Asymptotic and numerical analysis of thermoelastic waves in an isotropic plate

^** Email: n.scott{at}uea.ac.uk This paper considers the propagation of a plane thermoelasticwave in an infinite homogeneous isotropic plate subjected toeither isothermal or thermally insulated traction-free boundaryconditions. The primary concerns are the derivation and numericalexamination of the dispersion relations along with the developmentof asymptotic models for the fundamental branches of the dispersionrelations. Thermal dissipation is also discussed.     

Nonlinear responses from the interaction of two progressing waves at an interface

For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.     

Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.     

2‐D solitary waves in thermal media with nonsymmetric boundary conditions

Optical solitary waves and their stability in focusing thermal optical media, such as lead glasses, are studied numerically and theoretically in (2 + 1) dimensions. The optical medium is a square cell and mixed boundary conditions of Newton cooling and fixed temperature on different sides of the cell are used. Nonlinear thermal optical media have a refractive index which depends on temperature, so that heating from the optical beam and heat flow across the boundaries can change the refractive index of the medium. Solitary wave solutions are found numerically using the Newton conjugate‐gradient method, while their stability is studied using a linearized stability analysis and also via numerical simulations. It is found that the position of the solitary wave is dependent on the boundary conditions, with the center of the beam moving toward the warmer boundaries, as the parameters are varied. The stability of the solitary waves depends on the symmetry of the boundary conditions and the amplitude of the solitary waves.     

Nonlinear electrodynamic lensing of electromagnetic waves in a magnetic dipole field

We consider propagation of electromagnetic waves in magnetic dipole and gravitational fields proceeding in accordance with the nonlinear vacuum electrodynamics laws. We derive formulas describing the effect of nonlinear electrodynamic lensing of electromagnetic waves in the magnetic dipole field. We show that rotation of the magnetic dipole moment about an axis noncoincident with this moment leads to a nonlinear electrodynamic modulation of the electromagnetic radiation intensity by frequencies that are multiples of the dipole rotation frequency. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 85–94, January, 2007.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Multi solitary waves for nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let R_k(t,x) be K arbitrarily given solitary waves of the equation with different speeds v₁,v₂,…,v_K. In this paper, we prove that there exists a solution u(t) of the equation such that

     

Ground States to the Generalized Nonlinear Schr(o)dinger Equations with Bernstein Symbols

This paper concerns with existence and qualitative properties of ground states to generalized nonlinear Schr(o)dinger equations (gNLS) with abstract symbols.Under some structural assumptions on the symbol,we prove a ground state exists and it satisfies several fundamental properties that the ground state to the standard NLS enjoys.Furthermore,by imposing additional assumptions,we construct,in small mass case,a nontrivial radially symmetric solution to gNLS with H1-subcritical nonlinearity,even if the natural energy space does not control the H1-subcritical nonlinearity.     

Nonlinear waves in two-dimensions generated by variable pressure acting on the free surface of a heavy liquid

The development of nonlinear waves on the free surface of a heavy liquid initially at rest is treated analytically in cases where the external pressure force of limited power is distributed over a large area in the free surface but is otherwise arbitrary. In [1] approximate (up to small terms of higher order) solution of the problem is obtained in the form of functional series. In the present article the convergence theorems for the series are proved. When the pressure varies with time sinusoidally, the sums of the series are found in closed form. By passing to the limit in the solution as time goes to infinity, the form of the nonlinear steady-state wave is found. According to the solution, when the steady-state wave gets away from the variable pressure zone, a long chain of structures develops similar to so called Kelvin-Helmholtz billows. The existence of nonlinear standing waves is discovered, which have a finite number of nodes in the free surface infinite in extent, and the frequency spectrum and the form of these waves are found explicitly.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.     

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Long Time Behavior of Nonlinear Strain Waves in Elastic Waveguides

The initial-boundary value problem of the propagation of nonlinear longitudinal elastic waves in an initially strained rod is considered. The rod is assumed to interact with the surrouding elastic and viscous external medium. The long time behavior of solutions is derived ancl global attractors in E_1 space is obtained.     

