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.     

Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan–Porsezian–Daniel Equations in Birefringent Optical Fibers

Nonlinear waves on periodic backgrounds play an important role in physical systems. In this study, nonlinear waves that include solitons, breathers, rogue waves, and semi-rational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations are investigated. Moreover, the interactions between different types of nonlinear waves are examined and their dynamic behaviors are studied. In particular, it is observed that bright-dark rogue waves interact with bright-dark breathers or solitons on periodic backgrounds, four-petaled breathers interact with two eye-shaped breathers on periodic backgrounds, and a four-petal rogue wave interplays with a rogue wave on periodic backgrounds. Furthermore, it is found that the value of the parameter γ₃ affects the weak and strong interactions of these nonlinear waves. These results may be useful in the study of nonlinear wave dynamics in coupled nonlinear wave models.     

On the Kudryashov-Sinelshchikov equation for waves in bubbly liquids

A nonlinear evolution equation for wave propagation in bubbly liquids, taking into account viscosity and heat transfer, has been derived by Kudryashov and Sinelshchikov. In the case of no dissipation the authors have provided analytical solutions representing undistorted waves. These results are cast into a simpler form and studied in more detail. In addition to the wave profiles the corresponding phase curves are presented. Depending on some parameter the solutions represent solitary or periodic waves. Some of the periodic waves exhibit peaks or cusps. From the periodic waves a new type of “meandering” solutions is constructed.     

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

The (2 1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a beta-plane channel is investigated by using the classical Lie symmetry approach. Some types of group-invariant wave solutions are expressed by means of the lower-dimensional similarity reduction equations. In addition to the known periodic Rossby wave solutions, some new types of exact solutions such as the ring solitary waves and the breaking soliton type of vorticity solutions with nonlinear and nonconstant shears are also obtained.     

A Model for Periodic Nonlinear Electric Field Structures in Space Plasmas

In this study, we present a physical model to explain the generation mechanism of nonlinear periodic waves with a large amplitude electric field structures propagating obliquely and exactly parallel to the magnetic field. The ``Sagdeev potential' fromthe MHD equations is derived and the nonlinear electric field waveforms are obtained when the Mach number, direction of propagation, and the initial electric field satisfy certain plasma conditions. For the parallel propagation, the amplitude of theelectric field waves with ion-acoustic mode increases with the increase of initial electric field and Mach number but its frequency decreases with the increase of Mach number. The amplitude and frequency of the electric field waves with ion-cyclotron mode decrease with the increase of Mach number and become less spiky, andits amplitude increases with the increase of initial electric field. For the oblique propagation, only periodic electric field wave with an ion-cyclotron mode obtained, its amplitude and frequency increase with the increase of Mach number and become spiky. From our model the electric field structures show periodic, spiky, and saw-tooth behaviours corresponding to different plasma conditions.     

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

The (2 1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a beta-plane channel is investigated by using the classical Lie symmetry approach. Some types of group-invariant wave solutions are expressed by means of the lower-dimensional similarity reduction equations. In addition to the known periodic Rossby wave solutions, some new types of exact solutions such as the ring solitary waves and the breaking soliton type of vorticity solutions with nonlinear and nonconstant shears are also obtained.     

Study on Solitary Waves of a General Boussinesq Model

In this paper, we employ the bifurcation method of dynamical systems to study the solitary waves and periodic waves of a generalized Boussinesq equations. All possible phase portraits in the parameter plane for the travelling wave systems are obtained. The possible solitary wave solutions, periodic wave solutions and cusp waves for the general Boussinesq type fluid model are also investigated.     

Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.     

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.     

PARAMETRIC SIMULTONS IN ONE-DIMENSIONAL NONLINEAR LATTICES

Parametric simultaneous solitary wave (simulton) excitations are shown to be possible in nonlinear lattices. Taking a one-dimensional diatomic lattice with a cubic potential as an example, we consider the nonlinear coupling between the upper cut-off mode of acoustic branch (as a fundamental wave) and the upper cut-off mode of optical branch (as a second harmonic wave). Based on a quasi-discreteness approach the Karamzin－Sukhorukov equations for two slowly varying amplitudes of the fundamental and the second harmonic waves in the lattice are derived when the condition of second harmonic generation is satisfied. The lattice simulton solutions are given explicitly and the results show that these lattice simultons can be nonpropagating when the wave vectors of the fundamental wave and the second harmonic waves are exactly at π/a (where a is the lattice constant) and zero, respectively.     

Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents

In the present paper, the random interfacial waves in N-layer density-stratified fluids moving at different steady uniform speeds are researched by using an expansion technique, and the second-order asymptotic solutions of the random displacements of the density interfaces and the associated velocity potentials in N-layer fluid are presented based on the small amplitude wave theory. The obtained results indicate that the wave-wave second-order nonlinear interactions of the wave components and the second-order nonlinear interactions between the waves and currents are described. As expected, the solutions include those derived by Chen （2006） as a special case where the steady uniform currents of the N-layer fluids are taken as zero, and the solutions also reduce to those obtained by Song （2005） for second-order solutions for random interfacial waves with steady uniform currents if N = 2.     

