Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.Comment: To appear in Inventiones Mathematica

A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial direction x and have the profile of a uni- or multipulse solitary wave in the other z. The waves are detected using a combination of Hamiltonian spatial dynamics and homoclinic Lyapunov-Schmidt theory. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a family of points Pk,k+1, k = 1, 2, . . . in its two-dimensional parameter space is identified at which a Hamiltonian 0202 resonance takes place (the zero eigenspace and generalised eigenspace are respectively two and four dimensional). The point Pk,k+1 is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratio k : k+1 simultaneously exist (‘Wilton ripples’). A reduction principle is applied to demonstrate that the problem is locally equivalent to a four-dimensional Hamiltonian system near Pk,k+1. It is shown that a Hamiltonian real semisimple 1 : 1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curve Rk,k+1 emanating from Pk,k+1. Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points of Rk,k+1 near Pk,k+1 are found by a scaling and perturbation argument, and the homoclinic Lyapunov-Schmidt method is applied to construct an infinite family of multipulse homoclinic solutions which resemble multiple copies of the unipulse solutions

A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form. We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature

Solvable cubic resonant systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.Comment: v2: 23 pages, 1 figure, minor corrections, published versio

Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of Benjamin-Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero. At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions). The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order. The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have nonconstant coefficients. The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash-Moser iteration in Sobolev spaces, and Lyapunov-Schmidt decomposition. (Version 2: small change in Section 2).Comment: 47 page

Topological Sigma-model, Hamiltonian Dynamics and Loop Space Lefschetz Number

We use path integral methods and topological quantum field theory techniques to investigate a generic classical Hamiltonian system. In particular, we show that Floer's instanton equation is related to a functional Euler character in the quantum cohomology defined by the topological nonlinear $\sigma$--model. This relation is an infinite dimensional analog of the relation between Poincar\'e--Hopf and Gauss--Bonnet--Chern formul\ae$~$ in classical Morse theory, and can also be viewed as a loop space generalization of the Lefschetz fixed point theorem. By applying localization techniques to path integrals we then show that for a K\"ahler manifold our functional Euler character coincides with the Euler character determined by the finite dimensional de Rham cohomology of the phase space. Our results are consistent with the Arnold conjecture which estimates periodic solutions to classical Hamilton's equations in terms of de Rham cohomology of the phase space.Comment: 10 pages, LaTEX. New title and some modifications in the text. Version to appear in Phys. Lett.

Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional $$J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.$$ We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.Comment: 36 pages, 4 figure

KdV equation under periodic boundary conditions and its perturbations

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations

Oscillatory spatially periodic weakly nonlinear gravity waves on deep water

A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves