Asymptotic stability of solitary waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation $$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$ is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, $$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 .$$ In particular, we study the case wheref(u)=u ^p+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C ⁴). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4^?, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.     

Coupled perturbed modes and internal solitary waves

Coupled perturbed mode theory combines conventional coupled modes and perturbation theory. The theory is used to directly calculate mode coupling in a range-dependent shallow water problem involving propagation through continental shelf internal solitary waves. The solitary waves considered are thermocline depressions, separating well-mixed upper and lower layers. The method is fast and accurate. Results highlight mode coupling associated with internal solitary waves, and mode capture or loss to and from the discrete mode spectrum.     

On the generation of internal solitary marine waves

Summary A nonlinear model of generation of internal ?solitary? marine waves is discussed: when a large surface wave—for instance a superficial wave or a bore of tidal origin—passes over a submarine mountain or crosses a strait, packets of internal waves may often be detected. We study this phenomenon taking into account the effect of the air-sea surface; we show that the phenomenon can be schematized, in an approximate but realistic way, by using the solutions of an inhomogeneous KdV equation. The forcing term depends on the air-sea surface elevation and on the bottom topography. We then apply our model to the marine currents, of tidal origin, flowing through the straits of Gibraltar.

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Riassunto Si studia un modello non lineare della generazione di onde interne marine. Quando una grande onda superficiale passa sopra una montagna sottomarina o attraversa uno stretto si osservano pacchetti di onde interne. Si studia questo fenomeno tenendo conto della interfaccia aria-mare, si mostra che il fenomeno può essere rappresentato in maniera approssimata dalle soluzioni di un'equazione di KdV non omogenea. Il termine forzante dipende dalla superficie aria-mare e dalla topografia del fondo. Infine il modello proposto è applicato alle correnti di marea dello stretto di Gibilterra.

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Резюме Обсуждается нелинейная модель образования внутренних ?одиночных? морских волн: когда большая поверхностная волна проходит над подводной горой или приливная волна пересекает пролив, часто могут детектироваться пакеты внутренних волн. Мы исследуем зто явление, учитывая влияние границы раздела воздух-море. Мы показываем, что это явление может быть схематизировано приближенным, но реалистическим способом, используя решения неоднородного уравнения Кортевега-де Вриса. Силовой член зависит от возвышения границы раздела воздух-море и от топографии дна. Затем мы применяем нашу модель для морских течений (приливного происхождения), проходязих через Гибралтарский пролив.

     

Generation, propagation, and breaking of internal solitary waves

Tidal, two-layer flow over topography generates a kink of the interface separating an upstream interfacial elevation from a depression above the topography. Upstream undular bores and solitary waves of large amplitude are generated from the interfacial kink. The waves propagate upstream when the tide turns. Interfacial simulations of this kind of generation process fit with the observations at Knight Inlet in British Columbia, in the Sulu Sea experiment, and undular bores generated by internal tides in the Strait of Gibraltar. Fully nonlinear interfacial computations compare successfully with experimental observations of solitary waves in the laboratory and in the field for wave amplitudes ranging from small to maximal values. The waves exhibit only minor sensitivity to a finite thickness of the pycnocline. Analytical solitary waves are recaptured in the small amplitude limit. Shear-induced breaking appears first in the top part of the pycnocline and is expressed in terms of the Richardson number. Convective breaking in the top part of the water column occurs beyond a threshold amplitude when a pronounced stratification continues all the way to the ocean surface.     

Anomalous resonance phenomena of solitary waves with internal modes

We investigate the nonparametric, pure ac driven dynamics of nonlinear Klein-Gordon solitary waves having an internal mode of frequency Omega(i). We show that the strongest resonance arises when the driving frequency delta = Omega(i)/2, whereas when delta = Omega(i) the resonance is weaker, disappearing for nonzero damping. At resonance, the dynamics of the kink center of mass becomes chaotic. As we identify the resonance mechanism as an indirect coupling to the internal mode due to its symmetry, we expect similar results for other systems.     

Large-eddy simulation of the generation and propagation of internal solitary waves

A modified large-eddy simulation model,the dynamic coherent eddy model(DCEM)is employed to simulate the generation and propagation of internal solitary waves(ISWs)of both depression and elevation type,with wave amplitudes ranging from small,medium to large scales.The simulation results agree well with the existing experimental data.The generation process of ISWs is successfully captured by the DCEM method.Shear instabilities and diapycnal mixing in the initial wave generation phase are observed.The dissipation rate is not equal at different locations of an ISW.ISW-induced velocity field is analyzed in the present study.The structure of the bottom boundary layer(BBL)of internal wave packets is found to be different from that of a single ISW.A reverse boundary jet instead of a separation bubble exists behind the leading internal wave while separation bubbles appear in other parts of the wave-induced velocity field.The boundary jet flow resulting from the adverse pressure gradients has distinctive dynamics compared with free shear jets.     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.     

A model system for strong interaction between internal solitary waves

A mathematical theory is mounted for a complex system of equations derived by Gear and Grimshaw that models the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. For the model in question, the Cauchy problem is of interest, and is shown to be globally well-posed in suitably strong function spaces. Our results make use of Kato's theory for abstract evolution equations together with somewhat delicate estimates obtained using techniques from harmonic analysis. In weak function classes, a local existence theory is developed. The system is shown to be susceptible to the dispersive blow-up phenomenon investigated recently by Bona and Saut for Korteweg-de Vries-type equations.     

