Nonlinear Waves

Monopiles able to support very large offshore wind turbines are slender structures susceptible to nonlinear resonant phenomena. With the aim to better understand and model the wave-loading on these structures in very steep waves where ringing occurs and the numerical wave-loading models tend to lose validity, this study investigates the distinct influences of nonlinearities in the wave kinematics and in the hydrodynamic loading models. Six wave kinematics from linear to fully nonlinear are modelled in combination with four hydrodynamic loading models from three theories, assessing the effects of both types of nonlinearities and the wave conditions where each type has stronger influence. The main findings include that the nonlinearities in the wave kinematics have stronger influence in the intermediate water depth, while the choice of the hydrodynamic loading model has larger influence in deep water. Moreover, finite-depth FNV theory captures the loading in the widest range of wave and cylinder conditions. The areas of worst prediction by the numerical models were found to be the largest steepness and wave numbers for second harmonic, as well as the vicinity of the wave-breaking limit, especially for the third harmonic. The main cause is the non-monotonic growth of the experimental loading with increasing steepness due to flow separation, which leads to increasing numerical overpredictions since the numerical wave-loading models increase monotonically.

Experimentally observed grey solitons are analytically extracted from a physically viable Gross-Pitaevskii equation. Associated Lieb and Bogoliubov modes are calculated for these class of solitons. It is observed that, these nonlinear excitations have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which responds weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through both trap modulation and temporal variation of the two-body interaction.

In a cigar shaped Bose-Einstein condensate, explicit solutions of the coupled meanfield equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The wave vector of the impurity is controlled by the velocity of the grey soliton, which has an upper bound. It is found that the presence of the impurity lowers the speed of the grey soliton, as compared to the defect free case, where it can reach the sound velocity. The grey soliton's energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics.

Quasiperiodicity is the concept of order without translation symmetry. The discovery of quasiperiodic order in natural materials transformed the way scientists examine and define ordered structure. We show and verify experimentally that quasiperiodicity can be observed by scattering processes from a periodic structure, provided the interaction area is of finite width. This is made through a momentum conservation condition, physically realizing a geometrical method used to model quasiperiodic structures by projecting a periodic structure of a higher dimension.

The recurrence coe#cients of generalized Charlier polynomials satisfy a system of nonlinear recurrence relations. We simplify the recurrence relations, show that they are related to certain discrete Painleve equations, and analyze the asymptotic behaviour.

From a dipole lattice model, we derive a Boussinesq-type equation that governs the propagation of a longitudinal strain wave in a uniform pre-strained bar with constant cross section. We then describe an experiment based on high-speed, single-point photoelasticity to measure strain waves in a Polymethylmethacrylate (PMMA) bar generated when it breaks whilst undergoing a tensile test. The experimental results are finally compared to the model predictions, showing a good correspondence in terms of wave propagation, strain rate at the leading edge and amplitude and frequency of the following ripple.

New nonlinear evolution equations are derived that generalize the system by Matsuno and a terrain-following Boussinesq system by Nachbin . The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters .

A numerical wave tank based on fully nonlinear potential flow theory is used to calculate changes in local properties of periodic waves shoaling over barred-beaches (wave height, celerity, frontto-back asymmetry). Results show that strongly nonlinear wave decomposition phenomena occur in a modulation region beyond the bars. These are analyzed in detail, and discussed in the paper.

