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1.
We investigate numerically the impact of the self-frequency shift effect on isolated and bound states of some soliton-like
(plain and composite) solutions of the complex Ginzburg–Landau equation. In the absence of the self-frequency shift effect,
stable bound states are found for a phase difference of ±π/2 between the constituent plain, respectively, composite, pulses. In the presence of such effect, the corresponding stationary
points remain symmetrically located in the interaction plane, but the line joining them is rotated in the counterclockwise
direction. Moreover, we verify that one of these points remains a stable stationary point, whereas the other one turns out
to be unstable. It is shown that the bound states propagate with the same velocity as the single pulses. 相似文献
2.
We study a class of dissipative nonlinear PDE's forced by a random force ηomega(
t
,
x
), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions),
and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form
where the η
k
's are smooth bounded identically distributed random fields. The equation in question defines a Markov chain in an appropriately
chosen phase space (a subset of a function space) that contains the zero function and is invariant for the (random) flow of
the equation. Concerning this Markov chain, we prove the following main result (see Theorem 2.2): The Markov chain has a unique invariant measure. To prove this theorem, we present a construction assigning, to any invariant measure, a Gibbs measure for a 1D system with
compact phase space and apply a version of Ruelle–Perron–Frobenius uniqueness theorem to the corresponding Gibbs system. We
also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced PDE.
Received: 24 January 2000 / Accepted: 17 February 2000 相似文献
3.
Andrei Khrennikov 《Foundations of Physics Letters》2006,19(4):299-319
We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale. 相似文献
4.
The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR
4 and a large class of more complicated topologies and for self-dual metrics. We show that ifR
0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric. 相似文献
5.
S. T. Swift R. A. d'Inverno J. A. G. Vickers 《General Relativity and Gravitation》1986,18(11):1093-1103
We define an everywhere invariant space of metricsU to be one for which the set of diffeomorphisms, which leaveU invariant, contains all the isometries of the individual metrics inU. We also generalize Killing's equation to a new equation, the invariance equation, which has as solutions those vector fields which leaveU invariant. By combining these ideas we give a new method for finding the isometries of a given metric. 相似文献
6.
Ricardo Weder 《Communications in Mathematical Physics》2000,215(2):343-356
In this paper we study the following nonlinear Schr?dinger equation on the line,
where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x→±∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0, ∞). The solutions to this equation
have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution
at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all
small solutions approach a particular periodic orbit in the center manifold as t→±∞. In general, the periodic orbits are different for t→±∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with
initial data near a nonlinear bound state is asymptotic as t→±∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→±∞.
Received: 20 January 2000 / Accepted: 1 June 2000 相似文献
7.
Aurelio Patelli Stefano Ruffo 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2014,68(11):1-13
Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states. 相似文献
8.
An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation. 相似文献
9.
Dario Bambusi 《Communications in Mathematical Physics》1997,189(1):205-226
We consider small perturbations of the Zakharov–Shabat nonlinear Schr?dinger equation on [0,π] with vanishing or periodic
boundary conditions; we prove a Nekhoroshev type result for solutions starting in the neighbourhood (in the H
1 topology) of the majority of small amplitude finite dimensional invariant tori of the linearized system. More precisely we
will prove that along the considered solutions all the actions of the linearized system are approximatively constant up to
times growing exponentially with the inverse of a suitable small parameter.
Received: 10 December 1996 / Accepted: 17 March 1997 相似文献
10.
In this paper we consider a nonlocal evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean field approximation. In the presence of a small magnetic field, it admits two stationary and homogeneous solutions, representing the stable and metastable phases of the physical system. We prove the existence of an invariant, one dimensional manifold connecting the stable and metastable phases. This is the unstable manifold of a distinguished, spatially nonhomogeneous, stationary solution, called the critical droplet.(4, 10) We show that the points on the manifold are droplets longer or shorter than the critical one, and that their motion is very slow in agreement with the theory of metastable patterns. We also obtain a new proof of the existence of the critical droplet, which is supplied with a local uniqueness result. 相似文献
11.
