Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

2017 ◽  
Vol 29 (12) ◽  
pp. 127104 ◽  
Author(s):  
Zeng Liu ◽  
Dali Xu ◽  
Shijun Liao
Keyword(s):  
Steady State ◽  
Energy Flux ◽  
Wave Groups ◽  
State Wave ◽  
Linear And Nonlinear ◽  
Flux Conservation

Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

2019 ◽  
Vol 867 ◽  
pp. 348-373 ◽  
Author(s):  
Z. Liu ◽  
D. Xie
Keyword(s):  
Steady State ◽  
Water Depth ◽  
Deep Water ◽  
High Frequency ◽  
Finite Amplitude ◽  
Wave Groups ◽  
State Wave ◽  
Resonant Interactions ◽  
Frequency Components ◽  
Finite Water Depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.

Finite amplitude steady-state wave groups with multiple near resonances in deep water

2017 ◽  
Vol 835 ◽  
pp. 624-653 ◽  
Author(s):  
Z. Liu ◽  
D. L. Xu ◽  
S. J. Liao
Keyword(s):  
Steady State ◽  
Linear Operator ◽  
Deep Water ◽  
Wave Equations ◽  
Finite Amplitude ◽  
Wave Groups ◽  
Homogeneous Solutions ◽  
State Wave ◽  
Resonant Interactions ◽  
Auxiliary Linear Operator

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

On the near resonances of collinear steady-state wave groups in finite water depth

2019 ◽  
Vol 182 ◽  
pp. 584-593 ◽  
Author(s):  
Zeng Liu ◽  
Jianglong Sun ◽  
De Xie ◽  
Zhiliang Lin
Keyword(s):  
Steady State ◽  
Water Depth ◽  
Wave Groups ◽  
State Wave ◽  
Finite Water Depth

A PREY-PREDATOR MODEL WITH DIFFUSION AND A SUPPLEMENTARY RESOURCE FOR THE PREY IN A TWO‐PATCH ENVIRONMENT

2005 ◽  
Vol 9 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J. Dhar
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Prey Species ◽  
Steady State Solution ◽  
State Solution ◽  
Prey Population ◽  
Predator Species ◽  
Additional Resource ◽  
Linear And Nonlinear ◽  
Predator Model

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases. Šiame straipsnyje modeliuojama ir analizuojama plešr‐unu ir auku dinamika, laikant, kad plešr-unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas. Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

Linear and nonlinear components of human electroretinogram

1984 ◽  
Vol 51 (5) ◽  
pp. 952-967 ◽  
Author(s):  
C. L. Baker ◽  
R. F. Hess
Keyword(s):  
Steady State ◽  
Spatial Frequency ◽  
Step Response ◽  
Linear Component ◽  
Fast Wave ◽  
Uniform Field ◽  
Nonlinear Component ◽  
Symmetric Component ◽  
Linear And Nonlinear ◽  
Nonlinear Components

We compared electroretinographic (ERG) responses to uniform-field and a variety of pattern stimuli using both transient and steady-state analyses. Evidence is provided that for all of these stimuli, the peak at high temporal frequencies in the steady-state response corresponds to the fast wave of the transient response and that the peak at low temporal frequencies corresponds to the slow wave of the step response. A variety of contrast-modulated grating stimuli were used to demonstrate that the fast, high-frequency response can be regarded as the sum of two components, an "odd-symmetric" component, which behaves linearly and is independent of spatial frequency, and an "even-symmetric" component, which behaves nonlinearly and has a band-pass spatial-frequency dependence. The prevailing distinction that is made between pattern and uniform-field ERGs is a consequence of the fact that the uniform-field ERG is dominated by the odd-symmetric (linear) component, whereas the so-called pattern (contrast reversal) ERG reveals the even-symmetric (nonlinear) component in isolation. Since a uniform field can also drive the nonlinear component, the present dichotomy ("luminance" versus "pattern") can be better understood in terms of the linear and nonlinear components of the response rather than in terms of the stimuli that produce them.

Balance-Equation Approach to Hot-Carrier Transport in Semiconductors

1992 ◽  
Vol 06 (07) ◽  
pp. 805-936 ◽  
Author(s):  
X.L. Lei ◽  
N.J.M. Horing
Keyword(s):  
Steady State ◽  
Balance Equation ◽  
Carrier Transport ◽  
Thermal Noise ◽  
Transport Theory ◽  
Equation Approach ◽  
Balance Equations ◽  
Hot Carrier ◽  
Linear And Nonlinear ◽  
And Diffusion

The balance-equation approach to nonlinear hot-carrier transport theory, formulated by Lei and Ting (1984), is addressed in this comprehensive review. A central feature is the role of strong electron-electron interactions in promoting rapid thermalization about the drifted transport state and the concomitant substantial simplification of the transport theory. This physical feature is embodied in the initial density matrix chosen to represent the unperturbed carrier system. Force and energy balance equations are formulated for the dc steady state, ac dynamic and transient cases of charge conduction, including memory effects. The scattering mechanisms include impurity and phonon interactions along with dynamic nonlocal screening effects due to carrier-carrier interactions. Both linear and nonlinear resistivities are discussed in the degenerate and nondegenerate statistical regimes. Interesting phenomena such as electron cooling and thermal noise and diffusion are discussed as well. Semiconductor microstructure transport is described for both linear and nonlinear hot carrier conduction. In this connection, quasi-2D-systems, heterojunctions, and quantum well superlattices are treated with attention to steady state, transient and high frequency transport, including, for example, superlattice plasmon resonance structure. Type-II superlattice transport is reviewed as well as type-I, and electron-hole drag is treated in conjunction with negative minority electron mobility in a quantum well. Multivalley semiconductors are discussed in some detail. Furthermore, attention is also focused on the center-of-mass velocity fluctuation, Langevin-type equation and thermal noise and diffusion for microstructures and multivalley systems. A number of particularly important phenomena are examined from the balance-equation point of view, such as nonequilibrium phonons, higher order scattering effects and weak localization, hydrodynamic equations for weakly nonuniform systems, and the intracollisional field effect. Alternative formulations and interpretations of the balance-equation approach are reviewed. The distinction between this many-particle, isothermal, balance-equation theory and the noninteracting (single-particle) adiabatic transport theory is discussed to clarify issues subject to controversy in the literature. Finally, we give a brief review of recent developments in the balance-equation approach: investigation of the distribution function in balance-equation theory, improved calculations for GaAs/AlGaAs heterojunctions, extension of the balance equations to an abitrary energy band and recent work on superlattice miniband transport.

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