非均匀交换各向异性铁磁介质的非线性表面自旋波

利用Landau-Lifshitz 方程，研究了具有非均匀交换各向异性的半无限大铁磁体的非线性表 面自旋波理论。导出了部分钉扎纯交换铁磁介质的磁化强度所满足的边界条件和非线性表面 自旋波的色散关系，并获得了自旋波振幅沿z方向驻波的一维非线性Schrdinger方程和包 络振幅沿平面传播的二维非线性Schrdinger方程，结果表明铁磁体磁化强度的包络振幅随时空变化的性质是由二维非线性Schrdinger方程决定的。因此预言铁磁介质的表面非线性激发应是二维孤波的形式。对于弱非线性表面自旋波，对非线性Schrdinger方程存在孤子形式解的可能性作了讨论. 关键词： 表面自旋波 Landau-Lifshitz方程 非线性Schrdinger方程 孤子     

Nonlinear theory of electrostatic baryonic waves in ambiplasma

A collisionless nonmagnetized ambiplasma consisting of Maxwellian gases of protons, antiprotons, electrons, and positrons is considered. The dispersion relation for electrostatic baryonic waves is derived and analyzed and exact expressions for the linear wave phase velocities are obtained. Two types of such waves are shown to be possible in ambiplasma: acoustic and plasma ones. Analysis of the dispersion relation has allowed the ranges of parameters in which nonlinear solutions should be sought in the form of solitons to be found. A nonlinear theory of baryonic waves is developed and used to obtain and analyze the exact solution to the basic equations. The analysis is performed by the method of a fictitious potential. The ranges of phase velocities of periodic baryonic waves and soliton velocities (Mach numbers) are determined. It is shown that in the plasma under consideration, these ranges do not overlap and that the soliton velocity cannot be lower than the linear velocity of the corresponding wave. The profiles of physical quantities in a periodic wave and a soliton (wave scores) are plotted.     

Wavy film flow down a vertical plate: Comparisons between the results of integral approaches and full-scale computations

This paper is devoted to a theoretical analysis of nonlinear two-dimensional waves using the Navier-Stokes equations in their full statement. Steady-state travelling wave regimes have been found and an analysis of their linear stability has been carried out. It is shown that the flow regimes obtained using the Navier-Stokes equations are qualitatively different from the solutions of Shkadov’s integral approach starting from some values of the Kapitza number. It is also found that the wave regimes of the Navier-Stokes equations have an internal vortex at moderate Reynolds numbers. The results obtained using “the regularized integral model” are in excellent agreement with the Navier-Stokes calculations for Re/Ka ≤2. Unlike the solutions found using an integral approach, it is shown that only a few types of nonlinear waves exist when the full Navier-Stokes equations are considered. The text was submitted by the author in English.     

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

The Fermi-Pasta-Ulam recurrence and related phenomena for 1<Emphasis Type="Italic">D</Emphasis> shallow-water waves in a finite basin

Different regimes of the Fermi-Pasta-Ulam (FPU) recurrence are simulated numerically for fully nonlinear “one-dimensional” potential water waves in a finite-depth flume between two vertical walls. In such systems, the FPU recurrence is closely related to the dynamics of coherent structures approximately corresponding to solitons of the integrable Boussinesq system. A simplest periodic solution of the Boussinesq model, describing a single soliton between the walls, is presented in analytic form in terms of the elliptic Jacobi functions. In the numerical experiments, it is observed that depending on the number of solitons in the flume and their parameters, the FPU recurrence can occur in a simple or complicated manner, or be practically absent. For comparison, the nonlinear dynamics of potential water waves over nonuniform beds is simulated, with initial states taken in the form of several pairs of colliding solitons. With a mild-slope bed profile, a typical phenomenon in the course of evolution is the appearance of relatively high (rogue) waves, while for random, relatively short-correlated bed profiles it is either the appearance of tall waves or the formation of sharp crests at moderate-height waves.     

Nonlinear waves in a rotating plasma cylinder

Previous work on exact cylindrical surface waves in a nonneutral cold-electron plasma bounded by a dielectric is extended to include plasma rotation. A set of nonlinear rate equations describing the temporal behavior of the system is derived by first determining an appropriate spatial wave structure. Physically relevant periodic solutions are obtained     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波     

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.     

近海波-流相互作用的缓坡方程理论体系

在对近海一大类水波模型——缓坡方程的总体考察、分析和判断基础上,引进一个表征近海普遍波-流相互作用机制的算子,进而借助普适的Hamilton水波变分原理,建立了一个关联于时间的近海波-流相互作用缓坡方程理论体系,从形式到内容上达到了一种高度协调和统一.