We study the dynamics of geometric spin system on the torus with long-range interaction. As the number of particles goes to infinity, the process converges to a deterministic, dynamical magnetization field that satisfies an Euler equation (law of large numbers). Its stable steady states are related to the limits of the equilibrium measures (Gibbs states) of the finite particle system. A related equation holds for the magnetization densities, for which the property of propagation of chaos also is established. We prove a dynamical central limit theorem with an infinite-dimensional Ornstein-Uhlenbeck process as a limiting fluctuation process. At the critical temperature of a ferromagnetic phase transition, both a tighter quantity scaling and a time scaling is required to obtain convergence to a one-dimensional critical fluctuation process with constant magnetization fields, which has a non-Gaussian invariant distribution. Similarly, at the phase transition to an antiferromagnetic state with frequencyp
0, the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequencyp
0 and has a non-Gaussian invariant distribution on these fields. Finally, we compute the critical fluctuation process in the infinite particle limit at a triple point, where a ferromagnetic and an antiferromagnetic phase transition coincide.Work supported by Deutsche Forschungsgemeinschaft 相似文献
12.
Recently Auberson, Mahoux, Roy and Singh have proved a long standing conjecture of Roy and Singh: In 2N-dimensional phase space, a maximally realistic quantum mechanics can have quantum probabilities of no more than N+1 complete commuting cets (CCS) of observables coexisting as marginals of one positive phase space density. Here I formulate
a stationary principle which gives a nonperturbative definition of a maximally classical as well as maximally realistic phase
space density. I show that the maximally classical trajectories are in fact exactly classical in the simple examples of coherent
states and bound states of an oscillator and Gaussian free particle states. In contrast, it is known that the de Broglie-Bohm
realistic theory gives highly nonclassical trajectories. 相似文献
13.
K. Kundu B.C. Gupta 《The European Physical Journal B - Condensed Matter and Complex Systems》1998,3(1):23-33
We study the formation of stationary localized states using the discrete nonlinear Schr?dinger equation in a Cayley tree with
connectivity K. Two cases, namely, a dimeric power law nonlinear impurity and a fully nonlinear system are considered. We introduce a transformation
which reduces the Cayley tree into an one dimensional chain with a bond defect. The hopping matrix element between the impurity
sites is reduced by . The transformed system is also shown to yield tight binding Green's function of the Cayley tree. The dimeric ansatz is used
to find the reduced Hamiltonian of the system. Stationary localized states are found from the fixed point equations of the
Hamiltonian of the reduced dynamical system. We discuss the existence of different kinds of localized states. We have also
analyzed the formation of localized states in one dimensional system with a bond defect and nonlinearity which does not correspond
to a Cayley tree. Stability of the states is discussed and stability diagram is presented for few cases. In all cases the
total phase diagram for localized states have been presented.
Received: 18 September 1997 / Revised: 31 October and 17 november 1997 / Accepted: 19 November 1997 相似文献
14.
L. Gelens E. Knobloch 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2010,59(1):23-36
The supercritical complex Swift-Hohenberg equation models pattern formation
in lasers, optical parametric oscillators and photorefractive oscillators.
Simulations of this equation in one spatial dimension reveal that much of
the observed dynamics can be understood in terms of the properties of exact
solutions of phase-winding type. With real coefficients these states take
the form of time-independent spatial oscillations with a constant phase
difference between the real and imaginary parts of the order parameter and
may be unstable to a longwave instability. Depending on parameters the
evolution of this instability may or may not conserve phase. In the former
case the system undergoes slow coarsening described by a Cahn-Hilliard
equation; in the latter it undergoes repeated phase-slips leading either
to a stable phase-winding state or to a faceted state consisting of an array
of frozen defects connecting phase-winding states with equal and opposite
phase. The transitions between these regimes are studied and their location
in parameter space is determined. 相似文献
15.
József Fritz Carlangelo Liverani Stefano Olla 《Communications in Mathematical Physics》1997,189(2):481-496
The set of stationary measures of an infinite Hamiltonian system with noise is investigated. The model consists of particles
moving in with bounded velocities and subject to a noise that does not violate the classical laws of conservation, see [OVY]. Following
[LO] we assume that the noise has also a finite radius of interaction, and prove that translation invariant stationary states
of finite specific entropy are reversible with respect to the stochastic component of the evolution. Therefore the results
of [LO] imply that such invariant measures are superpositions of Gibbs states.
