Linear Stability of Solitary Waves for the One‐Dimensional Benney–Luke and Klein–Gordon Equations

The linear stability of the solitary waves for the one‐dimensional Benney–Luke equation in the case of strong surface tension is investigated rigorously and the critical wave speeds are computed explicitly. For the Klein–Gordon equation, the stability of the traveling standing waves is considered and the exact ranges of the wave speeds and the frequencies needed for stability are derived. This is achieved via the abstract stability criteria recently developed by Stanislavova and Stefanov.     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

耦合BBM方程组孤立波解的轨道稳定性(英文)

本文研究具有Hamilton形式的耦合BBM方程组孤立波解的轨道稳定性.首先找到两族显式孤立波解.然后通过详细的谱分析证明出孤立波解的轨道稳定性.     

Ground state solitary waves for the planar Schrödinger-Poisson system

In this paper, we investigate the planar Schrödinger–Poisson System. Based on fixed point argument, Riesz’s rearrangement, Hardy–Littlewood–Sobolev inequality and critical point theory, we prove the existence and symmetry properties of ground state solitary waves. In addition to their existence, we also obtain the orbital stability of solitary waves.     

Existence,orbital stability and instability of solitary waves for coupled BBM equations

This paper is concerned with the orbital stability/instability of solitary waves for coupled BBM equations which have Hamiltonian form. The explicit solitary wave solutions will be worked out first. Then by detailed spectral analysis and decaying estimates of solutions for the initial value problem, we obtain the orbital stability/instability of solitary waves.     

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Stability of multipeakons

The Camassa–Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610]. We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons.     

Stability of smooth solitary waves for the generalized Korteweg–de-Vries equation with combined dispersion

The problem of orbital stability of smooth solitary waves in the generalized Korteweg–de-Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for any speed of wave propagation.     

Orbital stability for periodic standing waves of the Klein–Gordon–Zakharov system and the beam equation

The existence and stability of spatially periodic waves ${\left(e^{i{\omega}t}\varphi_\omega, \psi_\omega\right)}$ in the Klein–Gordon–Zakharov (KGZ) system are studied. We show a local existence result for low regularity initial data. Then, we construct a one-parameter family of periodic dnoidal waves for (KGZ) system when the period is bigger than ${\sqrt{2}\pi}$ . We show that these waves are stable whenever an appropriate function satisfies the standard Grillakis–Shatah–Strauss (Grillakis et al. J Funct Anal 74(1):160–197, 1987; Grillakis et al. J Funct Anal 94(2):308–348, 1990) type condition. We compute the intervals for the parameter ω explicitly in terms of L and by taking the limit L → ∞ we recover the previously known stability results for the solitary waves in the whole line case. For the beam equation, we show the existence of spatially periodic standing waves and show that orbital stability holds if an appropriate functional satisfies Grillakis–Shatah–Strauss type condition.     

Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation

We establish the local well-posedness for a generalized Dullin-Gottwald-Holm equation by using Kato’s theory. Furthermore, the orbital stability of the peaked solitary waves is also proved.     

Orbital stability of solitary waves for the Klein-Gordon-Zakharov equations

1.IntroductionInthetheoreticalinvestigationsofthedynamicsofstrongLangmuirturbulenceinplasmaphysics,varioustypesofZakharovequationstakeanimportantrole(see[3--8]).Intillspaper,weconsiderthefollowingKlein-Gordon-Zakharovequations:{:ti:::::;.;;:~"aiR,(11)withuacomplexfunctionandnarealfunction.Thelocalandglobalekistenceoftheinitialvalueproblemfor(1.l)wasconsideredin[4,6].Inthispaper,weconsidertheorbitalstabilityofthesolitarywavesof(1.l).Byapplyingtheabstracttheoryof[1,2]anddetailedspectralanalys…     

Travelling waves of the KP equations with transverse modulations

Kadomtsev-Petviashvili (KP) equations arise genetically in modelling nonlinear wave propagation for primarily unidirectional long waves of small amplitude with weak transverse dependence. In the case when transverse dispersion is positive (such as for water waves with large surface tension) we investigate the existence of transversely modulated travelling waves near one-dimensional solitary waves. Using bifurcation theory we show the existence of a unique branch of periodically modulated solitary waves. Then, we briefly discuss the case when the transverse dispersion is negative (such as for water waves with zero surface tension).     

Model Equations for Gravity-Capillary Waves in Deep Water

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity-capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.     

Orbital Stability of Solitary Waves of the Nonlinear Schrodinger-KdV Equation

This paper concerns the orbital stability of solitary waves of the system of KdV equation coupling with nonlinear Schrödinger equation. By applying the abstract results of Grillakis et al. [1- 2] and detailed spectral analysis, we obtain the stability of the solitary waves.     

Exact solitary water waves with capillary ripples at infinity

We prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.     

Benjamin-Ono方程孤立波解的轨道稳定性

本文给出了Benjamin-Ono方程的孤立波解,并应用M.Grillakis[4,5]等的抽象理论,通过谱分析,证明了该孤立波解是轨道稳定的.     

Nonlinear temporal dynamics in magnetic fluids between undulatory parallel walls

The effects of undulatory parallel walls and a normal magnetic field on the stability of weakly nonlinear waves at a horizontal interface of two magnetic inviscid fluids are investigated. We assumed that the walls have a weak sinusoidal undulation. The frequency of the main waves is similar to a problem having smooth boundaries. The breaker surface tension and the breaker magnetic field are obtained. The stability analysis concerns the interaction of two propagation wave numbers satisfying the resonance condition imposed by the periodicity of the sinusoidal walls. The first-resonance case occurs whenever the wall wave number is nearly equal to twice the propagation wave number while the second-resonance case occurs whenever the two kinds of wave numbers are nearly equal. When the wave number of the undulation is far from the propagation wave number, the sinusoidal walls have the same effect of the smooth walls on the stability criterion. The stability conditions and the transition curves in the two resonance cases are treated away from the critical state. The existence conditions and stability of Stokes waves near the critical state are discussed. Numerous illustrations and graphs amplify the work.     

Three Dimensional Flexural–Gravity Waves

Waves propagating on the surface of a three–dimensional ideal fluid of arbitrary depth bounded above by an elastic sheet that resists flexing are considered in the small amplitude modulational asymptotic limit. A Benney–Roskes–Davey–Stewartson model is derived, and we find that fully localized wavepacket solitary waves (or lumps) may bifurcate from the trivial state at the minimum of the phase speed of the problem for a range of depths. Results using a linear and two nonlinear elastic models are compared. The stability of these solitary wave solutions and the application of the BRDS equation to unsteady wave packets is also considered. The results presented may have applications to the dynamics of continuous ice sheets and their breakup.