Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

Bifurcation of solitons,peakons, and periodic cusp waves for \varvec{\theta }-equation

In this paper, we study the bifurcation of traveling wave solutions for \(\theta \) -equation using the bifurcation method and qualitative theory of dynamical systems. Not only smooth solitons but also explicit peakons and periodic cusp waves are obtained. In addition, we show that a new kind of phase portrait may admit peakons and infinitely many periodic cusp waves, in contrast to the traditional phase portraits. To the best of our knowledge, until now, this phenomenon has not appeared in any other literature. Some results of the previous studies are extended for this study.     

Global Hopf bifurcation of two-parameter flows

The behavior of center-indices, as introduced by J. Mallet-Paret & J. Yorke, is analyzed for two-parameter flows. The integer sum of center-indices along a one-dimensional curve in parameter space is called the H-index. A nonzero H-index implies global Hopf bifurcation. The index H is not a homotopy invariant. This fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero, which we call B-points. To each B-point we assign an integer B-index, such that the H-index relates to the B-indices by a formula such as occurs in the calculus of residues.This formula is easily applied to study global bifurcation of periodic solutions in diffusively coupled two-cells of chemical oscillators and to treat spatially heterogeneous time-periodic oscillations in porous catalysts.Dedicated to the memory of Charlie Conley     

Analytic Description of Layer Undulations in Smectic A Liquid Crystals

We investigate the layer undulations that appear in smectic A liquid crystals when a magnetic field is applied in the direction parallel to the smectic layers. In an earlier work (García-Cervera and Joo in J Comput Theor Nanosci 7:795–801, 2010) the authors characterized the critical field using the Landau–de Gennes model for smectic A liquid crystals. In this paper, we obtain an asymptotic expression of the unstable modes using Γ-convergence theory, and a sharp estimate of the critical field. Under the assumption that the layers are fixed at the boundaries, the maximum layer undulation occurs in the middle of the cell and the displacement amplitude decreases near the boundaries. Our estimate of the critical field is consistent with the Helfrich–Hurault theory. When natural boundary conditions are considered, the displacement amplitude does not diminish near the boundary, in sharp contrast with the Dirichlet case, and the critical field is reduced compared to the one calculated in the classical theory. This is consistent with the experiments carried out by Ishikawa and Lavrentovich (Phys Rev E 63:030501(R), 2001). Furthermore, we prove the existence and stability of the solution to the nonlinear system of the Landau–de Gennes model using bifurcation theory. Numerical simulations are used to illustrate the predictions of the analysis.     

Multiple bifurcation analysis in a ring of delay coupled oscillators with neutral feedback

Spatiotemporal periodic patterns, including phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or anti-phase oscillations are investigated in a ring of bidirectionally coupled oscillators with neutral delay feedback. It is confirmed that neutral feedback makes Hopf bifurcation occur in a larger domain of parameters. We calculate the normal forms near Hopf bifurcation, D _N equivariant Hopf bifurcation and double-Hopf bifurcation in this neutral equation by using the method of multiple scales. Theoretically, the appearance of the in-phase, anti-phase and phase-locked oscillations that we observed in the simulation about a ring of delay coupled Hindmarsh–Rose neurons with neutral feedback is explained.     

Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.     

Dynamics and bifurcations in the weak electrolyte model for electroconvection of nematic liquid crystals: a Ginzburg–Landau approach

In this paper we present the results of a bifurcation study of the weak electrolyte model for nematic electroconvection, for values of the parameters including experimentally measured values of the nematic I52. The linear stability analysis shows the existence of primary bifurcations of Hopf type, involving normal as well as oblique rolls. The weakly nonlinear analysis is performed using four globally coupled complex Ginzburg–Landau equations for the waves' envelopes. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with $O(2)\times O(2)$ symmetry. A rich variety of stable waves, as well as more complex spatiotemporal dynamics is predicted at onset. A temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, is identified. Eckhaus stability boundaries for travelling waves are also determined. The methods developed in this paper provide a systematic investigation of nonlinear physical mechanisms generating the patterns observed experimentally, and can be generalized to any two-dimensional anisotropic systems with translational and reflectional symmetry.     

A microstructure of martensite which is not a minimiser of energy: the X-interface

In this paper we investigate the X-interface, a microstructure observed in Indium-Thallium byBasinski &Christian [5].Ball &James [3] have shown howsimple martensitic microstructures can be represented by sequences of elastic deformations which minimise a free-energy functional. In contrast we show that the X-interfacecannot be represented by such a sequence. In an attempt to understand this result we develop a less restrictive theory based onEricksen's ideas about low-energy modes of deformation for martensitic materials. This theory has some interesting conclusions for the X-interface and Indium-Thallium, for the wedge-like microstructures analysed recently byBhattacharya [6], and for the general problem of microstructures which cannot be represented by minimising sequences. The calculations in this paper apply only to cubic-to-tetragonal transformations.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

We investigate the band-gap structure of the frequency spectrum for elastic waves in a high-contrast, two-component periodic elastic medium. We consider two-dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. In this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large. The main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator-valued functions in the complex plane, such as the generalized Rouché’s theorem. We establish a connection between the band structures and the Dirichlet eigenvalue problem on the elementary hole. We also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is, the wider the band-gap is, provided that the criterion is fulfilled. This phenomenon was reported by Economou and Sigalas (J Acoust Soc Am 95:1734–1740, 1994) who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps. Our analysis in this paper agrees with this experimental finding.     

Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.     

The Regularity and Local Bifurcation of¶Steady Periodic Water Waves

Steady periodic water waves on infinite depth, satisfying exactly the kinematic and dynamic boundary conditions on the free surface, with or without surface tension, are given by solutions of a rather tidy nonlinear pseudo-differential operator equation for a 2π-periodic function of a real variable. Being an Euler-Lagrange equation, this formulation has the advantage of gradient structure, but is complicated by the fact that it involves a non-local operator, namely the Hilbert transform, and is quasi-linear. This paper is a mathematical study of the equation in question. First it is shown that its W ^1,2 solutions are real analytic. Then bifurcation theory for gradient operators is used to prove the existence of (non-zero) small amplitude waves near every eigenvalue (irrespective of multiplicity) of the linearised problem. Finally it is shown that when surface tension is absent there are no sub-harmonic bifurcations or turning points at the outset of the branches of Stokes waves which bifurcate from the trivial solution. Accepted: December 6, 1999     

Symmetry-breaking in periodic gravity waves with weak surface tension and gravity-capillary waves on deep waterBrisure de symétrie dans les vagues de gravité avec une faible tension de surface et les vagues de gravité-capillarité périodiques en eau profonde

The method developed by Debiane and Kharif for the calculation of symmetric gravity-capillary waves on infinite depth is extended to the general case of non-symmetric solutions. We have calculated non-symmetric steady periodic gravity-capillary waves on deep water. It is found that they appear via bifurcations from a family of symmetric waves. On the other hand we found that the symmetry-breaking bifurcation of periodic steady class 1 gravity wave on deep water is possible when it approaches the limiting profile, if it is very weakly influenced by surface tension effects. To cite this article: R. Aider, M. Debiane, C. R. Mecanique 332 (2004).     

Alternate pacing of border-collision period-doubling bifurcations

Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise C ¹ map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations are not well understood. This paper considers a particular class of border-collision bifurcations, i.e., border-collision period-doubling bifurcations. We apply a subharmonic perturbation to the bifurcation parameter, which is also known as alternate pacing, and we investigate the response under such pacing near the original bifurcation point. The resulting behavior is characterized quantitatively by a gain, which is the ratio of the response amplitude to the applied perturbation amplitude. The gain in a border-collision period-doubling bifurcation has a qualitatively different dependence on parameters from that of a classical period-doubling bifurcation. Perhaps surprisingly, the differences are more readily apparent if the gain is plotted versus the perturbation amplitude (with the bifurcation parameter fixed) than if plotted versus the bifurcation parameter (with the perturbation amplitude fixed). When this observation is exploited, the gain under alternate pacing provides a useful experimental tool to identify a border-collision period-doubling bifurcation.     

Bifurcation from Discrete Rotating Waves

Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries. We present a systematic approach to the study of local bifurcation from discrete rotating waves. The approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range. Our results are valid for dynamical systems with finite symmetry group, and more generally, for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group.     

On the existence of the low-frequency surface waves in a porous mediumSur l'existence des ondes de surface basse fréquence en milieux poreux

The existence and propagation of the surface waves at a vacuum/porous medium interface are investigated in the low frequency range. Two types of surface waves are shown to be possible: the generalized Rayleigh wave, which always exists, and the Stoneley wave, which exists for a limited range of wave numbers. Moreover, within the k-domain of existence the Stoneley wave cannot appear for certain values of elastic parameters of the solid phase. The bifurcation behavior of both the Stoneley wave and the Biot (P2) bulk wave, depending on the wave number, is revealed. The asymptotic formulas for the phase velocities of the surface waves are derived. To cite this article: I. Edelman, C. R. Mecanique 332 (2004).     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999     

Analytics of heteroclinic bifurcation in a 3:1 subharmonic resonance

Analytical approximation of heteroclinic bifurcation in a 3:1 subharmonic resonance is given in this paper. The system we consider that produces this bifurcation is a harmonically forced and self-excited nonlinear oscillator. This bifurcation mechanism, resulting from the disappearance of a stable slow flow limit cycle at the bifurcation point, gives rise to a synchronization phenomenon near the 3:1 resonance. The analytical approach used in this study is based on the collision criterion between the slow flow limit cycle and the three saddles involved in the bifurcation. The amplitudes of the 3:1 subharmonic response and of the slow flow limit cycle are approximated and the collision criterion is applied leading to an explicit analytical condition of heteroclinic connection. Numerical simulations are performed and compared to the analytical finding for validation.