Received: 26 September 1996 / Accepted: 3 January 1997 相似文献
16.
Using the quantum kinematic approach of Mukunda and Simon, we propose a geometric phase in Bohmian mechanics. A reparametrization and gauge invariant geometric phase is derived along an arbitrary path in configuration space. The single valuedness of the wave function implies that the geometric phase along a path must be equal to an integer multiple of 2π. The nonzero geometric phase indicates that we go through the branch cut of the action function from one Riemann sheet to another when we locally travel along the path. For stationary states, quantum vortices exhibiting the quantized circulation integral can be regarded as a manifestation of the geometric phase. The bound-state Aharonov-Bohm effect demonstrates that the geometric phase along a closed path contains not only the circulation integral term but also an additional term associated with the magnetic flux. In addition, it is shown that the geometric phase proposed previously from the ensemble theory is not gauge invariant. 相似文献
17.
In the preceding paper under the same title we have formulated a theorem which describes the set of states (i.e., probability measures on phase space of an infinite system of particles inR
v) corresponding to stationary solutions of the BBGKY hierarchy. We have proved the following statement: ifG is a Gibbs measure (Gibbs random point field) corresponding to a stationary solution of the BBGKY hierarchy, then its generating function satisfies a differential equation which is conjugated to the BBGKY hierarchy. The present paper deals with the investigation of the conjugated equation for the generating function in particular cases. 相似文献
18.
G.L.S. Paula W. Figueiredo 《The European Physical Journal B - Condensed Matter and Complex Systems》1998,1(4):519-522
The stationary states of the random-field Ising model are determined through the master equation approach, where the contact
with the heat bath is simulated by the Glauber stochastic dynamics. The phase diagram of the model is constructed from the
stationary values of the magnetization as a function of temperature and field amplitude. The continuous phase transitions
coincide with the equilibrium ones, while the first-order transitions occur at fields larger than the corresponding values
at equilibrium. The difference between the fields at the limit of stability of the ordered phase and that of the equilibrium
is maximum at zero temperature and vanishes at the tricritical point. We also find the mean field time auto-correlation function
at the stationary states of the model.
Received: 4 June 1997 / Revised: 5 August 1997 / Accepted: 10 November 1997 相似文献
19.
A. Zaviyalov R. Iliew O. Egorov F. Lederer 《Applied physics. B, Lasers and optics》2011,104(3):513-521
We numerically investigate the formation of soliton pairs (bound states) in mode-locked fiber ring lasers in the normal dispersion domain. In the distributed mathematical model (complex cubic-quintic Ginzburg–Landau equation), we observe a discrete
family of soliton pairs with equidistantly increasing peak separation. We show that stabilization of previously unstable bound
states can be achieved when the finite relaxation time of the saturable absorber is taken into account. The domain of stability
can be controlled by varying this relaxation time. Furthermore, we investigate the parameter domain where the region of stable
bound states does not shrink to zero for vanishing absorber recovery time corresponding to a laser with an instantaneous saturable absorber. For a certain domain of the small-signal gain, we obtain a robust first level bound state with almost
constant separation where the phase of the two pulses evolves independently. Moreover, their phase difference can evolve either
periodically or chaotically depending on the small signal gain. Interestingly, higher level bound states exhibit a fundamentally
different dynamics. They represent oscillating solutions with a phase difference alternating between zero and π. 相似文献
20.
In a stationary case and for any potential, we solve the three-dimensional quantum Hamilton-Jacobi equation in terms of the solutions of the corresponding Schrödinger equation. Then, in the case of separated variables, by requiring that the conjugate momentum be invariant under any linear transformation of the solutions of the Schrödinger equation used in the reduced action, we clearly identify the integration constants successively in one, two and three dimensions. In each of these cases, we analytically establish that the quantum Hamilton-Jacobi equation describes microstates not detected by the Schröodinger equation in the real wave function case. 相似文献