Pair Of Solitary Waves Research Articles

Waves in strongly nonlinear Gardner-like equations on a lattice

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka–Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.

Numerical investigation of solitary wave interaction with a flapper wave energy converter using incompressible SPH method

In this research, mesh-free incompressible smoothed particle hydrodynamics (ISPH) was used to study the oscillatory behavior of a flapper wave energy converter located at the end of a numerical tank under the impact of solitary waves. A two-step projection approach including prediction and correction steps was employed to conduct temporal discretization of the governing equations for inviscid fluid flows. At different times, the pressure fields were implicitly calculated by the ISPH method to solve a pressure Poisson equation (PPE) derived upon the enforcement of the incompressibility conditions. To the authors’ knowledge, not many mesh-less numerical studies have been accomplished addressing different aspects of absorption of wave power using oscillating flappers. In this study, effects of different parameters, including solitary wave height, flapper spring stiffness, moment of inertia of the wall withstanding wave loads, initial angle of the flapper and initial longitudinal wave crest–flapper distance for a pair of solitary waves, were examined to analyze the wave energy absorption performance of the flapper. The results showed the necessity of adopting optimum spring stiffness to maximize the wave power absorption. Otherwise, wave overtopping may occur for sub-optimal spring stiffness values, while the absorbed wave power was lower than maximal for too rigid flappers. It was further found that the flapper oscillation period decreased with increasing the spring stiffness, with the other parameters imposing no significant effect on the period. As another finding, it was figured out that the moment of inertia of the downstream wall must be optimal to capture maximum wave power. The initial angle of the flapper was also found to be an important factor contributing to wave power absorption. The maximal absorbed wave power was achieved when a pair of solitary waves was flowing toward the flapper in such a way that the second wave hit the flapper at the end of the flapper period, causing some resonance phenomenon. Damping process of the oscillatory motion due to the flow viscosity and flapper friction was also considered. The results of the present work can be used to maximize the wave power absorption by adjusting the wave converter design considering the wave height, spring stiffness, moment of inertia and longitudinal distance of wave crest between two successive solitary waves.

Nonlinear dispersive regularization of inviscid gas dynamics

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in one dimension, singularities in the Hopf equation can be non-dissipatively smoothed via Korteweg–de Vries (KdV) dispersion. In this paper, we develop a minimal conservative regularization of 3D ideal adiabatic flow of a gas with polytropic exponent γ. It is achieved by augmenting the Hamiltonian by a capillarity energy β(ρ)(∇ρ)2. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy, and entropy using the standard Poisson brackets is β(ρ) = β*/ρ for constant β*. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of ρ, which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary waves, and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine–Hugoniot conditions. Nevertheless, in one-dimension, for γ = 2, numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves, which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocusing nonlinear Schrödinger equation (cubically nonlinear for γ = 2), with β* playing the role of ℏ2. Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV and nonlinear Schrödinger equations to include the adiabatic dynamics of density, velocity, pressure, and entropy in any dimension.

Error Estimates for Galerkin Approximations of the Serre Equations

We consider the Serre system of equations, a nonlinear dispersive system that models two-way propagation of long waves of not necessarily small amplitude on the surface of an ideal fluid in a channel. We discretize in space the periodic initial-value problem for the system using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove an optimal-order $L^{2}$-error estimate for the resulting semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge--Kutta scheme for time stepping, we construct a highly accurate fully discrete scheme in order to approximate solutions of the system, in particular, solitary-wave solutions, and study numerically phenomena such as the resolution of general initial profiles into sequences of solitary waves, and overtaking collisions of pairs of solitary waves propagating in the same direction with different speeds.

Interaction of solitary waves for the generalized KdV equation

We consider a class of generalized KdV equations with a small parameter and nonlinearities of the type um. We create a finite differences scheme to simulate the solution of the Cauchy problem and present some numerical results for the problem of the solitary waves interaction. In particular, we consider sufficient condition under which pairs of solitary waves interact, in the asymptotic sense, in accordance with the soliton scenario.

Dynamics of Solitary Wave Scattering in the Fermi-Pasta-Ulam Model

In this Letter, we study the scattering dynamics of a pair of solitary waves in the Fermi-Pasta-Ulam model with interaction potential V(x)=alphax(2)/2+x(4)/4 and establish a quantitative connection between the scattering property and the energy transport behavior. The energy and momentum conservation laws are obtained and the scattering rates of solitary waves are calculated. Our studies suggest that the anharmonic limit model with alpha=0 can be taken as a paradigm model for studying lattice solitary waves.

An optical soliton pair among absorbing three-level atoms

The motion of a pair of solitons propagating through an absorbing three-level system in the lambda configuration is analysed. In the course of this analytic analysis, the existence condition for solitary wave pairs is derived. This condition allows for two possible velocities of the soliton pair.

