Search

This paper is concerned with the robust stability and stabilization problems of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations. A proper sufficient condition via the fixed-point principle is proposed to guarantee that the given system is in a standard form. Then, based on the singular perturbation approach, a linear matrix inequality (LMI) based sufficient condition is presented such that the original system is standard and input-to-state stable (ISS) simultaneously. Thus, it can be easily verified for it only depends on the solution of an LMI. After that, for the case where the nominal system is unstable, the problem of designing a control law to make the resulting closed-loop system ISS is addressed. To achieve this, a sufficient condition is proposed via LMI techniques for the purpose of implementation. The criteria presented in this paper are independent of the small parameter and the stability bound can be derived effectively by solving an optimal problem. Finally, the effectiveness of the obtained theoretical results is illustrated by two numerical examples.

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting., Comment: Comments are welcomed

The electroacoustic waves, particularly ion-acoustic waves (IAWs), and their expansion in the medium of a magnetized collision-free plasma system has been investigated theoretically. The plasma system is assumed to be composed of both positively and negatively charged mobile ion species and kappa-distributed hot electron species. In the nonlinear perturbation regime, the magnetized Korteweg–de Vries (KdV) and magnetized modified KdV (mKdV) equations are derived by using reductive perturbation method. The prime features (i.e., amplitude, phase speed, width, etc.) of the IAWs are studied precisely by analyzing the stationary solitary wave solutions of the magnetized KdV and magnetized mKdV equations, respectively. It occurs that the basic properties of the IAWs are significantly modified in the presence of the excess superthermal hot electrons, obliqueness, the plasma particle number densities, etc. It is also observed that, in case of magnetized KdV solitary waves, both compressive and rarefactive structures are formed, whereas only compressive structures are found for the magnetized mKdV solitary waves. The implication of our results in some space and laboratory plasma situations is concisely discussed.

In this paper, we study the following nonlinear Schrodinger equation \begin{document}$ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $\end{document} where \begin{document}$ V* $\end{document} is the Fourier multiplier defined by \begin{document}$ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $\end{document} and \begin{document}$ f(x) $\end{document} is Gevrey smooth. It is shown that for \begin{document}$ 0\leq|\epsilon|\ll1 $\end{document} , there is some \begin{document}$ (V_n)_{n\in\mathbb{Z}} $\end{document} such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [ 7 ] and Cong-Liu-Shi-Yuan [ 8 ] to the case that the nonlinear perturbation depends explicitly on the space variable \begin{document}$ x $\end{document} . The main difficulty here is the absence of zero momentum of the equation.

We consider the mimetic tachyon model in the Lagrange multiplier approach. We study both the linear and non-linear perturbations and find the perturbation and non-gaussianity parameters in this setup. By adopting two types of the scale factor as the power-law ($a=a_{0}\,t^{n}$) and intermediate ($a=a_{0}\exp(bt^{\beta})$) scale factors, we perform a numerical analysis on the model which is based on Planck2018 TT, TE, EE+lowE+lensing +BAO +BK14 and Planck2018 TTT, EEE, TTE and EET data sets. We show that the mimetic tachyon model with both the power-law and intermediate scale factors, in some ranges of its parameter space is instabilities-free and observationally viable. The power-law mimetic tachyon model with $26.3, Comment: 14 pages, 15 figures, 6 tables

We study linear perturbations of the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model in the Regge-Wheeler formalism which is a standard framework to study perturbations of spherically-symmetric black holes. In particular, we show that the general solution of linear perturbation equations can be given in terms of two copies of a master scalar satisfying scalar wave equation on the FLRW background (with a Regge-Wheeler/Zerilli type potential) thus representing two gravitational degrees of freedom, and one scalar satisfying a transport type equation representing (conformal) matter perturbation. We expect the Regge-Wheeler formalism to be easily extended to include nonlinear perturbations, akin to to the recent work [Phys. Rev. D 96, 124026 (2017)]., Comment: 5 pages, no figures

We review the theoretical efforts to develop models to analyze fiber-optic coherent systems using perturbation analysis. We start with models for the nonlinear signal-signal distortions and continue to address nonlinear signal-noise interactions and SOA-induced distortions.

