Fifth order evolution equation for long wave dissipative solitons

Third and fifth order nonlinear wave equations which arise in the theory of water waves possess solitary and periodic traveling waves. Solitary waves also arise in systems with dissipation and instability where a balance between these effects allows the existence of dissipative solitons. Here we search for a model equation to describe long wave dissipative solitons including fifth order dispersion. The equation found includes quadratic and cubic nonlinearities. For periodic solutions in a small box we characterize the rate of growth, and show that they do not blow up in finite time. Analytic solutions are constructed for special parameter values.     

On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation

We study the stability of a four-parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation under two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability under perturbation for both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a theory of nonlinear stability under periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.     

A new compact difference scheme for second derivative in non-uniform grid expressed in self-adjoint form

A single-parameter family of self-adjoint compact difference (SACD) schemes is developed for discretizing the Laplacian operator in self-adjoint form. Developed implicit scheme is formally second-order accurate (with respect to truncation error) with a free parameter, α which helps control the numerical properties in the spectral plane. The SACD scheme is analyzed in the spectral plane for its resolution properties for both periodic and non-periodic problems using the matrix spectral analysis [T.K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind schemes, J. Comput. Phys. 192 (2) (2003) 677–694]. The major objective here is to identify the advantages of the new scheme over the traditional explicit second order CD2 scheme, in discretizing the Laplacian operator in self-adjoint form. This appears in Navier–Stokes equation and in other transport equations and boundary value problems (bvp) expressed in orthogonal co-ordinate systems, either in physical or in transformed plane. We also compare the developed method with the higher order compact schemes for non-uniform grids. To demonstrate the accuracy of SACD scheme we have tested it for: (i) bi-directional wave propagation problem, given by the second order wave equation and (ii) an elliptic bvp, as in the Stommel ocean model for the stream function. These examples help infer the properties of SACD scheme when solving different types of partial differential equations. Most importantly the effects of grid-stretching and choice of value of the free parameter (α) are investigated here. We also compare the present implicit compact method with explicit compact method known as the higher order compact (HOC) method.Also, the practical applications of the SACD scheme are explored by solving the Navier–Stokes equation for the cases of: (a) a flow inside a lid-driven cavity and (b) the receptivity and instability of an external adverse pressure gradient flow over a flat plate. In the former, unsteadiness of the flow is captured and in the latter, the receptivity of the flow is studied in causing flow instability by triggering Tollmien–Schlichting waves. The new scheme shows a marked improvement over the explicit scheme for low Reynolds number steady flow in lid driven cavity. Whereas for the flow in the same geometry at higher Reynolds numbers, efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures. Presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer.     

Evolution of acoustic waves in microinhomogeneous media with quadratic elastic nonlinearity and relaxation

A nonlinear wave equation for the velocity “relaxator” is derived in the framework of the rheological model and the corresponding equation of state of a microinhomogeneous medium containing viscoelastic defects with quadratic nonlinear elasticity. The equation is qualitatively analyzed, and numerical solutions to it are presented for a stationary symmetric shock wave and the evolution of initially harmonic waves.     

Application of higher-order KdV——mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere

Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV--mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.     

Long wave-short wave resonance in nonlinear negative refractive index media

We show that long wave-short wave resonance can be achieved in a second-order nonlinear negative refractive index medium when the short wave lies on the negative index branch. With the medium exhibiting a second-order nonlinear susceptibility, a number of nonlinear phenomena such as solitary waves, paired solitons, and periodic wave trains are possible or enhanced through the cascaded second-order effect. Potential applications include the generation of terahertz waves from optical pulses.     

Wave Resonance in Media with Modular,Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.     

有流存在时三层流体界面波的二阶Stokes波解

以小振幅波理论为基础，利用摄动方法研究了有背景流场存在时密度三层成层状态下的界面内波，得到了各层流体速度势的二阶渐近解及界面内波波面位移的二阶Stokes波解，并讨论了界面波的Kelvin-Helmholtz不稳定性.结果表明：有流存在的情况下三层密度成层流体界面内波的一阶渐近解(线性波解)、频散关系及二阶渐近解不仅依赖于各层流体的厚度和密度，也依赖于各层流体的背景流场；界面内波波面位移的二阶Stokes波解不仅描述了界面波之间的二阶非线性相互作用，也描述了背景流与界面波之间的二阶非线性相互作用；当每层流 关键词： 界面波 均匀流 二阶Stokes波解 Kelvin-Helmholtz不稳定性     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlevé truncated expansion method. And we investigate interactive properties of solitons and periodic waves.     

On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.     

Bifurcation methods of dynamical systems for generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

By applying the bifurcation theory of dynamical system to the generalized KP-BBM equation, the phase portraits of the travelling wave system are obtained. It can be shown that singular straight line in the travelling wave system is the reason why smooth periodic waves converge to periodic cusp waves. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.      

Rogue internal waves in the ocean: Long wave model

Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.     

Modulated periodic stokes waves in deep water

Modulated deep-water 1D Stokes waves are considered experimentally and theoretically. Wave trains are modulated in a controlled fashion and their evolution is recorded. Data from repeated laboratory experiments are reproducible near the wave maker, but diverge away from the wave maker. Numerical integration of a perturbed nonlinear Schrodinger equation and an associated linear spectral problem indicate that under suitable conditions modulated periodic Stokes waves evolve chaotically. Sensitive spectral evolution in the neighborhood of homoclinic manifolds of the unperturbed nonlinear Schrodinger equation is found.     

Solitons and spectral renormalization methods in nonlinear optics

Localized wave solutions, often referred to as solitary waves or solitons, are important classes of solutions in nonlinear optics. In optical communications, weakly nonlinear, quasi-monochromatic waves satisfy the “classical” and the “dispersion-managed” nonlocal nonlinear Schrödinger equations, both of which have localized pulses as special solutions. Recent research has shown that mode-locked lasers are also described by similar equations. These systems are variants of the classical nonlinear Schrödinger equation, appropriately modified to include terms which model gain, loss and spectral filtering that are present in the laser cavity. To study their remarkable properties, a computational method is introduced to find localized waves in nonlinear optical systems governed by these equations.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system

Abstract

In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation.     

Dynamics of the plane and solitary waves in a Noguchi network:Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrodinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al.(Phys. Rev. E 91 022925(2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.     

Extreme waves statistics for the Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h ? 1, the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical nonlinear Schrödinger equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h ～ 1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical nonlinear Schrödinger equation.