She received her Ph.D. degree in Mechanics from ITU in 2003. She completed her postdoctoral studies at the University of Colorado at Boulder, USA. She has published 18 research papers in peer-reviewed journals in SCI, 4 book chapters, 21 conference proceedings in the fields of perturbation methods, nonlinear wave propagation in arteries, optical solitons and wave collapse in optics and water waves problems. She is the editor of the book "Perturbation Methods with Applications in Science and Engineering" by IntechOpen. Dr. Bakırtaş is a member of the Scientific Committee of TUMTMK (Turkish National Committee of Theoretical and Applied Mechanics) and she was awarded "Dr. Serhat Ozyar, Young Scientist Of The Year Award" in 2004 and "Best Ph.D. Dissertation Award" by TUMTMK in 2003. Dr. Nalan Antar is working as a Professor of applied mathematics for the Istanbul Technical University, Department of Mathematical Engineering. She received her Ph.D. degree in mechanics from the Istanbul Technical University in 1999. She has completed her post-doctoral studies at the University of Alberta, Department of Mathematics and Statistics, Canada and later she participated in academic research projects at the University of Colorado at Boulder, USA. She has published 32 research papers in peer-reviewed journals in SCI, 13 conference proceedings in the fields of nonlinear wave propagation in arteries, optical solitons in nonlinear optics and water waves problems, in particular gravity currents. She has been the supervisor of many graduate students in applied mathematics. Dr. Antar is a member of the Scientific Committee of TUMTMK (Turkish National Committee of Theoretical and Applied Mechanics). XIII Contents Preface Chapter 1 Water Waves and Light: Two Unlikely Partners by Georgios N. Koutsokostas, Theodoros P. Horikis, Dimitrios J. In the tenth chapter, the evolution of light beams in a cubic-quintic-septic-nonical medium is investigated. As the model equation, an extended form of the well-known nonlinear Schrödinger equation is taken into account. By using a special ansatz, exact analytical solutions describing bright/dark and kink solitons are constructed. In the last chapter, the authors investigated the modulational instability (MI) in Kerr media, induced by cross-phase modulation of two optical beams in nonlinear fiber including the effect of higher-order dispersion. This book is intended to reach researchers, scientists, and postgraduate students in academia, as well as in industry, to give impressions and suggestions on a variety of topics and is published as an open access book in order to increase the impact of the contained information.

Les transferts de chaleur à travers un film ruisselant sur une plaque chauffée dépendent non seulement des caractéristiques thermiques du fluide mais aussi des caractéristiques de l'écoulement : les travaux expérimentaux de la littérature montrent que la présence d'instabilités hydrodynamiques peuvent mener à des intensifications de transfert non négligeable par rapport au cas d'un film d'épaisseur uniforme. Un modèle asymptotique à 4 équations fondé sur une méthode aux résidus pondérés a été formulé précédemment dans le cas des transferts de chaleur . Il s'agira de valider ce modèle en le comparant à une résolution directe de l'équation de Fourier par une méthode pseudo-spectrale, puis d'identifier les phénomènes responsables de cette intensification.

We propose to localize spin mixing dynamics in a spin-1 Bose-Einstein condensate by a temporal modulation of spin exchange interaction, which is tunable with optical Feshbach resonance. Adopting techniques from coherent control, we demonstrate the localization/freezing of spin mixing dynamics, and the suppression of the intrinsic dynamic instability and spontaneous spin domain formation in a ferromagnetically interacting condensate of 87 Rb atoms. This work points to a promising scheme for investigating the weak magnetic spin dipole interaction, which is usually masked by the more dominant spin exchange interaction.

A rigorous and most general covariant kinetic formalism is developed to study the nonlinear waves interaction in relativistic Vlasov plasmas. The typical nonlinear plasma reaction is a nonlinear current measured by the nonlinear plasma conductivity, and these quantities are derived here on the basis of relativistic Vlasov-Maxwell equations. Knowing the nonlinear plasma conductivity allows us to determine all plasma modes nonlinearly excited in plasma. The general covariant form of nonlinear conductivity is provided first for any value of plasma temperature and for the whole complex frequency plane by a correct analytical continuation. The further analysis is restricted to a correct relativistic particle distribution which is vanishing for particle speeds greater than speed of light. In the limit of nonrelativistic plasma temperatures the covariant nonlinear conductivity is significantly different from the standard noncovariant nonrelativistic results which are reached only in the formal limit of an infinitely large speed of light c → ∞.

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Several theoretical and numerical aspects concerning the highly accurate Boussinesq-type equations of are discussed. A re-derivation of the model recently presented by Bingham et al. (2009) is outlined. This provides a more general framework for the model establishing the correct relationship between a velocity formulation and a velocity potential formulation and correcting previous errors in the potential formulation. A new shoaling enhancement operator is introduced which enables the derivation of new models which differ from the existing ones at O(∇h). The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. A new stable discretization scheme around structural corners within the fluid domain is also presented.

