Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation

Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smales amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L²() that stays away from the boundary of the neighbourhood of 0 W^2,2() in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L²() to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. Energetically means that the distance to the set of all minimisers is defined in terms of the total energy, and conditionally means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W^2,2() that contains the quiescent state and we are not interested in the fate of solutions that leave this set.     

Stability and Instability of Fourth-Order Solitary Waves

We study ground-state traveling wave solutions of a fourth-order wave equation. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. The analysis depends on the variational characterization of the ground states rather than information about the linearized operator.     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

Asymptotic Stability of N-Solitary Waves of the FPU Lattices

We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Bäcklund transformation and use the result to analyze the linearized FPU equation.     

Stability of Solitary Waves for the Generalized Higher-Order Boussinesq Equation

This work studies the stability of solitary waves of a class of sixth-order Boussinesq equations.     

Particle Trajectories in Linear Water Waves

We prove that in linear periodic gravity water waves there are no closed orbits for the water particles in the fluid. Each particle experiences per period a backward-forward motion that leads overall to a forward drift. This paper was written while both authors participated in the program “Wave Motion” at the Mittag-Leffler Institute, Stockholm, in the Fall of 2005.     

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

二阶线性随机微分方程的渐近稳定性

对刚度系数是遍历过程的二阶线性随机微分方程，本文研究了其平凡解几乎处处渐近稳定性问题。利用刚度系数导数过程的性质，给出了平凡解几乎处处渐近稳定的充分条件。当刚度系数是遍历高斯过程或周期过程时，还具体计算了其渐进稳定区域。结果表明，本文结果改进了目前有关的渐近稳定性的条件。     

Viscous Effects Can Destabilize Linear and Nonlinear Water Waves

Long waves on a running stream in shallow water are shown theoretically to be susceptible, in some circumstances, to a viscous instability, which can lead to rapid linear and nonlinear growth. The theory is based on high Reynolds numbers and involves viscous-inviscid interplay, leading in effect to a viscosity-modified version of the classical nonlinear K dV equation. This is with a pre-existing mean flow present. The modification is due to a Stokes wall layer and it can cause severe linear and nonlinear instability. A model profile for the original mean flow is studied first, followed by a smooth realistic profile, the latter provoking a nonlinear critical layer in addition. The theory is linked with interactive-boundary-layer analysis and linear and nonlinear Tollmien-Schlichting waves and there is some analogy with the recent findings (in work by the authors) of nonlinear break-ups occurring in any unsteady interactive boundary layer, including the external boundary layer and internal channel or pipe flows.     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

A Multiplicity Result for Solitary Gravity-Capillary Waves in Deep Water via Critical-Point Theory

. (Accepted May 29, 1998)     

Solitary Elastic Waves and Elastic Wavelets

A new group of wavelets that have the form of solitary waves and are the solutions of the wave equations for dispersive media is proposed to call elastic wavelets. That this group includes well-known Mexican-hat wavelets is proved. It is proposed to use elastic wavelets to study local features of the profile evolution of a solitary wave in an elastic dispersive medium     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

Stability of Concatenated Traveling Waves

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is individually asymptotically stable. We show that the sequence of traveling waves is itself asymptotically stable: as \(t\rightarrow \infty \), the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Essentially the same result was previously proved by Wright (J Dyn Differ Equ 21:315–328, 2009) and Selle (Decomposition and stability of multifronts and multipulses, 2009), who regarded the concatenated wave pattern as a sum of traveling waves. In contrast to their work, we regard the pattern as a sequence of traveling waves restricted to subintervals of \(\mathbb {R}\) and separated at any finite time by small jump discontinuities. Our proof uses spatial dynamics and Laplace transform.     

Solitary Waves in a Collisionless Plasma with Isothermal Pressure

Wave resonances in the hydrodynamic model of an isotropic collisionless quasi-neutral hot plasma with isothermal ions and electrons are considered. These resonances lead to the formation of two types of solitary waves: solitary waves proper and generalized solitary waves. The latter result from the nonlinear resonance of the proper solitary waves with magnetosonic and Alfvén periodic waves. The possibility of observing these waves in the Earth's magnetospheric plasma is discussed.     

Highly Nonlinear Solitary Waves for the Inspection of Adhesive Joints

In this paper we propose the use of highly nonlinear solitary waves (HNSWs) to monitor the curing process of an adhesive layer utilized to bond two aluminum sheets, and to inspect an adhesively-bonded aluminum lap-joint. HNSWs are mechanical waves that can form and travel in highly nonlinear systems, such as a chain of spherical particles where they are generated by means of a mechanical impact. They are characterized by a constant spatial wavelength and possess the important property that their speed, amplitude, and duration can be tuned by modifying the particles?? material or size, or the velocity of the impact. In the study presented in this paper, we investigate the feasibility of HNSWs for the nondestructive testing of adhesively-bonded structures. Two experiments are illustrated. In the first experiment we observe the curing process of a commercial 2-Ton Clear epoxy used to bond two aluminum sheets. In the second experiment, six types of bond quality were created on an aluminum lap-joint. In both experiments we noted that certain characteristics of the HNSWs such as time-of-flight and amplitude are affected by the physical conditions of the test specimen.     

On Discontinuity Waves in Linear Piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation. This revised version was published online in July 2006 with corrections to the Cover Date.