Nonlinear Waves of Vorticity

This is a review of solutions of the vorticity equation for two-dimensional flow of an inviscid incompressible fluid that represent nonlinear waves. Geophysical applications are emphasized. Some of the solutions are valid in the beta-plane of Rossby. Some are related to weakly nonlinear perturbations of basic parallel flows and axisymmetric flows, to initial-value problems of hydrodynamic instability and to variational principles of minimal enstrophy or maximal entropy. Some have been found by exploiting well-known ideas of the theory of solitons. In addition to listing known solutions and presenting a synthesis of their relationship to other fluid dynamic results, we report a few new ideas and new solutions for strongly nonlinear waves.     

快变海底和自由表面流共振生成的弱非线性Stokes波

运用WKBJ型摄动逼近法,对环境流同沙纹海底共振产生的自由表面水波的非线性效应进行了研究。沙纹海底由缓变平均水深部分和快变海底部分叠加构成。根据对快变海底波长的不同选取,可以相应地激发环境流同非平整海底的同步共振、超谐波共振和次谐波共振,由此产生自由表面波运动。对次谐波共振进行了详细考察。对于定常流自治动力系统,对可能出现的非线性各种稳态及其稳定性进行了探讨。假如环境流具有一个小振动分量,动力系统成为非自治的,则将发生混沌现象。     

Resonant Interactions between Vortical Flows and Water Waves. Part II: Shallow Water

Any weak, steady vortical flow is a solution, to leading order, of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of long irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic and have comparable length scales, resonant interactions can occur between the various components of the flow. The interaction is described by two coupled Korteweg-de Vries equations and a two-dimensional streamfunction equation.     

Resonant Interactions between Vortical Flows and Water Waves. Part I: Deep Water

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by “triad-like” ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly mentioned.     

Weakly Nonlinear Waves in Nonideal Fluids

We study the propagation of weakly nonlinear waves in nonideal fluids, which exhibit mixed nonlinearity. A method of multiple scales is used to obtain a transport equation from the Navier–Stokes equations, supplemented by the equation of state for a van der Waals fluid. Effects of van der Waals parameters on the wave evolution, governed by the transport equation, are investigated.     

Weakly Nonlinear Internal Waves in Shear

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg-deVries equation when the depth is small compared to the wavelength, and to the Benjamin-Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a waveform.     

Some Properties of Long Nonlinear Waves

This paper is concerned with certain mathematical aspects of long nonlinear wave propagation on a free surface. Some special solutions are noted and the equations are shown to have an infinite number of conservation laws.     

Partially Coherent Waves in Nonlinear Periodic Lattices

We study the propagation of partially coherent (random-phase) waves in nonlinear periodic lattices. The dynamics in these systems is governed by the threefold interplay between the nonlinearity, the lattice properties, and the statistical (coherence) properties of the waves. Such dynamic interplay is reflected in the characteristic properties of nonlinear wave phenomena (e.g., solitons) in these systems. While the propagation of partially coherent waves in nonlinear periodic systems is a universal problem, we analyze it in the context of nonlinear photonic lattices, where recent experiments have proven their existence.     

Long Nonlinear Waves in Resonance with Topography

The evolution of periodic long surface waves over a periodic bottom topography resonant with the waves is studied. Coupled Korteweg–de Vries equations are derived and describe the evolution in terms of interaction between right- and left-traveling waves. The coupling arises from the cumulative effect of wave scattering. We discuss the various conserved quantities of the system and compute solutions for the initial value problem and for the time-periodic problem of fluid "sloshing" in a tank. Some three-dimensional extensions are discussed.     

Nonlinear Dynamics of Marine Systems in Random Waves

The dynamics of ships or offshore structures is influenced by several different effects, some of which have a distinctly nonlinear characteristic. Even though in many situations the motion can sufficiently be described by linear models, nonlinear phenomena play a crucial role in the investigation of some more critical operating conditions: Large amplitude motions, sudden jumps in the dynamical behavior and sensitivity to the initial conditions are likely to occur under some circumstances. The response of floating systems such as moored buoys and barges in regular waves can be approximated by analytical or numerical techniques. These analyses reveal the characteristics of different periodic motions. In order to determine how these responses change under a more general forcing, the motion of floating structures under the influence of random disturbances is described by probability distributions. Different mathematical tools can efficiently be applied to models with few degrees of freedom. The localized statistical linearization used here is also promising for larger systems. Modelling aspects of offshore structures and random waves are discussed as well as the determination of probability distributions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Superposition of Delta Waves in Multi-dimensional Space

We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.     

Instability of Solitary Waves in Nonlinear Composite Media

In this paper,we investigate a class of Hamiltonian systems arising in nonlinear composite media.By detailed analysis and computation we obtain a decaying estimates on the semigroup and prove the orbitalinstability of two families of explicit solitary wave solutions (slow family in anisotropic case and solitary wavesin isotropic case),which theoretically verify the related guess and numerical results.     

Instabilities Associated with Forced Nonlinear Waves

This paper is concerned with the response of nonlinear wave systems to external forcing by a uniform train of sinusoidal waves. In particular the stability of the forced solution is studied and other resonant situations are investigated.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Integrability and Linear Stability of Nonlinear Waves

It is well known that the linear stability of solutions of \(1+1\) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general \(N\times N\) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for \(N=3\) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.     

On the Computation of Nonlinear Planetary Waves

A numerical method is developed to solve a class of nonlinear, nonlocal eigenvalue problems defined in an infinite strip, and is applied to compute solitary planetary waves in a sheared zonal current on the beta-plane. This method, an iterative procedure derived from the natural variational structure of these problems, is implemented in the physical case when the ambient parallel flow has a linear or a quadratic velocity profile. The results of the numerical experiments establish rigorous limits on the range of validity of the formal asymptotic theory of weakly nonlinear long waves, and also reveal some new phenomena involving strongly nonlinear waves. The iterative procedure is analyzed in a general setting, and is shown to be globally convergent without restriction on the wave amplitude.     

有限变形弹性杆中的几何非线性波

利用有限变形理论的Lagrange描述,借助非保守系统的Hamilton型变分原理,导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解,计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程,这个方程在相平面上对应着异宿鞍-焦轨道,其解为振荡孤波解．如果略去粘性效应或横向惯性,方程将分别退化为KdV方程或Bergers方程,由此得到孤波解或冲击波解,它们在相平面上对应着同宿轨道或异宿轨道．     

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Summary. In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables. Received November 8, 2000; accepted January 6, 2001 Online publication March 23, 2001     

A New Class of Nonlinear Waves in Parallel Flows

Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.