A universal bifurcation mechanism arising from progressive hydroelastic waves

A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schr?dinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

Invariant Manifolds for Steady Boltzmann Flows and Applications

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of ?1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.     

A new method for the calculation of steady periodic capillary-gravity waves on water of arbitrary uniform depth

This paper presents a method for the calculation of steady periodic capillary-gravity waves on water of arbitrary uniform depth. The method developed by Debiane and Kharif in 1997 for infinite depth is extended to finite depth. The water-wave problem is reduced to a system of nonlinear algebraic equations which is solved using Newton's method. For the resonant configurations, the method does not suffer from the Wilton's failures and is valid for all depths. In addition, it is shown that the method allows the computation of solitary waves and generalized solitary waves.     

Hopf Bifurcation and Exchange of Stability in Diffusive Media

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized perturbations.     

非线性参数激励系统的动力分叉研究

本文针对弹性梁动力曲屈分叉问题,建立了系统的非线性Mathiue方程,较全面地讨论了此类参数激励系统的1/2亚谐分叉特性,指出以往对此类问题的研究得到的只是一种退化情形下的分叉特性,阐述了分叉方程的截断对分叉结果的影响,得到了一些新的结果。文中还介绍了一个模型弹性梁系统分叉响应特性的实测结果,证实了理论分析的可靠性。     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN VARIANT BOUSSINESQ EQUATIONS

The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.     

Free surface flow under gravity and surface tension due to an applied pressure distribution: I Bond number greater than one-third

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behavior of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ > 1/3, and the magnitude and sign of the pressure forcing parameter ɛ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F < 1 but not too close to 1, linear theory gives a good prediction for the numerical solution of the nonlinear problem in the case of bifurcation from the uniform flow. As F approaches 1, the nonlinear terms need to be taken account of. In this case the forced Korteweg-de Vries equation is found to be an appropriate model to describe bifurcations from an unforced solitary wave. In general, it is found that for given values of F < 1 and τ > 1/3, there exists both elevation and depression waves. In some cases, a limiting configuration in the form of a trapped bubble occurs in the depression wave solutions.     

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

     

Nonlinear waves on the surface of a charged liquid. Instability,bifurcation, and nonequilibrium shapes of the charged surface

Plane nonlinear waves in shallow water are described by the Kortewegde Vries equation [1–3]. The present paper contains theoretical investigations of nonlinear waves and nonlinear equilibrium shapes on the surface of a charged liquid. The influence of the field on the velocity and shape of a hydrodynamic soliton is considered. The bifurcation of the equilibrium shapes is investigated. Problems of the equilibrium shapes of a charged liquid are solved in the nonlinear formulation of the dynamics of nonlinear solitary forms (lunes, trenches) on the surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–102, May–June, 1984.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Standing waves for a two-way model system for water waves

In this paper, we prove the existence of a large family of nontrivial bifurcating standing waves for a model system which describes two-way propagation of water waves in a channel of finite depth or in the near shore zone. In particular, it is shown that, contrary to the classical standing gravity wave problem on a fluid layer of finite depth, the Lyapunov–Schmidt method applies to find the bifurcation equation. The bifurcation set is formed with the discrete union of Whitney's umbrellas in the three-dimensional space formed with 3 parameters representing the time-period and the wave length, and the average of wave amplitude.     

移动载荷作用下浮冰的非线性动力学响应

本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 三阶截断模型的优势在于其对应的三阶NLS具有准确的非线性项系数, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.