On interface waves in misoriented pre-stressed incompressible elastic solids

Some relationships, fundamental to the resolution of interfacewave problems, are presented. These equations allow for thederivation of explicit secular equations for problems involvingwaves localized near the plane boundary of anisotropic elastichalf-spaces, such as Rayleigh, Scholte, or Stoneley waves. Theyare obtained rapidly, without recourse to the Stroh formalism.As an application, the problems of Stoneley wave propagationand of interface stability for misaligned predeformed incompressiblehalfspaces are treated. The upper and lower half-spaces aremade of the same material, subject to the same prestress, andare rigidly bonded along a common principal plane. The principalaxes in this plane do not, however, coincide, and the wave propagationis studied in the direction of the bisectrix of the angle betweena principal axis of the upper half-space and a principal axisof the lower half-space.     

Effect of vertical stratification on characteristics and energy of nonlinear internal solitary waves from a numerical model

A numerical model is set up to study the impact of changes in vertical stratification on the properties of internal solitary waves (ISWs) generated by tidal flow over a ridge. Based on modifications of the observed stratification with a secondary thermocline over a main one in the South China Sea, the effects of five kinds of stratification on the characteristics and energy conversion of ISWs are investigated. In general, the isopycnal undergoing maximum displacement in ISWs is from slightly below the main thermocline. When the stratification below the ridge crest is reduced, the wave amplitude and the number of ISWs in a wave packet increase, while the phase speed, the wave half-width, the sum of ISW kinetic energy (KE) and available potential energy (APE) and the ratio of KE to APE decrease. When the stratification in the upper layer is reduced, the ISW amplitude, the number of ISWs, the phase speed and the sum of KE and APE decrease, while the wave half-width and the ratio of KE to APE increase. If the main thermocline is over the secondary one, the ISW amplitude, the wave half-width, the sum of KE and APE and the ratio of KE to APE increase, while the phase speed reduces. For stratification with two thermoclines, the ISW phase speed increases but the half-width decreases. In addition, the ratio of baroclinic to barotropic energy is found to be between 10% and 40%, and the ratio of ISW KE to APE is between 1.30 and 1.65. It is also shown that the ratio of KE to APE for the stratification with two thermoclines is about 2–6% larger than that for the stratification with only one thermocline. If the thermocline is lowered by about 30 m (0.064 of the total water depth), the ratio of KE to APE reduces by about 10%.     

Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic function , which is essentially the Jacobian elliptic function with one non‐trivial parameter . More precisely, we show that this function fulfills equations of the form for all . We give explicit expressions for the coefficients and for given s. Moreover, we show that for any s the set of functions constitutes a basis for . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.     

Semiclassical stationary states of Nonlinear Schrödinger equations with an external magnetic field

In this paper we obtain multiple solutions of the nonlinear Schrödinger equation with an external magnetic field

     

一个非线性积分微分方程的数值研究

本文采用Ｆｏｒｎｂｅｒｇ和Ｗｈｉｔｈａｍ的拟谱方法，数值研究了一个非线性积分微分方程的初值问题：Ａｔ＋６ＡＡｘ＋１２｜ｌｎε｜∫＋∞－∞Ａ（ｘ′，ｔ）（ｘ′－ｘ）２＋ε２｛｝１／２ｄｘ′＝０发现当ε很小时，其解与ＫｄＶ方程的解接近·较大的ε和初始条件对解的影响是很大的·     

On the stability of the solitary waves to the rotation Benjamin‐Ono equation

In this paper, we study several aspects of solitary wave solutions of the rotation Benjamin‐Ono equation. By solving a minimization problem on the line, we construct a family of even travelling waves ψ_c,γ. We then prove the uniqueness of even ground states associated with large speed and their orbital stability. Note that this improves the results in Esfahani and Levandosky, where only the stability of the set of ground states is proven.