Periodic solitary waves for two coupled nonlinear Klein–Gordon and Schr dinger equations

Six new periodic analytic solitary-wave solutions of two coupled nonlinear Klein–Gordon and Schrodinger equations are presented. Besides their application to optical solitary wavepairs due to cross-phase modulation or modal birefringence, their application to the (presently) unphysical regime and the significance of symmetry are discussed. For two related special sets of the two coupled equations, additional possible analytic solutions are presented.

Bifurcation of gap solitons through catastrophe theory.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic band gap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical "morphing" process.

Soliton interaction in a nonlinear waveguide in the presence of resonances.

The simultaneous propagation of two optical pulses through a nonlinear dispersive medium composed of a resonant three-level system is investigated. By choosing a soliton of area 4 pi and order N=2 at the pump frequency, together with a weaker pulse with a sech profile at the signal frequency, we show that the pump soliton breaks up into a pair of solitary waves which are cloned to the signal frequency. Due to a combination of coherent population trapping and nonlinear dispersive effects, the pair interacts in a repulsive fashion so that the taller wave travels faster than the shorter one.

Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.

Matched pulse propagation in a three-level system

The B\"{a}cklund transformation for the three-level Maxwell-Bloch equation is presented in the matrix potential formalism. By applying the B\"{a}cklund transformation to a constant electric field background, we obtain a general solution for matched pulses (a pair of solitary waves) which can emit or absorb a light velocity solitary pulse but otherwise propagate with their shapes invariant. In the special case, this solution describes a steady state pulse without emission or absorption, and becomes the matched pulse solution recently obtained by Hioe and Grobe. A nonlinear superposition rule is derived from the B\"{a}cklund transformation and used for the explicit construction of two solitons as well as nonabelian breathers. Various new features of these solutions are addressed. In particular, we analyze in detail the scattering of "invertons", a specific pair of different wavelength solitons one of which moving with the velocity of light. Unlike the usual case of soliton scattering, the broader inverton changes its sign through the scattering. Surprisingly, the light velocity inverton receives time advance through the scattering thereby moving faster than light, which however does not violate causality.

Solitary waves and atomic population transfer via the continuum

Exact analytic results are presented that represent a pair of solitary waves that can propagate through an atomic medium with their shapes invariant while producing transfer of atoms or molecules from one bound state to another via a continuum. The speed of the waves and the amount of population transfer are given in terms of the wave amplitudes. A general (not solitary wave) solution for the case of a single wave is presented, together with a nonlinear Beer's law of absorption.

Matched optical solitary waves for three- and five-level systems.

Exact analytic results are presented that give a general solution for a pair of solitary waves which can propagate through a three- and a five-level system with their shapes invariant. These solitary waves vary widely in shape and form: from ones for which the pulses have similar shape to ones which have very different but complementary shapes. A general type of solitary-wave pair which is insensitive to small perturbations is identified.

Bright–dark solitary wave pairs resulting from stimulated Raman scattering and loss

We show that under the combined effects of stimulated Raman scattering and loss, the pump and Stokes waves can form a bright-dark solitary wave pair. Our analysis takes into account dispersive and walk-off effects as well as the processes of self-phase modulation and cross-phase modulation. The physical origin of this formation is discussed.

Solitary waves in the resonant phenomenon between a surface gravity wave packet and an internal gravity wave

A two-layer inviscid incompressible fluid system of intermediate depth is considered. A multiple-scales perturbation technique is applied to the basic equations and boundary conditions for a two-layer fluid system to derive a system of weakly nonlinear partial integrodifferential equations governing the resonant interaction between a surface gravity wave packet and an internal gravity wave at an intermediate depth, providing a bridge between the existing shallow and deep fluid theories. The convolution integral term in these equations accounts for the dispersion in the lower-layer fluid. An iterative fast Fourier transform scheme is developed to find solitary wave solutions to this system of equations. The overtaking collision of two pairs of solitary waves, simulated using a spectral method, is found to be inelastic. It is found that the amplitude of the solitary waves changes slightly after the collision. The phase shifts these solitary waves undergo was calculated numerically.

On long nonlinear internal waves over slope-shelf topography

An experimental and theoretical study of the propagation and stability of long nonlinear internal waves over slope-shelf topography is presented. A generalised Korteweg-de Vries (KdV) equation, including the effects of nonlinearity, dispersion, dissipation and varying bottom topography, is formulated and solved numerically for single and rank-ordered pairs of solitary waves incident on the slope. The results of corresponding laboratory experiments in a salt-stratified system are reported. Very good agreement between theory and experiment is obtained for a range of stratifications, topography and incident-wave amplitudes. Significant disagreement is found in some cases if the effects of dissipation and higher-order (cubic) nonlinearity are not included in the theoretical model. Weak shearing and strong breaking (overturning) instabilities are observed and found to depend strongly on the incident-wave amplitude and the stratification on the shelf. In some cases the instability of the lowest-mode wave leads to the generation of a second-mode solitary wave. The application of these findings to the prediction and interpretation of field data is discussed.