This paper describes the extension of a finite difference model based on a recently derived highly accurate Boussinesq formulation to include domains having arbitrary piecewise-rectangular bottom-mounted structures. The resulting linearized system is analyzed for stability on a structurally divided domain, and it is shown that exterior corner points pose potential stability problems, as well as other numerical difficulties. These are mainly due to the discretization of high-order mixed-derivative terms near these points, where the flow is theoretically singular. Fortunately, the system is receptive to dissipation, and these problems can be overcome in practice using high-order filtering techniques. The resulting model is verified through numerical simulations involving classical linear wave diffraction around a semi-infinite breakwater, linear and nonlinear gap diffraction, and highly nonlinear deep water wave run-up on a vertical plate. These cases demonstrate the applicability of the model over a wide range of water depth and nonlinearity.

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Wave conditions in extratropical regions have seasonal variations. The wave condition in a period of one or more years exhibits variation, as displayed by samples of data for such a period. It is important to have a measure of the variability in the wave-induced extreme response and fatigue damage for marine structures due to this statistical uncertainty for sea states in various service periods. For instance, this variability needs to be considered when assessing the safety of a structure until next inspection or repair after damage. Moreover, this variability illustrates the uncertainty due to limited data, say in connection with measuring service behaviour in periods of limited duration. Scatter diagrams for the northern North Sea for 1-, 2-and 4-year periods as well as a 29-year period are applied in this study to determine the typical wave-induced response of an FPSO and a semisubmersible. The wave conditions are considered representative for extratropical climate. The marginal distribution of the significant wave height is modelled with a hybrid lognormal-Weibull model and the conditional distribution of peak period is fitted by a lognormal distribution. The longterm response of fatigue stress ranges is fitted by a generalized gamma distribution, which includes the frequently used two-parameter Weibull distribution as a special case. The extreme values are described by a two-parameter Weibull tail model. Compared with the 29-year distribution, the variation in the annual distributions is quite considerable as measured either by the distribution parameters, or the predicted extreme value of significant wave height, or the long-term response for extreme and fatigue loading. The correlation between the extreme and fatigue response and various measures of the significant wave height, H S , is also studied. High correlation is for instance found

Deformations of the focusing non-linear Schrödinger model (NLS) are considered in the context of the quasi-integrability concept. We strengthen the results of JHEP 09 (2012) 103 for bright soliton collisions. We addressed the focusing NLS as a complement to the one in JHEP 03 (2016) 005, in which the modified defocusing NLS models with dark solitons were shown to exhibit an infinite tower of exactly conserved charges. We show, by means of analytical and numerical methods, that for certain two-bright-soliton solutions, in which the modulus and phase of the complex modified NLS field exhibit even parities under a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved during the scattering process of the solitons. We perform extensive numerical simulations and consider the bright solitons with deformed potential V = 2η 2+ |ψ| 2 2+ , ∈ R, η < 0. However, for two-soliton field components without definite parity we also show numerically the vanishing of the first non-trivial anomaly and the exact conservation of the relevant charge. So, the parity symmetry seems to be a sufficient but not a necessary condition for the existence of the infinite tower of conserved charges. The model supports elastic scattering of solitons for a wide range of values of the amplitudes and velocities and the set {η, }. Since the NLS equation is ubiquitous, our results may find potential applications in several areas of non-linear science.

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schrödinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential V = ηI 2 -6 I 3 and the saturable type potential satisfying V [I] = 2ηI -I q 1+I q , q ∈ Z Z + , with a deformation parameter ∈ IR and I = |ψ| 2 . The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of {η, , q}. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

We consider a real Lagrangian off-critical submodel describing the soliton sector of the so-called conformal affine sl(3)(1) Toda model coupled to matter fields. The theory is treated as a constrained system in the context of Faddeev–Jackiw and the symplectic schemes. We exhibit the parent Lagrangian nature of the model from which generalizations of the sine-Gordon (GSG) or the massive Thirring (GMT) models are derivable. The dual description of the model is further emphasized by providing the relationships between bilinears of GMT spinors and relevant expressions of the GSG fields. In this way we exhibit the strong/weak coupling phases and the (generalized) soliton/particle correspondences of the model. The sl(n)(1) case is also outlined.

