Waves in microstructured solids and the Boussinesq paradigm

The emergence of soliton trains and interaction of solitons are analyzed by using a Boussinesq-type equation which describes the propagation of bi-directional deformation waves in microstructured solids. The governing equation in the one-dimensional setting is based on the Mindlin model. This model includes scale parameters which show explicitly the influence of the microstructure in wave motion. As a result the governing equation has a hierarchical structure. The analysis is based on numerical simulation using the pseudospectral method. It is shown how the number of solitons in emerging trains depends on the initial excitation. The head-on collision of emerged solitons is not fully elastic due to radiation but the solitons preserve their identity after collision and the speed of solitons is retained while the radiation keeps a certain mean value. That is why we have kept through this paper the notion of solitons.     

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志,特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义.本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先,考虑几何非线性效应和压电耦合效应,利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次,采用Ja...     

On the theory of three-dimensional multihump solitons in active-dissipative media

Stable localized nonlinear coherent structures, i.e. solitons, play a key role in the stochastization of the processes occurring in active-dissipative media. In this study, three-dimensional multi-hump solitons are investigated for a model equation which qualitatively describes the wave processes in some physical systems. The existence of 3D multihump solitons is demonstrated numerically and the soliton behavior is studied. The results are generalized to describe multihump solitons in descending viscous-fluid layers [1]. An unusual physical phenomenon observed in experiments [1], namely, stable two-hump coherent structures on the surface of a downflowing viscous-fluid layer, is explained qualitatively.     

Mechanism of the Emergence of Rogue Waves As a Result of the Interaction between Internal Solitary Waves in a Stratified Basin

The features of the interaction between internal solitary waves are investigated within the framework of the completely integrable Gardner equation with positive cubic nonlinearity. It is shown that the soliton polarity affects radically the result of the interaction between the solitons. The role of the pair interactions between solitons of different polarities proceeding when rogue waves emerge in the soliton fields in a stratified basin is demonstrated. The effect of such interactions on the higher-order moments of the wave field is studied.     

Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

In nonlinear science, the interactions among solitons are well studied because the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, taking the Kadomtsev–Petviashvili (KP) equation as an illustration model, a new method is established to find interactions among different types of nonlinear waves. The nonlocal symmetries related to the Darboux transformation (DT) of the KP equation is localized after embedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions. It is shown that the essential and unique role of the DT is to add an additional soliton on a Boussinesq-type wave or a KdV-type wave, which are two basic reductions of the KP equation.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

KdV方程孤立子及其相互作用的Petrov-Galerkin有限元数值模拟

本文给出一种Petrov-Galerkin有限元方法,并用以求Kdv方程各种初值问题的数值解,包括孤立波进波解,多个孤立于的相互作用,孤立子与振荡尾波等,所得结果与分析解及其它数值结果作了比较,表明本方法精度高、稳定性好,几乎没有高频伪振荡,计算程序简洁、明瞭,经济实用。     

Study on the head-on collisions of two-dimensional trains of solitons

The head on collisions of trains of solitons induced by a two-dimensional submerged elliptical cylinder at critical speed in shallow water are studied based on velocity potential theory. The boundary value problems are solved through boundary element method (BEM). The nonlinear free surface boundary conditions are satisfied. The mixed Euler–Lagrangian method is adopted to track the free surface through a time stepping scheme. The effects of thickness and velocity of the elliptical cylinder on the evolution of solitary waves have been investigated. Two sets of solitons are truncated from these trains of solitary waves. The head-on collisions of these solitons have been simulated. The wave profiles and velocity fields during collision have been analysed. The propagation of solitary waves is the transmissions of kinetic energy and the collision processes are the results of the dynamic balance of potential energy and kinematic energy.     

Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-α model of single-phase lossless fluids

A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg-de Vries equation from shallow-water wave theory.     

