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Zhou Gang

Publications and source records attributed to Zhou Gang.

31 records · Page 2Linked to original sources
arXiv2013

Ballistic Motion of a Tracer Particle Coupled to a Bose gas

We study the motion of a heavy tracer particle weakly coupled to a dense interacting Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We derive the effective dynamics of the tracer particle, which is described by a non-linear integro-differential equation with memory, and prove that if the initial speed of the tracer particle is below the speed of sound in the Bose gas the motion of the particle approaches an inertial motion at constant velocity at large times.

math-ph↗
arXiv2012

Hamiltonian dynamics of a particle interacting with a wave field

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.

math-ph↗
arXiv2012

On the theory of slowing down gracefully

We discuss the transport of a tracer particle through the Bose Einstein condensate of a Bose gas. The particle interacts with the atoms in the Bose gas through two-body interactions. In the limiting regime where the particle is very heavy and the Bose gas is very dense, but very weakly interacting ("mean-field limit"), the dynamics of this system corresponds to classical Hamiltonian dynamics. We show that, in this limit, the particle is decelerated by emission of gapless modes into the condensate (Cerenkov radiation). For an ideal gas, the particle eventually comes to rest. In an interacting Bose gas, the particle is decelerated until its speed equals the propagation speed of the Goldstone modes of the condensate. This is a model of "Hamiltonian friction". It is also of interest in connection with the phenomenon of "decoherence" in quantum mechanics. It is based on work we have carried out in collaboration with D Egli, IM Sigal and A Soffer.

math-ph↗
arXiv2011

Friction in a Model of Hamiltonian Dynamics

We study the motion of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle's trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest.

math-ph↗
arXiv2010

Some Hamiltonian Models of Friction

Mathematical results on some models describing the motion of a tracer particle through a Bose-Einstein condensate are described. In the limit of a very dense, very weakly interacting Bose gas and for a very large particle mass, the dynamics of the coupled system is determined by classical non-linear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction (with memory) caused by the emission of Cerenkov radiation of gapless modes into the gas. Precise results are stated and outlines of proofs are presented. Some technical details are deferred to forthcoming papers.

math.AP↗
arXiv2010

Equipartition of Mass in Nonlinear Schrödinger / Gross-Pitaevskii Equations

We study the infinite time dynamics of a class of nonlinear Schrödinger / Gross-Pitaevskii equations. In our previous paper, we prove the asymptotic stability of the nonlinear ground state in a general situation which admits degenerate neutral modes of arbitrary finite multiplicity, a typical situation in systems with symmetry. Neutral modes correspond to purely imaginary (neutrally stable) point spectrum of the linearization of the Hamiltonian PDE about a critical point. In particular, a small perturbation of the nonlinear ground state, which typically excites such neutral modes and radiation, will evolve toward an asymptotic nonlinear ground state soliton plus decaying neutral modes plus decaying radiation. In the present article, we give a much more detailed, in fact quantitative, picture of the asymptotic evolution. Specificially we prove an equipartition law: The asymptotic soliton which emerges has a mass which is equal to the initial soliton mass plus one half the mass contained in the initially perturbing neutral modes.

math.AP↗
arXiv2008

Dynamics of Nonlinear Schrodinger / Gross-Pitaevskii Equations; Mass Transfer in Systems with Solitons and Degenerate Neutral Modes

Nonlinear Schrodinger / Gross-Pitaevskii equations play a central role in the understanding of nonlinear optical and macroscopic quantum systems. The large time dynamics of such systems is governed by interactions of the nonlinear ground state manifold, discrete neutral modes (``excited states'') and dispersive radiation. Systems with symmetry, in spatial dimensions larger than one, typically have degenerate neutral modes. Thus, we study the large time dynamics of systems with degenerate neutral modes. This requires a new normal form (nonlinear matrix Fermi Golden Rule) governing the system's large time asymptotic relaxation to the ground state (soliton) manifold.

math-ph↗
arXiv2007

Neck Pinching Dynamics Under Mean Curvature Flow

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.

math.AP↗
arXiv2006

Blow-up in Nonlinear Heat Equations

We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions. The approach is analogous to one used in bifurcation theory and our techniques can be regarded as a time-dependent version of the Lyapunov-Schmidt decomposition.

math.AP↗
arXiv2006

Relaxation of Solitons in Nonlinear Schrodinger Equations with Potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to $\mathcal{L}^2-$unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.

math-ph↗
arXiv2006

On Soliton Dynamics in Nonlinear Schrodinger Equations

In this paper we announce the result of asymptotic dynamics of solitons of nonlinear Schrodinger equations with external potentials. To each local minima of the potential there is a soliton centered around it. Under some conditions on the nonlinearity, the potential and the datum, we prove that the solution can be decomposed into two parts: the soliton and the term dissipating to infinity.

math-ph↗