We study nonlinear gravitational perturbations of vacuum Einstein equations, with $��2$, generalizing previous studies for $n=2$. We follow the formalism by Ishibashi, Kodama and Seto to decompose the metric perturbations into tensor, vector and scalar sectors, and simplify the Einstein equations. The tensor perturbations are the new feature of higher dimensions. We render the metric perturbations asymptotically anti-de Sitter by employing a suitable gauge choice for each of the sectors. Finally, we analyze the resonant structure of the perturbed equations at second order for the five dimensional case, by a partial study of single mode tensor-type perturbations at the linear level. For the cases we studied, resonant terms vanish at second order., 43 pages, discussion expanded, numerical factor missed in a Mathematica file related to previous version corrected, thanks to referees for their comments that led to correction. Resonant terms now vanish at second order for cases studied

We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation x'=A(t)x,t≠τi,∆x|t=τi=Bix(τi),i ∈ Z in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general dichotomy on R called the nonuniform (h,k,µ,ν)-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlinear perturbations, we establish a new version of the GrobmanHartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive differential equations x'=A(t)x+f(t,x), t≠τi,∆x|t=τi=Bix(τi) + gi(x(τi)),i ∈ Z with the help of nonuniform (h,k,µ,ν)-dichotomies.

Due to the strong nonlinear perturbations near the libration points, the continuous low-thrust technique has many potential applications in stationkeeping relative motions. The Hamiltonian structure-preserving (HSP) control is employed in this paper to stabilize formation flying on quasi-periodic orbits near L L 1 of the restricted Sun–Earth–Moon–spacecraft system. In the bi-circular model (BCM), a multiple shooting corrector is developed to refine quasi-periodic orbits as the chief spacecraft's reference trajectories. The linearized variation equation in BCM is used to design the stationkeeping control. A HSP controller is constructed to change the topology of the equilibrium from hyperbolic to elliptic using only relative position feedbacks consisting of stable, unstable and center manifolds. The critical control gains for transient and long-term stabilities are presented to guide the selection of control gains.

We develop a nonlinear perturbation theory of Reissner-Nordstr\"om black holes. We show that, at each perturbation level, Einstein-Maxwell equations can be reduced to four inhomogeneous wave equations, two for polar and two for axial sector. Gravitational part of these equations is similar to Regge-Wheeler and Zerilli equations with source and additional coupling to the electromagnetic sector. We construct solutions to the inhomogeneous part of wave equations in terms of sources for Einstein-Maxwell equations. We discuss $\ell=0$ and $ \ell=1$ cases separately., Comment: 9 pages

In the present paper, discussion of perturbations against a Q-ball solution is continued. It is shown that in order to correctly describe perturbations containing nonoscillation modes, it is also necessary to consider nonlinear equations of motion for the perturbations, like in the case of oscillation modes only. It is also shown that the additivity of the charge and the energy of different modes holds for the most general nonlinear perturbation consisting of oscillation and nonoscillation modes., 15 pages. v2: typos corrected, minor changes in the text

The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, $T_{eq}$, follows a universal law; i.e., $T_{eq}\sim \epsilon^{-2}$, where the perturbation strength, $\epsilon$, characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where $2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3$, $0, Comment: 21 pages

This paper employs the modified Kudryashov's approach and addendum to Kudryashov's scheme for extracting optical soliton solutions for highly dispersive nonlinear perturbation Schrodinger equation having Kudryashov's arbitrary form with sextic-power law refractive index and generalized non-local laws. Bright, dark, singular solitons and combo bright-singular solitons are recovered. The modified Kudryashov's scheme gives dark, singular solitons and combo bright-singular solitons, while the addendum to Kudryashov's approach secures bright and singular solitons.

We consider a system described by the linear heat equation with adiabatic boundary conditions. We impose a nonlinear perturbation determined by a family of interval maps characterized by a certain set of parameters. The time instants of the perturbation are determined by an additional dynamical system, seen here as part of the external interacting system. We analyse the complex behaviour of the system, through the scope of symbolic dynamics, and the dependence of the behaviour on the time pattern of the perturbation, comparing it with previous results in the periodic case.

