Unforced Problem Research Articles

Particle paths beneath forced small amplitude periodic waves in a shallow channel with constant vorticity

Particle paths beneath small amplitude periodic forced waves in a shallow water channel are investigated. The problem is formulated in the forced Korteweg-de Vries equation framework which allows to approximate the velocity field in the bulk fluid. We show that the flow can have zero, one or three stagnation points. Besides, differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast.

Comment on "Slow passage through resonance".

In a recent numerical and analytical study, Park etal. [Phys. Rev. E 84, 056604 (2011)PLEEE81539-375510.1103/PhysRevE.84.056604] presented the statement that a linearly damped harmonic oscillator subject to a linear frequency chirp ω_{f}(t)=ω_{0}+εt experiences "an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value ω_{0} and the natural frequency ω_{n} for resonance in the unforced problem," i.e., the resonance occurs when the instantaneous frequency ω_{inst} approaches (ω_{0}+ω_{n})/2. This statement is not valid because the instantaneous frequency of the forcing function actually grows twice as fast as stated in the paper, and the resonance actually occurs for instantaneous forcing frequency ω_{inst} approximately equal to the natural frequency ω_{n}. We also highlight the difference between the critical frequency ramp rate for resonance amplitude measurement and the critical frequency ramp rate for resonance frequency and damping measurements.

Vertically forced stably stratified cavity flow: instabilities of the basic state

The linear stability of a stably stratified fluid-filled cavity subject to vertical oscillations is determined via Floquet analysis. Retaining the viscous and diffusion terms in the Navier–Stokes–Boussinesq equations, with no-slip velocity boundary conditions, no-flux temperature conditions on the sidewalls and constant temperatures on the top and bottom walls, we find that the instabilities are primarily subharmonic (as is typical in many parametrically forced systems), except for in a few low-forcing-frequency ranges where the instabilities are synchronous. When the viscosity is small, the Floquet modes resemble the inviscid eigenmodes of the unforced problem, except in boundary layers. We establish scaling laws quantifying how viscosity regularizes the degeneracy associated with the inviscid idealization, and how it scales the thickness and intensity of the boundary layers. The product of boundary layer thickness and intensity remains constant with decreasing viscosity, leading to a delta distribution of vorticity on the walls in the limit of zero viscosity. This is in contrast to the zero wall vorticity in the inviscid case.

Nonlinear Dirac equation solitary waves in external fields

We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalar-scalar self-interaction g2/κ+1(Ψ[over ¯]Ψ)κ+1 in the presence of various external electromagnetic fields. We find exact solutions for special external fields and we study the behavior of solitary-wave solutions to the NLDE in the presence of a wide variety of fields in a variational approximation depending on collective coordinates which allows the position, width, and phase of these waves to vary in time. We find that in this approximation the position q(t) of the center of the solitary wave obeys the usual behavior of a relativistic point particle in an external field. For time-independent external fields, we find that the energy of the solitary wave is conserved but not the momentum, which becomes a function of time. We postulate that, similarly to the nonlinear Schrödinger equation (NLSE), a sufficient dynamical condition for instability to arise is that dP(t)/dq[over ̇](t)<0. Here P(t) is the momentum of the solitary wave, and q[over ̇] is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials that we always have dP(t)/dq[over ̇](t)>0, so, when instabilities do occur, they are due to a different source. We investigate the accuracy of our variational approximation using numerical simulations of the NLDE and find that, when the forcing term is small and we are in a regime where the solitary wave is stable, that the behavior of the solutions of the collective coordinate equations agrees very well with the numerical simulations. We found that the time evolution of the collective coordinates of the solitary wave in our numerical simulations, namely the position of the average charge density and the momentum of the solitary wave, provide good indicators for when the solitary wave first becomes unstable. When these variables stop being smooth functions of time (t), then the solitary wave starts to distort in shape.

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.

Slow passage through resonance

The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A~ε(-1/2), ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

Hydroelastic solitary waves in deep water

The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves.