We investigate the existence and stability of discrete breathers in a chain of masses connected by linear springs and subjected to vibro-impact on-site potentials. The latter are comprised of harmonic springs and rigid constraints limiting the possible motion of the masses. Local dissipation is introduced through a non-unit restitution coefficient characterizing the impacts. The system is excited by uniform time-periodic forcing. The present work is aimed to study the existence and stability of similar breathers in the space of parameters, if additional harmonic potentials are introduced. Existence-stability patterns of the breathers in the parameter space and possible bifurcation scenarios are investigated analytically and numerically. In particular, it is shown that the addition of harmonic on-site potential can substantially extend the stability domain, at least close to the anti-continuum limit. This result can be treated as an increase in the robustness of the breather from the perspective of possible practical applications.

In the present paper a Korteweg-de Vries (KdV) type evolution equation, including the third-and fifth order dispersion and the fourth order nonlinearity is used for modelling the wave propagation in microstructured solids. The model equation is solved numerically under localised initial conditions. Possible solution types are introduced and discussed.

The propagation of solitary waves in dilatant granular materials is studied using the hierarchical Korteweg-de Vries type evolution equation. The model equation is solved numerically under localized initial conditions by the pseudospectral method. The behaviour of the solution is analysed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). Five solution types are introduced. Special attention is paid to the solitonic character of solutions.

The Mindlin-type model is used for describing the longitudinal deformation waves in microstructured solids. The evolution equation (one-wave equation) is derived for the hierarchical governing equation (two-wave equation) in the nonlinear case using the asymptotic (reductive perturbation) method. The evolution equation is integrated numerically under harmonic as well as localized initial conditions making use of the pseudospectral method. Analysis of the results demonstrates that the derived evolution equation is able to grasp essential effects of microinertia and elasticity of a microstructure. The influence of these effects can result in the emergence of asymmetric solitary waves.

A coupled system involving a nonlinear scalar PDE and a linear ODE is theoretically investigated. This hypebolic system with relaxation models the propagation of nonlinear waves in a waveguide connected to Helmholtz resonators, this device being an example of a nonlinear acoustic metamaterial. In a previous paper [Sugimoto, J. Fluid. Mech. 1992], it has been shown that this device allows also the propagation of acoustic solitons. In the present paper, the mathematical properties of the coupled system are analysed: formation of singularity in finite time, existence of entropy solutions in fractional BV spaces and uniqueness with a single family of entropies. New results are also deduced about weakly coupled systems. Numerical simulations illustrate these findings.

When waves propagate from deep water to shallow water, they transform. Extended mild slope equation includes wave transformations such as refraction, diffraction, shoaling, reflection and dissipations due to bottom friction and wave breaking and harbour resonance. Extended mild slope equation can be applied to the rapidly varying topographies through higher order bottom effetcs. Nonlinear wave celerity and group velocity have been considered in the calculations. In this study, extended mild slope equation has been reduced to Helmholtz equation and solved with finite volume method. Numerical model has been tested on semicircular shoaling area and compared with the physical experiment measurements given in literarure. Numerical model has been applied to the Fethiye Bay in the Mediterranean Sea in Turkey.

- Antalya Bay is located in the Mediterranean Sea of Turkish coasts. City of Antalya is one of the major tourism cities of Turkey with an increasing population. The sea outfall was constructed in the Antalya Bay. In this study, the currents and wind have been observed and ...