In this paper we consider the hyperbolic formulation of the constraints introduced by R\'acz. Using the numerical framework recently developed by us we construct initial data sets which can be interpreted as nonlinear perturbations of Schwarzschild data in Kerr-Schild coordinates and investigate their asymptotics. Our results suggest that, unless one finds a way to exploit the freedom to pick the free part of the initial data in some suitable way, generic initial data sets obtained by this method may violate fundamental asymptotic conditions., Comment: 26 pages, 10 figures

In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter)., Comment: 20 pages, v5: several minor typos corrected

Наведено математичну модель формування каверни при дії збурень різної природи, в основі якої лежать теорії потенціальних течій з вільними границями і малих збурень. Виконано аналіз результатів конкретних кавітаційних течій за конусами. Розглянуто метод розрахунку кавітаційних течій за конусами з урахуванням поля гравітації. Використано метод малих збурень вісесиметричної каверни. Диференціальні рівняння задачі лінеаризуються в околі незбуреної поверхні каверни. Побудовано математичну модель задачі у вигляді нескінченної системи нелінійних диференціальних рівнянь другого порядку. Задачу Коші для цієї системи розв’язано за допомогою пакету прикладної математики Matlab. На конкретних течіях показано процес деформування форми каверни під впливом поля гравітації

In this work, the propagation of higher-dimensional nonlinear Rossby waves under the generalized beta effect is considered. Using the methods of weak nonlinear perturbation expansions and the multiple scales, we obtain a new (2 + 1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation for the first time. Furthermore, a new dispersion relation for the linear Rossby waves is given corresponding to the linearized Boussinesq equation. More importantly, based on the methods of the traveling wave setting and the Jacobi elliptic function expansions, several kinds of exact traveling wave solutions for the higher-dimensional nonlinear Rossby waves, including the periodic solutions, solitary solutions and others are obtained. Finally, we simulate the solitary solutions obtained by using the method of the Jacobi elliptic function. The numerical results show that the amplitude of the Rossby solitary waves is decreasing with the increase of generalized beta effect.

We provide a concise review on multi-field inflation and cosmological perturbations. We discuss convenient and physically meaningful bases in terms of which perturbations can be systematically studied. We give formal accounts on the gauge fixing conditions and present the perturbation action in two gauges. We also briefly review non-linear perturbations., (v1) 59 pages, 8 figures, invited review for International Journal of Modern Physics D; (v2) 60 pages, typos corrected and references added

For the nonlinear kinetic equation describing the one-dimensional motion of a quasineutral collisionless plasma, perturbation velocities are determined and conditions of generalized hyperbolicity are formulated. Exact (in particular, periodical) solutions of the model are constructed and interpreted physically for the class of traveling waves. Differential conservation laws approximating the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation, which show the possibility of turnover of the kinetic distribution function.

We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.

In this Letter, pinning synchronization of a directed network with Markovian jump (DNMJ) and nonlinear perturbations is considered. By analyzing the structure of the network, a detailed pinning scheme is given to ensure the synchronization of all nodes in a DNMJ. This pinning scheme can overcome those difficulties on deciding which nodes needs to be pinned. This scheme can also identify the exact least number of pinned nodes for a DNMJ model. In addition, the time-varying polytopic directed network with Markovian jump is discussed. Finally, examples are provided to illustrate the effectiveness of the gained criteria.

We study the influence of additive and parametric slowly varying harmonic (at the Chandler frequency and doubled frequency) and stochastic Gaussian broadband perturbations on mathematical expectations, variances, and covariations of oscillations of the Earth’s pole. The influence of perturbations on both regular and irregular stochastic oscillations is considered in detail. Results of numerical experiments are presented. The developed models and software are included into information resources on the fundamental problem “Statistical dynamics of the Earth’s rotation” of Russian Academy of Sciences.

We consider a nonlinear Dirichlet problem driven by the $p$--Laplacian differential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is $p$--linear and resonant with respect to $\lambda_1>0$ (the principal eigenvalue of $(-\Delta_p,W^{1,p}_0(Z))$) at infinity and the other when the perturbation is $p$--superlinear at infinity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper--lower solutions and with suitable truncation techniques.

For nonautonomous linear delay equations $v'=L(t)v_t$ admitting a nonuniform exponential contraction, we establish the nonuniform exponential stability of the equation $v'=L(t) v_t +f(t,v_t)$ for a large class of nonlinear perturbations.

A coupled system of dynamic hyperbolic equations in electroma- gnetic-elasticity theory in the exterior of an open bounded obstacle $\mathcal{O}$ in 3-D is considered. In the presence of dissipative effects we obtain uniform decay rates of the solution as $t \rightarrow +\infty$. We do not require geometric assumptions on the obstacle or extra assumptions on the initial data. We apply our results to study the above system with a nonlinear perturbation, showing that the solutions hold the same rate of decay provided the initial data is "small" in a suitable sense. Previous results of this type have recently been obtained for the scalar wave equation by M. Nakao [18, 19] and R. Ikehata [8].

