33,422 research outputs found
Orbital stability of periodic waves for the nonlinear Schroedinger equation
The nonlinear Schroedinger equation has several families of quasi-periodic
travelling waves, each of which can be parametrized up to symmetries by two
real numbers: the period of the modulus of the wave profile, and the variation
of its phase over a period (Floquet exponent). In the defocusing case, we show
that these travelling waves are orbitally stable within the class of solutions
having the same period and the same Floquet exponent. This generalizes a
previous work where only small amplitude solutions were considered. A similar
result is obtained in the focusing case, under a non-degeneracy condition which
can be checked numerically. The proof relies on the general approach to orbital
stability as developed by Grillakis, Shatah, and Strauss, and requires a
detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure
Small BGK waves and nonlinear Landau damping
Consider 1D Vlasov-poisson system with a fixed ion background and periodic
condition on the space variable. First, we show that for general homogeneous
equilibria, within any small neighborhood in the Sobolev space W^{s,p}
(p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial
travelling wave solutions (BGK waves) with arbitrary minimal period and
traveling speed. This implies that nonlinear Landau damping is not true in
W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and any spatial period.
Indeed, in W^{s,p} (s<1+(1/p)) neighborhood of any homogeneous state, the long
time dynamics is very rich, including travelling BGK waves, unstable
homogeneous states and their possible invariant manifolds. Second, it is shown
that for homogeneous equilibria satisfying Penrose's linear stability
condition, there exist no nontrivial travelling BGK waves and unstable
homogeneous states in some W^{s,p} (p>1,s>1+(1/p)) neighborhood. Furthermore,
when p=2,we prove that there exist no nontrivial invariant structures in the
H^{s} (s>(3/2)) neighborhood of stable homogeneous states. These results
suggest the long time dynamics in the W^{s,p} (s>1+(1/p)) and particularly, in
the H^{s} (s>(3/2)) neighborhoods of a stable homogeneous state might be
relatively simple. We also demonstrate that linear damping holds for initial
perturbations in very rough spaces, for linearly stable homogeneous state. This
suggests that the contrasting dynamics in W^{s,p} spaces with the critical
power s=1+(1/p) is a trully nonlinear phenomena which can not be traced back to
the linear level
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
Integrable turbulence generated from modulational instability of cnoidal waves
We study numerically the nonlinear stage of modulational instability (MI) of
cnoidal waves, in the framework of the focusing one-dimensional Nonlinear
Schrodinger (NLS) equation. Cnoidal waves are the exact periodic solutions of
the NLS equation and can be represented as a lattice of overlapping solitons.
MI of these lattices lead to development of "integrable turbulence" [Zakharov
V.E., Stud. Appl. Math. 122, 219-234 (2009)]. We study the major
characteristics of the turbulence for dn-branch of cnoidal waves and
demonstrate how these characteristics depend on the degree of "overlapping"
between the solitons within the cnoidal wave.
Integrable turbulence, that develops from MI of dn-branch of cnoidal waves,
asymptotically approaches to it's stationary state in oscillatory way. During
this process kinetic and potential energies oscillate around their asymptotic
values. The amplitudes of these oscillations decay with time as t^{-a},
1<a<1.5, the phases contain nonlinear phase shift decaying as t^{-1/2}, and the
frequency of the oscillations is equal to the double maximal growth rate of the
MI, s=2g_{max}. In the asymptotic stationary state the ratio of potential to
kinetic energy is equal to -2. The asymptotic PDF of wave amplitudes is close
to Rayleigh distribution for cnoidal waves with strong overlapping, and is
significantly non-Rayleigh one for cnoidal waves with weak overlapping of
solitons. In the latter case the dynamics of the system reduces to two-soliton
collisions, which occur with exponentially small rate and provide up to
two-fold increase in amplitude compared with the original cnoidal wave.Comment: 36 pages, 25 figure
Chaotic behaviour of nonlinear waves and solitons of perturbed Korteweg - de Vries equation
This paper considers properties of nonlinear waves and solitons of
Korteweg-de Vries equation in the presence of external perturbation. For
time-periodic hamiltonian perturbation the width of the stochastic layer is
calculated. The conclusions about chaotic behaviour in long-period waves and
solitons are inferred. Obtained theoretical results find experimental
confirmation in experiments with the propagation of ion-acoustic waves in
plasma.Comment: 7 pages, LaTeX, 2 Postscript figures, submitted to Reports on
Mathematical Physic
Stability of a chain of phase oscillators
We study a chain of N + 1 phase oscillators with asymmetric but uniform coupling. This type of chain possesses 2 N ways to synchronize in so-called traveling wave states, i.e., states where the phases of the single oscillators are in relative equilibrium. We show that the number of unstable dimensions of a traveling wave equals the number of oscillators with relative phase close to π . This implies that only the relative equilibrium corresponding to approximate in-phase synchronization is locally stable. Despite the presence of a Lyapunov-type functional, periodic or chaotic phase slipping occurs. For chains of lengths 3 and 4 we locate the region in parameter space where rotations (corresponding to phase slipping) are present
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