Experimental investigation of cylindrically diverging solitons in an electric lattice

The results of an experimental investigation of cylindrical solitons in a two-dimensional electric LC-lattice are given. It is shown that in the continuum limit, propagation of cylindrical waves far from the center of symmetry in such a lattice may be described for each ray tube by a known modification of the Korteweg-de Vries equation which takes account of the cylindrical divergence. The dispersion term in this equation depends on the direction of wave propagation relative to the direction of the main axes of the lattice. Formation of solitons from non-soliton-shaped pulses was observed. The variations of soliton amplitude and duration with distance have been determined. They agree well with the numerical calculations by Maxon & Viecelli [2] and Dorfman [9]. Comparison of the obtained experimental data with the known theoretical laws of amplitude attenuation for diverging solitons [2, 12, 14] seems to favor the validity of the law A r^−2/3 rather than A r^−1/2.     

“Leapfrogging” solitons in a system of coupled KdV equations

A system of two KdV equations coupled by small linear dispersive terms is considered. This system describes, for example, resonant interaction of two transverse gravity internal wave modes in a shallow stratified liquid. In the framework of an approach based on Hamilton's equations of motion, evolution equations for parameters of two solitons belonging to different wave modes are obtained in the adiabatic approximation. It is demonstrated that when the solitons' velocities are sufficiently lose, the solitons may form a breather-like oscillatory bound state, which provides a natural explanation for recent numerical experiments demonstrating “leapfrogging” motion of the two solitons. The frequency and the maximum amplitude of the “breather”'s internal oscillations are obtained. For the case when the relative velocity of the solitons is not small, perturbation-induced phase shifts of the two colliding free solitons are calculated. Then emission of radiation (small-amplitude quasilinear waves) by an oscillating “breather,” also detected in the numerical experiments, is investigated in the framework of the perturbation theory based on the inverse scattering transform. The intensity of the emission is calculated. Radiative effects accompanying collision of the free solitons are also investigated.     

Dynamics of optical solitons and nonautonomous complex wave solutions to the nonlinear Schrodinger equation with variable coefficients

Variable coefficients nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counter constant coefficients in some real-world problems. Under consideration is a nonlinear variable coefficients Schrödinger’s equation with spatio-temporal dispersion in the Kerr law media. We are aimed at constructing novel solutions to the equation under consideration. Bright and combined dark–bright optical solitons are successfully revealed with aid of the complex amplitude ansatz scheme. Using two test functions, two nonautonomous complex wave solutions in dark and bright optical solitons forms are successfully revealed. The effect of the variable coefficients on the reported results can be clearly seen on the 3-dimensional and contour graphs.

     

Interaction soliton–sable dans un canal en eau peu profonde

Interaction between solitons and a sandy bed in shallow water is investigated. In our experiments, solitons are generated on the background of a harmonic wave, in a wave flume used in resonant mode. It is found that the sand ripples formed by the solitons propagation induce a significant decrease of solitons amplitude and of the phase shift between the soliton and the harmonic wave. However, the amplitude of the harmonic wave is approximately constant. The possible physical processes of such behaviour for the soliton amplitude and for the harmonic wave amplitude are discussed. To cite this article: F. Marin et al., C. R. Mecanique 333 (2005).     

Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.     

Dark spatiotemporal optical solitary waves in self-defocusing nonlinear media

Dark three-dimensional spatiotemporal solitons or the “dark light bullets” in the self-defocusing nonlinear media with equal diffraction and dispersion lengths are demonstrated analytically. Our results show that the main characteristic of the dark light bullets can be described by the cylindrical Korteweg–de Vries (CKdV) equation. The dark wave packets are composed of the single-layer and multilayer toroidal rings. For the multilayer rings, there exist small inner rings enclosed in a large ring-shaped toroidal structure, when one chooses different orders of the soliton solutions of the CKdV equation. The radius of dark ring solitons increases with the propagation distance. Present results provide a feasible method for controlling the fundamental structure of these beams in the self-defocusing nonlinear media.     

The integrable Boussinesq equation and it’s breather,lump and soliton solutions

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

     

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.