Exciton-polariton solitary waves

Effects of the exciton and polariton dispersions and the nonlinear exciton and photon interactions on the properties of polariton solitons in molecular crystals are investigated. Higher-order terms and phase-modulation (chirp) are taken into account. Bright- and dark-soliton solutions of the resulting modified nonlinear Schr?dinger (NLS) equation are presented. Nonlinearity- and dispersion-induced critical points on the polariton dispersion curve are obtained, separating regions with different solutions. Received 2 October 2001 / Received in final form 23 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: Stoychev@issp.bas.bg     

Varieties of solitons and solitary waves

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrödinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.     

Time reversing solitary waves

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore, we present numerical experiments for the time-reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a new Boussinesq model, both in transmission and in reflection.     

Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation

Certain generalizations of one of the classical Boussinesq-type equations, $$u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx} $$ are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.     

Scattering of regularized-long-wave solitary waves

The Lagrangian density for the regularized-long-wave equation (also known as the BBM equation) is presented. Using the trial function technique, ordinary differential equations that describe the time dependence of the position of the peaks, amplitudes, and widths for the collision of two solitary waves are obtained. These equations are analyzed in the Born and “equal-width” approximations and compared with numerical results obtained by direct integration utilizing the split-step fast Fourier-transform method. The computations show that collisions are inelastic and that production of solitary waves may occur.     

Interaction of solitary spin waves

We present the results of a computer experiment devoted to the problem of the interaction of two magnetic solitary spin waves moving in the direction perpendicular to the axis of easy magnetization in an uniaxial ferromagnet. Such waves being particular solutions of the Landau-Lifshitz equations move like a domain wall under the influence of an external magnetic field. Our computer experiment shows that the two solitary spin waves during their interaction, behave as two solitons and thus the concerned Landau-Lifshitz equations allows N-soliton solutions.     

Theory of solitary plastic waves

The paper deals with the dislocation dynamics of coherently propagating modes of plastic shear (Lüders bands) in single crystals oriented for single slip, in terms of a generalized Fisher-Kolmogorov equation. The role of (1) cross-slip and (2) non-axial stresses as propagation mechnisms is investigated, and the problem of propagation velocity selection is addressed. The phenomenon of slip band clustering which was observed in sufficiently thick tensile specimens is traced back to a propagative instability owing to non-axial stresses.     

Random generation of coherent solitary waves from incoherent waves

The random generation of coherent solitary waves from incoherent waves in a medium with an instantaneous nonlinearity has been observed. One excites a propagating incoherent spin wave packet in a magnetic film strip and observes the random appearance of solitary wave pulses. These pulses are as coherent as traditional solitary waves, but with random timing and a random peak amplitude.     

Bifurcations and traveling waves in a delayed partial differential equation

Here cell population dynamics in which there is simultaneous proliferation and maturation is considered. The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters. The solution behavior depends on the initial function varphi(x) and a three component parameter vector P=(delta,lambda,r). For strictly positive initial functions, varphi(0) greater, similar 0, there are three homogeneous solutions of biological (i.e., non-negative) importance: a trivial solution u(t) identical with 0, a positive stationary solution u(st), and a time periodic solution u(p)(t). For varphi(0)=0 there are a number of different solution types depending on P: the trivial solution u(t), a spatially inhomogeneous stationary solution u(nh)(x), a spatially homogeneous singular solution u(s), a traveling wave solution u(tw)(t,x), slow traveling waves u(stw)(t,x), and slow traveling chaotic waves u(scw)(t,x). The regions of parameter space in which these solutions exist and are locally stable are delineated and studied.     

New standing solitary waves in water

By means of the parametric excitation of water waves in a Hele-Shaw cell, we report the existence of two new types of highly localized, standing surface waves of large amplitude. They are, respectively, of odd and even symmetry. Both standing waves oscillate subharmonically with the forcing frequency. The two-dimensional even pattern presents a certain similarity in the shape with the 3D axisymmetric oscillon originally recognized at the surface of a vertically vibrated layer of brass beads. The stable, 2D odd standing wave has never been observed before in any media.     

Nucleus-acoustic solitary waves in white dwarfs

The formation of nucleus-acoustic solitary waves (NASWs), and their basic properties in white dwarfs containing non-relativistically or ultra-relativistically degenerate electrons, non-relativistically degenerate light nuclei, and stationary heavy nuclei have been theoretically investigated. The reductive perturbation method, which is valid for small but finite amplitude NASWs, is used. The NASWs are, in fact, associated with the nucleus acoustic (NA) waves in which the inertia is provided by the light nuclei, and restoring force is provided by the degenerate pressure of electrons. On the other hand, stationary heavy nuclei maintain the background charge neutrality condition. It has been found that the presence of the heavy nuclei significantly modify the basic features (polarity, amplitude, width, and speed) of the NASWs. The basic properties are also found to be significantly modified by the effects of ultra-relativistically degenerate electrons and relative number densities of light and heavy nuclei. The implications of our results in white dwarfs are pinpointed.