PHYSICAL REVIEW LETTERS 2$ MaRCH 1981 Force Office of Scientific Research under Con- tract No. 78-36748 and by the Office of Naval Re- search under Contract No. N00014-76-C-0867. 0 ~FIG. 1. Commutability of the Backlund transform. which is different but seemingly similar to Eq. ( ) is known. " In the framework of the IST meth- od, I can generalize the customary Zakharov- Shabat scheme which generates usual soliton and "oscillating" soliton solutions to include the present similarity-type "ripplon" solutions as well. " Finally, I note that the present decay-mode solution does not have proper limit in n -0 (one-di- mensional KdV limit) at the present form. In this respect, it seems that much still remains to be studied about decay modes in the most general form.

In this paper, we discuss the time evolution of a forced KdV equation by using a standard reductive perturbation technique. Using conservation laws, we derive the time evolution a KdV equation with an external force. Further, we make use of our newly designed method of flux correction along with the predictor-corrector scheme to study the growth of field quantities with time over a spatial span. The results obtained by using our new technique agree perfectly with the findings from the standard reductive perturbation technique. Our method will find quick acceptance in dealing with nonlinear plasma phenomena. Once we set up the basic equations, we can feed them into our scheme, and the results will automatically follow. We can even check the dependence of different waveforms as the initial and the boundary conditions.

We consider the 2D stationary propagating solitary waves of the so-called Boussinesq Paradigm equation. The fourth-order elliptic boundary value problem on infinite interval is solved by a Galerkin spectral method. An iterative procedure based on artificial time (Talse transients') and operator splitting is used. Results are obtained for the shapes of the solitary waves for different values of the dispersion parameters for both subcritical and supercritical phase speeds.

Remote sensing data reveals tidal impacts on chlorophyll-a (CHL) concentration in the Brazilian Equatorial Margin. In the northwest shelf, where the turbid Amazon River plume dominates, sediment resuspension by barotropic tides during spring tides or reduced currents in neap tides likely decrease primary production, shown by negative CHL differences (∼-55%) in spring-neap tide composites. The northeastern shelf, with clearer waters, shows positive CHL anomalies (∼32%) likely due to nutrient-rich vertical mixing from barotropic tides. Along internal tide (IT) pathways, elevated CHL bands (∼30% above average) align with IT crests. The CHL distribution in the spring-neap composite shows wave-like patterns, with positive CHL differences (∼2.2%) linked to IT crests. IT interactions with background currents cause the shifting of the location of incoherent IT throughout the year, partially disrupting the CHL signals in the spring-neap tide composites. A 1-2 day lag between higher CHL variability and tidal potential suggests delayed nutrient mixing post-spring-neap tides.

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. it is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear SchrGdinger equation. The latter equation allows us to study the modulational instability conditions leading to diflerent zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.

Among a lot of fascinating nonlinear effects, there is a great deal of interest in self modulation of a plane wave, or modulational instability (MI), which occurs in nonlinear dispersive media. Qualitatively a MI is the tendency for amplitude of a modulated carrier wave to break into isolated structures or solitons. Energy originally resident in the carrier of wave packets or long pulse is gradually transferred by nonlinear interaction, in the medium, into the spectrum side bands. As the side-band energy grows in amplitude, the modulated wave breaks up into a series of localized objects or envelope solitons. Composite structures formed by a singly or doubly-curved shallow shell resting on a nonlinear elastic foundation are wave guides that enable one to focus a high energy density in order that nonlinearities be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional localized modes in elastodynamics. The purpose of the present work is to study the influences of the geometric dispersion, the prestress on the shallow shell and the material nonlinearity of the elastic foundation on the vibrations modes of the structure. The basic equations which govern the dynamics of the elastic structure are deduced from a variational principle. The analysis is restricted to signals which consist of a slowly varying envelope in space and time modulating a harmonic carrier wave. In the limit of low amplitude the coupled equations are solved by means of a reductive perturbation method. It is shown that the complex amplitude of the envelope is governed by a two-dimensional nonlinear Schrödinger equation. This equation allows us to study the modulational instability conditions leading to different zones of instability. The mechanism of the self generated nonlinear waves in the elastic structure beyond the birth of modulational instability is numerically investigated on the original equations.