Recent investigations have revealed powerful selection rules for resonant energy transfer between modes of non-linear perturbations in global anti-de Sitter (AdS) space-time. It is likely that these selection rules are due to the highly symmetric nature of the underlying AdS background, though the precise relation has remained unclear. In this article, we demonstrate that the equation satisfied by the scalar field mode functions in AdS(d+1) has a hidden SU(d) symmetry, and explicitly specify the multiplets of this SU(d) symmetry furnished by the mode functions. We also comment on the role this structure might play in explaining the selection rules., 12 pages; v2: minor improvements, published version

We employ logarithmic perturbation theory to analyze selected nonlinear perturbations that are remarkable on several grounds. Our analysis offers a direct and general route to arrive at such pathological perturbing potentials for any state. Some new examples are explicitly constructed. A few issues of convergence are clarified.

The breathers in the cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the solitonlike wave, the periodic, quasiperiodic and chaotic breathers can be observed with the increase of cubic nonlinear perturbation. Finally, we discuss the breathers in the cubic-quintic nonlinear Schrodinger equation with the increase of quintic nonlinear perturbation.

The optimal velocity model of traffic is extended to take the relative velocity into account. The traffic behavior is investigated numerically and analytically with this model. It is shown that the car interaction with the relative velocity can effect the stability of the traffic flow and raise critical density. The jamming transition between the freely moving and jamming phases is investigated with the linear stability analysis and nonlinear perturbation methods. The traffic jam is described by the kink solution of the modified Korteweg–de Vries equation. The theoretical result is in good agreement with the simulation.

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained.

Using the nonlinear perturbation method, we study a crossover between quantum and classical regimes for the escape rate. We present a general formula for determining whether the escape rate changes smoothly around the crossover temperature or not. Applying it to tetragonal and hexagonal symmetries, it is found that the crossover is mostly of first order.

Stationary solutions for the $1+1$ cubic nonlinear Schr\"odinger equation (NLS) modeling attractive Bose-Einstein condensates (BECs) in a small potential are obtained via a form of nonlinear perturbation. The focus here is on perturbations to the bright soliton solutions due to small potentials which either confine or repel the BECs: under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of the bright solitons. Importantly, we do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This is useful, in that it allows us to consider perturbations of bright solitons of arbitrary size. In some cases, exact solutions can be recovered, and these agree with known results from the literature. Several special cases are considered which involve confining potentials of specific relevance to BECs. We make several observations on the influence of the small potentials on the behavior of the perturbed bright solitons. The results demonstrate the difference between perturbed bright solitons in the attractive NLS and those results found in the repulsive NLS for dark solitons, as discussed by Mallory and Van Gorder, [Phys. Rev. E 88, 013205 (2013)]. Extension of these results to more spatial dimensions is mentioned.

The asymptotic solution of Ford and Llewellyn Smith [J. Fluid Mech. 386, 305 (1999)] which describes the dynamics of a Kelvin vortex disturbed by a single large wavelength acoustic excitation, is shown to be unstable to three-dimensional short-wave nonlinear perturbations. This result might have interesting physical applications such as the destabilization of vortices in geophysics or engineering.

Nonlinear growth of an unstable density interface from single-mode initial perturbations at high Mach number was studied using the Nova laser. A variety of initially nonlinear perturbations were used, having amplitude-to-wavelength ratios of a0/λ = 0.11, 0.22, and 0.43. The measured mix width data from these experiments were nondimensionalized and exhibit self-similar growth.

The car-following model of traffic is extended to take into account the car interaction before the next car ahead (the next-nearest-neighbor interaction). The traffic behavior of the extended car-following model is investigated numerically and analytically. It is shown that next-nearest-neighbor interaction stabilizes the traffic flow. The jamming transition between the freely moving and jammed phases occurs at a higher density than the threshold of the original car-following model. The traffic current is enhanced without jam by the stabilization effect. The jamming transition is analyzed with the use of the linear stability and nonlinear perturbation methods. The traffic jim is described by the kink solution of the modified Korteweg-de Vries (MKdV) equation. The theoretical coexisting curve is in good agreement with the simulation result.

Nonlinear perturbations of gaseous disks with Jeans instability are investigated and it is shown that interactions of exponentially growing perturbations in certain cases result in the establishment of nonlinear spiral structures. The problem of the spiral structure of galaxies is discussed from the standpoint of nonlinear wave theory.

