Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lε$L_{\epsilon }$ (where ε is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line  , with in addition a simple eigenvalue in 0. Moreover, for ε<0$\epsilon <0$, there is a pair of imaginary eigenvalues which meet in 0 when ε=0$\epsilon =0$ and which disappear in the essential spectrum for ε>0$\epsilon >0$. For ε>0$\epsilon >0$ small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T₀,∞)$(T_0,\infty )$ possibly except a discrete set of isolated points.     

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.     

On a two-component Degasperis–Procesi shallow water system

We consider a two-component Degasperis–Procesi system which arises in shallow water theory. We analyze some aspects of blow up mechanism, traveling wave solutions and the persistence properties. Firstly, we discuss the local well-posedness and blow up criterion; a new blow up criterion for this system with the initial odd condition will be established. Finally, the persistence properties of strong solutions will also be investigated.     

Propagation of shear waves in piezoelectric “superlattice-substrate” structures

The results of a computer simulation of the dispersion relations for the propagation of shear waves in piezoelectric “superlattice-substrate” structures are analyzed. The superlattice consists of a finite number of layers and is made up of materials with 6mm symmetry. The dispersion relations are obtained using a formulation for periodic hamiltonian systems. This approach makes it possible to account for the anisotropy, the piezoelectric interaction of the mechanical and electric fields, and an arbitrary number of layers in the superlattice. Numerical results are presented for CdS-ZnO layers. Selective spatial localization of acoustic modes is demonstrated for different spectral regions. The effects of the ordering of the superlattice layers, of their number, and of the boundary conditions on the dispersion spectra and on the form of the shear wave motion are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, and Institute of Metal Physics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 148–156, 1999.     

Residual saturated porous media – from oscillating water blobs to waves in ground earth

A model for wave propagation in residual saturated porous media is presented distinguishing enclosed fluid clusters with respect to their eigenfrequency and damping properties. The additional micro-structure information of cluster specific damping is preserved during the formal upscaling process and allows a stronger coupling between micro- and macro-scale than characterisation via eigenfrequencies alone. A numerical example of sandstone filled with air and liquid clusters of different eigenfrequency and damping distributions is given. If energy dissipation due to viscous damping dominates energy storage due to cluster oscillations, the damping distribution is more influential than the eigenfrequency distribution and vice versa. Spreading the damping distribution around a constant mean value supported the effect of capillary forces and spreading the eigenfrequency distribution around a constant mean value supported the effect of viscous damping in the investigated samples. For a wide distribution of the liquid clusters' damping properties, two damping mechanisms of propagating waves occur at the same time: damping due to viscous effects (for highly damped clusters) and energy storage by cluster oscillations (for underdamped clusters). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects. In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock. The paper develops the method of estimating attraction domain for traveling waves we have studied before.     

Comments on “Elasto-plastic solution for shallow tunnel in semi-infinite space”

This paper considers the approach proposed by Zou et al. for determining the plastic zone around a shallow circular tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. The paper aims to analyze the correctness of the analytic expressions for elastic and plastic stress, as well as the reasonableness of the procedure building for solving the elasto-plastic interface. Finally, a promising way of solving this problem more exactly is presented.     

Variational derivation of two-component Camassa–Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa–Holm system (1). We show that the two-component Camassa–Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.     

Transport solutions of the Lamé equations and shock elastic waves

The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds are presented.     

Stable standing waves for a class of nonlinear Schrödinger-Poisson equations

We prove the existence of orbitally stable standing waves with prescribed L ²-norm for the following Schrödinger-Poisson type equation

$i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \quad \rm{in} \quad \mathbb R^{3},$

when \({p\in \left\{ \frac{8}{3}\right\}\cup (3,\frac{10}{3})}\). In the case \({3 < p < \frac{10}{3}}\), we prove the existence and stability only for sufficiently large L ²-norm. In case \({p=\frac{8}{3}}\), our approach recovers the result of Sanchez and Soler (J Stat Phys 114:179–204, 2004) for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In the final section, a further application to the Schrödinger equation involving the biharmonic operator is given.

