadratic coefficient (the most important) may become small. In that case, nonusual waves such as fronts can propagate. The last part of our work studies the case when a smooth stratification converges towards a piecewise constant profile having one discontinuity. We observe formally that the small waves which propagate at the interface of two homogeneous fluids are limits at leading order of waves travelling in the region where the smooth density varies rapidly.     

Traveling waves in a generalized nonlinear dispersive–dissipative equation

D. Zeidan In this paper, we consider the existence of traveling waves in a generalized nonlinear dispersive–dissipative equation, which is found in many areas of application including waves in a thermoconvective liquid layer and nonlinear electromagnetic waves. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory and invariant manifold theory, Fredholm theory, and the linear chain trick, we construct a locally invariant manifold for the associated traveling wave equation and use this invariant manifold to obtain the traveling waves for the nonlinear dispersive–dissipative equation. Copyright © 2015 John Wiley & Sons, Ltd.     

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus.We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an a posteriori approach.The proof of the main theorems is based on an iterative method to solve the functional equation.The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map.The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on successive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.     

Pseudo-normal form near saddle-center or saddle-focus equilibria

In this paper we introduce the pseudo-normal form, which generalizes the notion of normal form around an equilibrium. Its convergence is proved for a general analytic system in a neighborhood of a saddle-center or a saddle-focus equilibrium point. If the system is Hamiltonian or reversible, this pseudo-normal form coincides with the Birkhoff normal form, so we present a new proof in these celebrated cases. From the convergence of the pseudo-normal form for a general analytic system several dynamical consequences are derived, like the existence of local invariant objects.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].