The nonlinear interaction between four electrostatic waves in a plasma media is investigated analytically in the presence of linear damping or growth and also frequency mismatch, using a nonlinear perturbation method. Depending on the various initial conditions of the wave amplitudes, solutions of different types are discussed.

Nonlinear perturbations of Friedmann-Lemaitre cosmologies with dust and a positive cosmological constant have recently attracted considerable attention. In this paper our first goal is to compare the evolution of the first and second order perturbations by determining their asymptotic behaviour at late times in ever-expanding models. We show that in the presence of spatial curvature K or a positive cosmological constant, the density perturbation approaches a finite limit both to first and second order, but the rate of approach depends on the model, being power law in the scale factor if the cosmological constant is positive but logarithmic if it is zero and and K, 23 pages, no figures

The dynamic properties of cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the trajectories in phase space will exhibit different behaviour (elliptic orbit or homoclinic orbit) with the increase of nonlinear perturbation. We illustrate this phenomenon by mean of linearized stability analysis. The theoretical analysis is consistent with the numerical results.

A new exact nonlinear Newtonian solution for a plane matter flow superimposed on the isotropic Hubble expansion is reported. The dynamical effect of cosmic vacuum is taken into account. The solution describes the evolution of nonlinear perturbations via gravitational instability of matter and the termination of the perturbation growth by anti-gravity of vacuum at the epoch of transition from matter domination to vacuum domination. On this basis, an `approximate' 3D solution is suggested as an analog of the Zeldovich ansatz.

Increasingly accurate observations of the cosmic microwave background and the large scale distribution of galaxies necessitate the study of nonlinear perturbations of Friedmann-Lemaitre cosmologies, whose equations are notoriously complicated. In this paper we present a new derivation of the governing equations for second order perturbations within the framework of the metric-based approach that is minimal, as regards amount of calculation and length of expressions, and flexible, as regards choice of gauge and stress-energy tensor. Because of their generality and the simplicity of their structure our equations provide a convenient starting point for determining the behaviour of nonlinear perturbations of FL cosmologies with any given stress-energy content, using either the Poisson gauge or the uniform curvature gauge., 30 pages, no figures. Changed title to the one in published version and some minor changes and additions

We study dyonic soliton and black hole solutions of the ${\mathfrak {su}}(2)$ Einstein-Yang-Mills equations in asymptotically anti-de Sitter space. We prove the existence of non-trivial dyonic soliton and black hole solutions in a neighbourhood of the trivial solution. For these solutions the magnetic gauge field function has no zeros and we conjecture that at least some of these non-trivial solutions will be stable. The global existence proof uses local existence results and a non-linear perturbation argument based on the (Banach space) implicit function theorem., Comment: 23 pages, 2 figures. Minor revisions; references added

The standard nonlinear perturbation theory of the gravitational instability is extended to incorporate the nonlocal bias, redshift-space distortions, and primordial non-Gaussianity. We show that local Eulerian bias is not generally compatible to local Lagrangian bias in nonlinear regime. The Eulerian and Lagrangian biases are nonlocally related order by order in the general perturbation theory. The relation between Eulerian and Lagrangian kernels of density perturbations with biasing are derived. The effects of primordial non-Gaussianity and redshift-space distortions are also incorporated in our general formalism, and diagrammatic methods are introduced. Vertex resummations of higher-order perturbations in the presence of bias are considered. Resummations of Lagrangian bias are shown to be essential to handle biasing schemes in a general framework., 20 pages, 17 figures, submitted to PRD, revised version, references are added

Near the liquid-gas critical point, thermal disturbances can generate sounds. We study the acoustic emission over four decades of reduced temperatures [defined as $\ensuremath{\varepsilon}=(T\ensuremath{-}{T}_{c})/{T}_{c}$, with ${T}_{c}$ the critical temperature] along the critical isochore, under linear and nonlinear temperature perturbations, respectively. We identify various thermoacoustic behaviors by numerically solving the governing equations. It is shown that a homogeneous thermoacoustic-wave pattern dominates in the linear case, largely independent of \ensuremath{\varepsilon}; whereas under the nonlinear perturbation, variation in \ensuremath{\varepsilon} could lead to severe wavefront deformation. The strong nonlinear effect is found to be of a transient nature because, in due time, both cases tend to converge in terms of the energy yield of the adiabatic process.