Resonant triad dynamics in weakly damped Faraday waves with two-frequency forcing

Many of the interesting patterns seen in recent multi-frequency Faraday experiments can be understood on the basis of three-wave interactions (resonant triads). In this paper we consider two-frequency forcing and focus on a resonant triad occurring near the bicritical point where two pattern-forming modes with distinct wavenumbers onset simultaneously. This triad has been observed directly (in the form of rhomboids) and has also been implicated in the formation of quasipatterns and superlattices. We show how the symmetries of the undamped unforced problem (time translation, time reversal, and Hamiltonian structure) can be used, when the damping is weak, to obtain general scaling laws and additional qualitative properties of the normal form coefficients governing the pattern selection process near onset; such features help to explain why this particular triad is seen only for certain “low” forcing ratios, and predict the existence of drifting solutions and heteroclinic cycles. We confirm the anticipated parameter dependence of the coefficients and investigate its dynamical consequences using coefficients derived numerically from a quasipotential formulation of the Faraday problem due to Zhang and Viñals.

Wave interaction with two-dimensional bodies floating in a two-layer fluid: uniqueness and trapped modes

Existing results on the linearized water-wave problem for a homogeneous fluid are extended to the case of a two-layer fluid. In particular, the appropriate form of Maz'ya's identity is presented and used to obtain results on the uniqueness of the solutions to forcing problems for a structure in a two-layer fluid. Further, examples of geometries are constructed for which trapped modes occur; such modes are finite-energy solutions of the unforced problem and provide examples of non-uniqueness.

The ‘Indian rope trick’ for a parametrically excited flexible rod: nonlinear and subharmonic analysis

Recently, Mullin has demonstrated experimentally that an upright column that is longer than its critical length for self–weight buckling can be stabilized by subjecting it to vertical harmonic excitation. This paper extends an earlier linearized analysis of a rod–mechanics model of this set–up to include three dimensionality and geometric nonlinearity. The stability of the upright state is then analysed using weakly nonlinear asymptotic expansions in the limit of small–amplitude excitation. First, the unforced problem is treated, extending the classical result by Greenhill to show that all bifurcations are supercritical. The main results are for the forced problem near the simplest among the potential infinity of dynamic instabilities. These correspond to pure bending modes, and to resonances between a vibration mode of the column and the first harmonic or subharmonic of the drive. The result is an asymptotic description of these instabilities including information on the stability of dynamically bifurcating states in terms of the three dimensionless parameters of the problem (bending stiffness and the amplitude and frequency of excitation). A qualitative explanation is offered of why the earlier linearized analysis fails to quantitatively match the experiments.

Dispersive effects in Rossby-wave hydraulics

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson &amp; Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

On the Global Dynamics of a Controlled Viscous Burgers' Equation

In this paper we consider a boundary control problem for a forced Burgers' equation in a Hilbert state space consisting of square integrable functions on a finite interval. Our first main result consists in proving the global in time existence of solutions of the closed loop boundary control system for arbitrary L2 initial data and quite general forcing terms (disturbances). For the unforced problem, our main interest is in stability of equilibria, while for the forced problem we are interested in steady state behavior of solutions. We note that the uncontrolled (open loop), unforced problem is not asymptotically stable on L2. However, for positive gains we show that the unforced closed loop system is globally Lyapunov stable and locally exponentially stable. In addition, for nonzero stationary forcing, we show that there is a local absorbing ball and that the corresponding nonlinear semigroup is compact for all t > 0. Using this fact, we are able to deduce several consequences regarding the existence and properties of local attractors. Our method of proof is based on a systematic investigation of the smoothing properties of the controlled dynamics. In particular we establish various regularity results for the dynamics of the closed loop system.

On the Interval of Existence for Nonlinear Two-Point Boundary Value Problems

The existence question for a nonlinear, two point boundary value problem (continuous and Lipschitzian) can be reduced to that of the corresponding unforced problem with one zero boundary condition. For fixed left-hand end point, the set of right-hand end points for which existence fails consists of isolated points and isolated intervals.Certain bounding curves arise in the case of failure of existence for an interval of right-hand end points, and smoothness and dependence on initial conditions of these curves are determined in part. Also some relationships between the given problem and the problem with initial and terminal points reversed are established.