     

Periodic two-phase “Rogue waves”

A family of periodic (in x and z) two-gap solutions of the focusing nonlinear Schrödinger equation is constructed. A condition under which the two-gap solutions exhibit the behavior of periodic “rogue waves” is obtained.     

Diagonally implicit Runge–Kutta methods for 3D shallow water applications

We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized spatial domain is not involved in the stability condition. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

The orbital stability of standing waves of nonlinear Schrödinger equations with a general nonlinear term is investigated in this paper. We study the corresponding minimizing problem with L ²-constraint: $$E_\alpha = \inf\left\{\frac{1}{2}\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx - \int\limits_{\mathbb{R}^N} F(|u|) dx; u \in H^1(\mathbb{R}^N), \|u\|_{L^2(\mathbb{R}^N)}^2=\alpha\right\}.$$ We discuss when a minimizing sequence with respect to E _α is precompact. We prove that there exists α ₀ ≥ 0 such that there exists a global minimizer if α > α ₀ and there exists no global minimizer if α < α ₀. Moreover, some almost critical conditions which determine α ₀ = 0 or α ₀ > 0 are established, and the existence results with respect to ${E_{\alpha_0}}$ under some conditions are obtained.     

Dusty shock wave interaction – formation of fully dispersed waves and particle focusing effect

The problem of regular (symmetric and asymmetric) interaction of plane shock waves in a steady-state dusty gas flow is considered. For near-sonic flows with a fairly high particle mass loading, the possibility of the formation of wave structures is revealed, in which either all or only some of the incident or reflected waves degenerate into so-called fully dispersed waves, i.e. zones in which no discontinuities appear in the parameters of each phase. For stronger shock waves and low particle mass concentration, the effect of aerodynamic particle focusing and the formation of a narrow high-concentration beam of particles behind the point of the interaction of the waves are detected on the basis of parametric numerical calculations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applicable symbolic computations on dynamics of small-amplitude long waves and Davey–Stewartson equations in finite water depth

Some classes of nonlinear partial differential equations can be reduced to more tractable single nonlinear equations via the lowest order of the perturbed reductive technique. The nonlinear and dispersive waves of the shallow-water model are investigated throughout a finite depth of fluid under the influence of surface tension and gravitational force in an attempt to derive the Davey–Stewartson equations (DSEs). Dispersion properties of the model and conservation laws of the DSEs are studied. We apply the Painlevé analysis to investigate the integrability of the DSEs and to construct the Bäcklund transformation via the truncation Painlevé expansion. By employing the Bäcklund transformation, the Hamiltonian approach and the $(G^{{}^{\prime }}/G)$-expansion method to the DSEs, new traveling solitary and kink wave solutions are obtained. It is revealed that the amplitudes of waves decrease with increasing Ursell parameter. The trend of the wave profile does not change with time. In addition, through the Hamiltonian approach, it is found that the amplitude of the waves increases with increasing energy constant. Furthermore, the phase portrait method is applied to the resulting nonlinear first-order differential equations of the DS model to reveal its stability.     

Dynamics of the breather waves,rogue waves and solitary waves in an extend Kadomtsev–Petviashvili equation

Under investigation in this work is an extend Kadomtsev–Petviashvili (eKP) equation, which appears in the study of multi-component plasmas. By considering Bell’s polynomials, an effective and straightforward way is presented to succinctly derive its bilinear form and soliton solutions. Moreover, the homoclinic breather limit method is employed to construct the breather wave and rogue wave solutions of the equation. Finally, the dynamic behaviors of breather waves, rogue waves and solitary waves are discussed with graphic analysis. It is hoped that our results can be useful for explaining and enriching the dynamic behavior of these KP-type equations.     

Erratum to “On the Cauchy problem for a model equation for shallow water waves of moderate amplitude” [Nonlinear Anal. RWA 14 (5) (2013) 2022–2026]

The local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime was proved in Duruk Mutlubaş (2013). In this paper, we correct the mistake made in the proof of the main result and give the appropriate assumptions and corresponding results.     

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

We study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.