On the Propagation of Weakly Nonlinear Random Dispersive Waves

We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data.     

On the Resonant Generation of Weakly Nonlinear Stokes Waves in Regions with Fast Varying Topography and Free Surface Current

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Evolution Equations for Plane Cubically Nonlinear Elastic Waves

Consideration is given to the nonlinear theory of elastic waves with cubic nonlinearity. This nonlinearity is separated out, and the interaction of four harmonic waves is studied. The method of slowly varying amplitudes is used. The shortened and evolution equations, the first integrals of these equations (Manley–Rowe relations), and energy balance law for a set of four interacting waves (quadruplet) are derived. The interaction of waves is described using the wavefront reversal scheme     

骑行波的非线性演化方程

从能量的角度出发,采用Ｈａｍｉｌｔｏｎ描述交结合变分原理和摄动分析,并借助于符号运算导出了骑行在大波上的小波的Ｈａｍｉｔｏｎ密度函数和非线性动力学方程。这里的大流和小波是对波高而言的。在Ｈａｍｉｌｔｏｎ描述中,正则变量取为波高和速度势。本文导出了描述小波演化的二阶方程,在一阶近下的方程与Ｈｅｎｙｅｙ等人（１９８８）的结果一致。     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Vertical Momentum Fluxes Induced by Weakly Nonlinear Internal Waves on the Shelf

Free inertia-gravity internal waves in a two-dimensional stratified flow of an ideal fluid with a vertical velocity shear are considered in the Boussinesq approximation. The boundary-value problem for the amplitude of the vertical velocity of internal waves has complex coefficients; therefore, the wave frequency has an imaginary correction and the eigenfunction is complex. It is shown that the wave is weakly damped, the vertical wave momentum fluxes being nonzero and can be greater than the turbulent fluxes. The Stokes drift velocity component transverse to the direction of wave propagation is nonzero and less than the longitudinal component by an order of magnitude. The dispersion curves of the first two modes are cut off in the low-frequency domain due to the influence of critical layers in which the wave frequency taken with the Doppler shift is equal to the inertial frequency.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

On the Proper Form of the Amplitude Modulation Equations for Resonant Systems

The complex amplitude modulation equations of a discrete dynamicalsystem are derived under general conditions of simultaneous internal andexternal resonances. Alternative forms of the real amplitude and phaseequations are critically discussed. First, the most popular polar formis considered. Its properties, known in literature for a multitude ofspecific problems, are here proven for the general case. Moreover, thedrawbacks encountered in the stability analysis of incomplete motions(i.e. motions containing some zero amplitudes) are discussed as aconsequence of the fact the equations are not in standard normal form.Second, a so-called Cartesian rotating form is introduced, which makesit possible to evaluate periodic solutions and analyze their stability,even if they are incomplete. Although the rotating form calls for theenlargement of the space and is not amenable to analysis of transientmotions, it systematically justifies the change of variables sometimesused in literature to avoid the problems of the polar form. Third, amixed polar-Cartesian form is presented. Starting from the hypothesisthat there exists a suitable number of amplitudes which do not vanish inany motion, it is proved that the mixed form leads to standard formequations with the same dimension as the polar form. However, if suchprincipal amplitudes do not exist, more than one standard form equationshould be sought. Finally, some illustrative examples of the theory arepresented.     

On the Constitutive Equations of Some Weakly‐Textured Materials

(Accepted May 16, 1997)     

Weakly Nonlinear Gravity Three-Dimensional Unbounded Interfacial Waves: Perturbation Method and Variational Formulation

Weakly non-linear behaviour of interfacial short-crested waves with current is presented in this paper. Two approaches are used to determine analytical solutions. First, a perturbation method was applied to determine the fifth-order solutions. The advantage of this method is that it allows for the determination of the harmonic resonance condition which is one of the major short-crested waves characteristics. The second method is Whitham’s Lagrangian approach. From this method, we obtained a quadratic dispersion equation. In the linear case, we have shown that there is a critical current beyond which steady wave solutions cannot exist. This critical current is associated with the emergence of instability. For the non-linear case, the critical current increases with the wave amplitude as in the two-dimensional case.

     

Solutions of Weakly Nonlinear Systems of Ordinary Differential Equations Bounded on R

We obtain conditions for the existence of solutions bounded on the entire axis R for weakly nonlinear systems of ordinary differential equations in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes R ₊ and R _–.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 40–51, May 2005.     

Nonlinear Capillary-Gravity Waves on the Charged Surface of an Ideal Fluid

A second-order asymptotic expression for the profile of a capillary-gravity wave traveling over the charged surface of an ideal incompressible fluid is calculated analytically. Two types of steady-state profiles of nonlinear periodic capillary-gravity waves are found. For a certain fixed dimensionless surface charge the shape of the tops of the nonlinear waves changes: from blunt to pointed for short waves and from pointed to blunt for long waves.