On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

Singular periodic waves of an integrable equation from short capillary-gravity waves

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.     

Bifurcation and chaos of thin circular functionally graded plate in thermal environment

A ceramic/metal functionally graded circular plate under one-term and two-term transversal excitations in the thermal environment is investigated, respectively. The effects of geometric nonlinearity and temperature-dependent material properties are both taken into account. The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power law distribution of the volume fraction of the constituents. Using the principle of virtual work, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic forcing excitation and thermal load are derived. For the circular plate with clamped immovable edge, the Duffing nonlinear forced vibration equation is deduced using Galerkin method. The criteria for existence of chaos under one-term and two-term periodic perturbations are given with Melnikov method. Numerical simulations are carried out to plot the bifurcation curves for the homolinic orbits. Effects of the material volume fraction index and temperature on the criterions are discussed and the existences of chaos are validated by plotting phase portraits, Poincare maps. Also, the bifurcation diagrams and corresponding maximum Lyapunov exponents are plotted. It was found that periodic, multiple periodic solutions and chaotic motions exist for the FGM plate under certain conditions.     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity

We investigate the KdV like equation with higher order nonlinearity ut + a(1 +bun)unux + uxxx = 0with n ≥ 1, a, b ∈ R and α≠ 0. The bifurcations and explicit expressions of solitary wave solutions for theequation are discussed by using the bifurcation method and qualitative theory of dynamical systems. Thebifurcation diagrams, existence and number of the solitary waves are given.     

非线性弹性杆波动方程的显式精确解北大核心CSCD

应用sine-cosine方法对非线性弹性杆波动方程进行了求解,得到了该方程的一些新的周期波解和孤波解(材料常数n为不等于1的常数).对部分结果通过数学软件得到了解的图像,获得的结果有助于非线性弹性杆中孤波存在性问题的进一步研究.     

Existence of solitons in the nonlinear beam equation

This paper concerns the existence of solitons, namely stable solitary waves in the nonlinear beam equation with a suitable nonlinearity. An equation of this type has been introduced in [P. J. McKenna and W. Walter, Arch. Ration. Mech. Anal., 98 (1987), 167-177] as a model of a suspension bridge. We prove both the existence of solitary waves for a large class of nonlinearities and their stability. As far as we know this is the first result about stability of solitary waves in nonlinear beam equation.     

Long distance wave computation using nonlinear solitary waves

A recently developed method is described to propagate short wave equation pulses over indefinite distances and through regions of varying indices of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation (pde) that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain ∼2-3 grid cells in thickness indefinitely with no numerical spreading. A new feature described in this paper involves computing scattering of short pulses from complex objects such as complete aircraft. A simple “immersed surface” approach is used, that utilizes the same uniform grid as the propagation and avoids complex, body fitted or adaptive grid schemes.The new method should be useful in areas of wave propagation, from radar scattering and long distance communications to cell phone transmission.     

Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.     

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.     

3+1 dimensional envelop waves and its stability in magnetized dusty plasma

It is well known that there are envelope solitary waves in unmagnetized dusty plasmas which are described by a nonlinear Schrodinger equation (NLSE). A three dimension nonlinear Schrodinger equation for small but finite amplitude dust acoustic waves is first obtained for magnetized dusty plasma in this paper. It suggest that in magnetized dusty plasmas the envelope solitary waves exist. The modulational instability for three dimensional NLSE is studied as well. The regions of stability and instability are well determined in this paper.     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.     

TARGET PATTERNS AND SPIRAL WAVES IN REACTION-DIFFUSION SYSTEMS WITH LIMIT CYCLE KINETICSAND WEAK DIFFUSION

1.IntroductionTargetpatternsandspiralwavesarecommonlyobservedincertainmodelsofchemicalandbiologicalsystemssuchastheBelousov-ZhabotinskiireactionandthesocialamoebasDictyosteliumdiscoideium(of.11--4]andthereferencestherein).Thesesystemsaregovernedbyachemicalorbiologicalreactionandspatialdiffusion,i.e.reaction-diffusionequations.Generallyspeaking,atargetpatternisasetofconcentricringsofconstantconcentrationeachmovingoutward.Alonganyradialline,thepatternbehavesasymptoticallylikeaperiodictravellin…     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equation:ut + c0ux + 3uux - α2(uxxt + uuxxx + 2uxuxx) + γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves: periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

Solitary-wave families of the Ostrovsky equation: An approach via reversible systems theory and normal forms

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Limited functional analytic results exist for the occurrence of one family of solitary-wave solutions of this equation, as well as their approach to the well-known solitons of the famous Korteweg–de Vries equation in the limit as the rotation becomes vanishingly small. Since solitary-wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary-wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Ion acoustic solitary wave solutions of two‐dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation in quantum plasma

Propagation of two‐dimensional nonlinear ion‐acoustic solitary waves and shocks in a dissipative quantum plasma is analyzed. By applying the reductive perturbation theory, the two‐dimensional ion acoustic solitary waves in a dissipative quantum plasma lead to a nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation. By implementing extended direct algebraic mapping, extended sech‐tanh, and extended direct algebraic sech methods, the ion solitary traveling wave solutions of the two‐dimensional nonlinear KPB equation are investigated. An analytical as well as numerical solution of the two‐dimensional nonlinear KPB equation is obtained and analyzed with the effects of external electric field and ion pressure. Copyright © 2016 John Wiley & Sons, Ltd.     

Evolution of Packets of Surface Gravity Waves over Strong Smooth Topography

Wave packets in a smoothly inhomogeneous medium are governed by a nonlinear Schrödinger (NLS) equation with variable coefficients. There are two spatial scales in the problem: the spatial scale of the inhomogeneities and the distance over which nonlinearity and dispersion affect the packet. Accordingly, there are two limits where the problem can be approached asymptotically: when the former scale is much larger than the latter, and vice versa. In this paper, we examine the limit where the spatial scale of (periodic or random) inhomogeneities is much smaller than that of nonlinearity/dispersion (i.e., the latter effects are much weaker than the former). In this case, the packet undergoes rapid oscillations of the geometric-optical type, and also evolves slowly due to nonlinearity and dispersion. We demonstrate that the latter evolution is governed by an NLS equation with constant (averaged) coefficients. The general theory is illustrated by the example of surface gravity waves in a channel of variable depth. In particular, it is shown that topography increases the critical frequency, for which the nonlinearity coefficient of the NLS equation changes sign (in such cases, no steady solutions exist, i.e., waves with frequencies lower than the critical one disperse and cannot form packets).     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.