Optimal perturbations are investigated in a magnetohydrodynamic flow bounded by perfectly insulating or conducting walls. The parallel channel flow submitted to uniform, normal magnetic field is taken as an example. The stability equations (linearized Navier–Stokes and Maxwell equations) are solved simultaneously, because of the natural existence of a coupling between them. Exponential instability is studied first to set ideas and to fix some reference magnetic Prandtl and magnetic Reynolds numbers. Then, optimal perturbations are searched for by employing the approach first proposed by Butler and Farrell [Phys. Fluids A 4, 1637 (1992)]. The shape of the optimally perturbed velocity is poorly affected by the magnetic field; however, the magnetic field is found to stabilize both exponential instability and algebraically growing perturbations. The critical Reynolds numbers in the presence of magnetic fields can be very large and it is thus possible to find very significant transient g...

It is shown that the problem of determining the equilibrium shape of bodies formed by freezing of flow in porous medium can be reduced to the Riemann problem with displacement. A solitary body and linear array of bodies are used as examples. Algorithms for determining its boundary is constructed and realized. All results are presented in the graphic form and they correspond to wide diapason of physical parameters.

 Answer question ONE and ANY other two questions. 1 (a) Briefly discuss the importance of Oscillations and Waves in mathematical modelling and technological applications. 2 Marks (b) Distinguish between elastic wave and Doppler effect. 1 Mark (c) You are provided with a forced harmonic oscillator with damping subject to an external sinusoidal force F(t) = F 0 cos(ωt). Derive the expression for the amplitude of the steady state oscillations using complex representation, and explain the concept of resonance. 6 Marks (d) State any three characteristics of simple harmonic motion (SHM). 3 Marks (e) A simple pendulum of length 1.0 m swings with a small amplitude. State three assumptions required in order to calculate its period. 3 Marks (f) i. How is the quality factor Q defined for a damped oscillator? ii. What does a low Q factor imply in an RLC circuit?

We study nonlinear dispersive wave systems described by hyperbolic PDE's in R d and difference equations on the lattice Z d . The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi-Pasta-Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

We study nonlinear dispersive wave systems described by hyperbolic PDE&#39;s in ℝdand difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for nu...

In a previous paper, sufficiently large‐amplitude and left‐handed “pump waves” propagating parallel to the background magnetic field were shown to stabilize a moderately dense beam in a proton plasma against the generation of waves drawing their energy from the differential streaming motion of the beam [Gomberoff, 2003]. We now examine the general case of both left‐hand and right‐hand pump waves and their effects on beam instability as a function of pump wave amplitude and frequency, beam speed, and plasma component temperatures. We find that the left‐hand pump wave always gives beam stability above a threshold amplitude. Larger threshold trend with increasing beam speed and lower ones with increasing temperature. It is also shown that they can stabilize left‐hand polarized instabilities in the case of large drift velocities. The right‐hand pump similarly suppresses beam instabilities when its pump frequency is below the linearly unstable range of frequencies. However, when its pump...

This study investigates the application of the Physics-Informed Neural Network (PINN) to simulate the forward and backward dynamics of the N-soliton Kortewegde Vries (KdV) equation, given an initial condition. The approach utilized the conditional well-posedness of the KdV equation's Initial Value Problem (IVP) on an infinite domain. In this study, no boundary conditions were imposed or trained on the problem, as they are unnecessary when working with an infinite domain. The results of the PINN model for the one-soliton, two-soliton, and three-soliton KdV equation showed that the solution followed the physical properties of the equation with an error of 3.3353%, 0.9317%, and 0.8537%, respectively, demonstrating the effectiveness of the method. These results indicate that the PINN model is a promising approach for solving nonlinear PDEs over infinite spatial domains.

To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support observationallybased intuition that spectral energy density of internal waves is the result of predominantly wavewave interactions.

Withdrawn. STUDIES OF TEMPERATURE-DEPENDENT EXCIMERMONOMER CONVERSION IN A DENDRIMERIC ANTENNA SUPERMOLECULE BY FLUORESCENCE SPECTROSCOPY. Youfu Cao, Raoul Kopelman, Department of Chemistry, University of Michigan, Ann Arbor, MI. Nanostar, a phenylacetylene (PA) dendrimer labeled with perylene, is discovered to exhibit temperature-dependent emission spectra in certain organic solvents over the temperature range of 10-65 C. The monomer signal is increasing exponentially with increased temperature, while the excimer signal decreases. Models of excimer formation and dimmer dissociation dynamics are included, and the equilibrium constants at di erent temperature are calculated. This behavior suggests nanostar&#39;s potential application in uorescencebased optical thermometry, which is being investigated.

