Wave modulation: the geometry, kinematics, and dynamics of surface-wave packets

2016 ◽  
Vol 803 ◽  
pp. 292-312 ◽  
Author(s):  
N. E. Pizzo ◽  
W. Kendall Melville
Keyword(s):  
Evolution Equations ◽  
Wave Packets ◽  
A Priori ◽  
Analytic Solutions ◽  
Kinematics And Dynamics ◽  
Linear Dispersion ◽  
Closed Set ◽  
Weakly Nonlinear ◽  
Numerical Predictions ◽  
Variational Structure

We examine the geometry, kinematics, and dynamics of weakly nonlinear narrow-banded deep-water wave packets governed by the modified nonlinear Schrödinger equation (Dysthe, Proc. R. Soc. Lond. A., vol. 369, 1979, pp. 105–114; MNLSE). A new derivation of the spatial MNLSE, by a direct application of Whitham’s method, elucidates its variational structure. Using this formalism, we derive a set of conserved quantities and moment evolution equations. Next, by examining the MNLSE in the limit of vanishing linear dispersion, analytic solutions can be found. These solutions then serve as trial functions, which when substituted into the moment evolution equations form a closed set of equations, allowing for a qualitative and quantitative examination of the MNLSE without resorting to numerically solving the full equation. To examine the theory we consider initially symmetric, chirped and unchirped wave packets, chosen to induce wave focusing and steepening. By employing the ansatz for the trial function discussed above, we predict, a priori, the evolution of the packet. It is found that the speed of wave packets governed by the MNLSE depends on their amplitude, and in particular wave groups speed up as they focus. Next, we characterize the asymmetric growth of the wave envelope, and explain the steepening of the forward face of the initially symmetric wave packet. As the packet focuses, its variance decreases, as does the chirp of the signal. These theoretical results are then compared with the numerical predictions of the MNLSE, and agreement for small values of fetch is found. Finally, we discuss the results in the context of existing theoretical, numerical and laboratory studies.

Modulation instability of weakly nonlinear long internal wave packets

2020 ◽  
Author(s):  
Tatiana Talipova ◽  
Efim Pelinovsky
Keyword(s):  
Evolution Equations ◽  
Nonlinear Term ◽  
Modulation Instability ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Density Stratification ◽  
Gardner Equation ◽  
Weakly Nonlinear ◽  
Whitham Equations

<p>We exam the problem of the modulation instability of long internal waves. Such weakly nonlinear weakly dispersive wave packets in one-modal approximation are described by the Gardner equation (Korteweg-de Vries equation with both, quadratic and cubic nonlinearity and necessity condition for modulation instability of such quasi-harmonic waves is the positive coefficient of cubic nonlinear term, which is realized for certain density stratification. Nevertheless the linear dispersive relation used within the Gardner equation is valid for very long waves and does not describe waves of moderate length. It is why some other nonlinear evolution equations are applied in the theory of long surface waves like the Benjamin-Bona-Mahony (BBM) and Whitham equations. We use the extended versions of these equations including cubic nonlinear term and express all  coefficients through modal functions and density stratification. Then, the modulational instability of weakly modulated wave packets is investigated after deriving the nonlinear Schrodinger equation. Improved dispersion relation influences on the increment and size of modulational instability. Obtained results are compared with those, which known within the Gardner model.</p>

Evolution of baroclinic wave packets in a flow with continuous shear and stratification

1981 ◽  
Vol 377 (1771) ◽  
pp. 379-404 ◽  
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Wave Packets ◽  
Baroclinic Instability ◽  
Wave Dispersion ◽  
Baroclinic Wave ◽  
Sine Gordon Equation ◽  
Weakly Nonlinear ◽  
Boundary Slope ◽  
Continuous Shear

Evolution equations are developed for weakly nonlinear baroclinic wave packets in a flow with continuous shear and stratification. When wave dispersion is forced by using sloping upper and lower boundaries it is shown that, for boundary slope equal to isothermal slope in the basic state, these evolution equations are transformable to the sine-Gordon equation. The relation between behaviour in this model and that in the discretelayer models of baroclinic instability is discussed, and finally solutions of the sine-Gordon equation appropriate to a periodic domain are presented and compared with experimental results.

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

A theory for Langmuir solitons

1982 ◽  
Vol 27 (1) ◽  
pp. 95-120 ◽  
Author(s):  
N. Nagesha Rao ◽  
Ram K. Varma
Keyword(s):  
Mach Number ◽  
A Priori ◽  
Low Frequency ◽  
Analytic Solutions ◽  
Smooth Transition ◽  
Langmuir Waves ◽  
Method Of Solution ◽  
Mach Numbers ◽  
Ion Waves ◽  
Langmuir Solitons

A systematic and self-consistent analysis of the problem of Langmuir solitons in the entire range of Mach numbers (0 < M < 1) has been presented. A coupled set of nonlinear equations for the amplitude of the modulated, high-frequency Langmuir waves and the associated low-frequency ion waves is derived without using the charge neutrality condition or any a priori ordering schemes. A technique has been developed for obtaining analytic solutions of these equations where any arbitrary degree of ion nonlinearity consistent with the nonlinearity retained in the Langmuir field can be taken into account self-consistently. A class of solutions with non-zero Langmuir field intensity at the centre (ξ = 0) are found for intermediate values of the Mach number. Using these solutions, a smooth transition from single-hump solitons to the double-hump solitons with respect to the Mach number has been established through the definitions of critical and cut-off Mach numbers. Further, under appropriate limiting conditions, various solutions discussed by other authors are obtained. Sagdeev potential analyses of the solutions for the Langmuir field as well as the ion field are carried out. These analyses confirm the transition from single-hump solitons to the double-hump solitons with respect to the Mach number. The existence of many-hump solitons for higher-order nonlinearities in the low-frequency ion wave potential has been conjectured. The method of solution developed here can be applied to similar equations in other fields.