The American Physical Society Anisotropic shear dispersion parameterization for ocean eddy transport SCOTT RECKINGER, Montana State University, BAYLOR FOX-KEMPER, Brown University -The effects of mesoscale eddies are universally treated isotropically in global ocean general circulation models. However, observations and simulations demonstrate that the mesoscale processes that the parameterization is intended to represent, such as shear dispersion, are typified by strong anisotropy. We extend the Gent-McWilliams/Redi mesoscale eddy parameterization to include anisotropy and test the effects of varying levels of anisotropy in 1-degree Community Earth System Model (CESM) simulations. Anisotropy has many effects on the simulated climate, including a reduction of temperature and salinity biases, a deepening of the southern ocean mixed-layer depth, impacts on the meridional overturning circulation and ocean energy and tracer uptake, and improved ventilation of biogeochemical tracers, particularly in oxygen minimum zones. A process-based parameterization to approximate the effects of unresolved shear dispersion is also used to set the strength and direction of anisotropy. The shear dispersion parameterization is similar to drifter observations in spatial distribution of diffusivity and high-resolution model diagnosis in the distribution of eddy flux orientation.

In this study, we consider partial di erential equation problems describing nonlinear wave phenomena, e.g., a fully nonlinear third order Korteweg-de Vries (KdV) equation, the fourth order Boussinesq equation, the ÿfth order Kaup-Kupershmidt equation and an extended KdV5 equation. First, we develop a method of lines solution strategy, using an adaptive mesh reÿnement algorithm based on the equidistribution principle and spatial regularization techniques. On the resulting highly nonuniform spatial grids, the computation of high-order derivative terms appears particularly delicate and we focus attention on the selection of appropriate approximation techniques. Finally, we solve several illustrative problems and compare our computational approach to conventional solution techniques.

We i n vestigate the dynamics of one dimensional mass-spring chain with non-monotone dependence of the spring force vs. spring elongation. For this strongly nonlinear system we nd a family of exact solutions that represent the nonlinear waves. We h a ve found numerically that this system displays a dynamical phase transition from the stationary phase when all masses are at rest to the twinkling phase when the masses oscillate in a wave motion. This transition has two fronts, which propagate with di erent speeds. We study this phase transition analytically and derive the relations between its quantitative c haracteristics.

A nonlinear microwave signal processor based on the parametric interaction of a microwave magnetostatic wave, propagating in an yttrium-iron garnet (YIG) film waveguide, with a non-stationary (pulsed) electromagnetic pumping is proposed and practically realized. The proposed processor amplifies input pulsed microwave signals by up to 30 dB, performs operations of phase conjugation, time profile inversion, and controlled time delay of up to 1 µs. .

The anisotropic nature of solar wind magnetic turbulence fluctuations is investigated scale-by-scale using high cadence in-situ magnetic field measurements from the Cluster and ACE spacecraft missions. The data span five decades in scales from the inertial range to the electron Larmor radius. In contrast to the inertial range, there is a successive increase towards isotropy between parallel and transverse power at scales below the ion Larmor radius, with isotropy being achieved at the electron Larmor radius. In the context of wave-mediated theories of turbulence, we show that this enhancement in magnetic fluctuations parallel to the local mean background field is qualitatively consistent with the magnetic compressibility signature of kinetic Alfvén wave solutions of the linearized Vlasov equation. More generally, we discuss how these results may arise naturally due to the prominent role of the Hall term at sub-ion Larmor scales. Furthermore, computing higher-order statistics, we show that the full statistical signature of the fluctuations at scales below the ion Larmor radius is that of a single isotropic globally scale-invariant process distinct from the anisotropic statistics of the inertial range.