W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations

2021 ◽  
pp. 2150468
Author(s):  
Youssoufa Saliou ◽  
Souleymanou Abbagari ◽  
Alphonse Houwe ◽  
M. S. Osman ◽  
Doka Serge Yamigno ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Singular Function ◽  
Ion Acoustic Waves ◽  
Weakly Nonlinear ◽  
Quantum Electron ◽  
Ion Acoustic ◽  
Trigonometric Function Solutions

By employing the Modified Sardar Sub-Equation Method (MSEM), several solitons such as W-shape bright, dark solitons, trigonometric function solutions and singular function solutions have been obtained in two famous nonlinear evolution equations which are used to describe waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in a plasma. These models are the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation and the (3+1)-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation, respectively. Comparing the obtained results with Refs. 32–34 and Refs. 43–46, additional soliton-like solutions have been retrieved and will be useful in future to explain the interaction between lower nonlinear ion-acoustic waves and the parameters of the MSEM and the obtained figures will have more physical explanation.

scholarly journals Motion and Trajectory Constraints Control Modeling for Flexible Surgical Robotic Systems

Micromachines ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 386
Author(s):  
Olatunji Mumini Omisore ◽  
Shipeng Han ◽  
Yousef Al-Handarish ◽  
Wenjing Du ◽  
Wenke Duan ◽  
...  
Keyword(s):  
Real Time ◽  
A Priori ◽  
Tumor Resection ◽  
Computational Effort ◽  
Robotic Systems ◽  
Kinematics And Dynamics ◽  
Target Point ◽  
Real Time Control ◽  
Control Models ◽  
Constraints Model

Success of the da Vinci surgical robot in the last decade has motivated the development of flexible access robots to assist clinical experts during single-port interventions of core intrabody organs. Prototypes of flexible robots have been proposed to enhance surgical tasks, such as suturing, tumor resection, and radiosurgery in human abdominal areas; nonetheless, precise constraint control models are still needed for flexible pathway navigation. In this paper, the design of a flexible snake-like robot is presented, along with the constraints model that was proposed for kinematics and dynamics control, motion trajectory planning, and obstacle avoidance during motion. Simulation of the robot and implementation of the proposed control models were done in Matlab. Several points on different circular paths were used for evaluation, and the results obtained show the model had a mean kinematic error of 0.37 ± 0.36 mm with very fast kinematics and dynamics resolution times. Furthermore, the robot’s movement was geometrically and parametrically continuous for three different trajectory cases on a circular pathway. In addition, procedures for dynamic constraint and obstacle collision detection were also proposed and validated. In the latter, a collision-avoidance scheme was kept optimal by keeping a safe distance between the robot’s links and obstacles in the workspace. Analyses of the results showed the control system was optimal in determining the necessary joint angles to reach a given target point, and motion profiles with a smooth trajectory was guaranteed, while collision with obstacles were detected a priori and avoided in close to real-time. Furthermore, the complexity and computational effort of the algorithmic models were negligibly small. Thus, the model can be used to enhance the real-time control of flexible robotic systems.

Gaussian wave packets in inhomogeneous media with curved interfaces

1987 ◽  
Vol 412 (1842) ◽  
pp. 93-123 ◽  
Keyword(s):  
Wave Equation ◽  
Large Scale ◽  
Evolution Equations ◽  
Wave Packets ◽  
Present Theory ◽  
Orthogonal Direction ◽  
Time Harmonic ◽  
Method Of Solution ◽  
Classical Wave Equation ◽  
The Time Domain

Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturb­ances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

scholarly journals Hamiltonian description of wave dynamics in nonequilibrium media

1994 ◽  
Vol 1 (4) ◽  
pp. 234-248 ◽  
Author(s):  
N. N. Romanova
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Normal Modes ◽  
Weak Nonlinearity ◽  
Hamiltonian Description ◽  
Weakly Nonlinear ◽  
Wave Dynamics ◽  
Canonical Variables ◽  
Stable Points ◽  
Stable Area

Abstract. We consider Hamiltonian description of weakly nonlinear wave dynamics in unstable and nonequilibrium media. We construct the appropriate canonical variables in the whole wavenumber space. The essentially new element is the construction of canonical variables in a vicinity of marginally stable points where two normal modes coalesce. The commonly used normal variables are not appropriate in this domain. The mater is that the approximation of weak nonlinearity breaks down when the dynamical system is written in terms of these variables. In this case we introduce the canonical variables based on the linear combination of modes belonging to the two different branches of dispersion curve. As an example of one of the possible applications of presented results the evolution equations for weakly nonlinear wave packets in the marginally stable area are derived. These equations cannot be derived if we deal with the commonly used normal variables.

scholarly journals SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

2006 ◽  
Vol 03 (02) ◽  
pp. 247-267 ◽  
Author(s):  
JOHN K. HUNTER
Keyword(s):  
Rayleigh Waves ◽  
Evolution Equations ◽  
Hyperbolic Surface ◽  
Smooth Solutions ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Hamiltonian Evolution ◽  
Short Time ◽  
Nonlinear Scale ◽  
Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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