Solitary waves and the structures of discontinuities in non-dissipative models with complex dispersion

Stationary solutions of reversible evolutionary equations of mechanics with higher derivatives are analysed. A two-dimensional graphical method for investigating the solutions of systems of ordinary differential equations is described, which enables one to find special types of solutions: periodic waves, solitary waves and the structures of discontinuities. At the same time, solitary waves can be obtained by taking the limit of sequences of periodic waves and the structures of discontinuities obtained by taking the limit of sequences of solitary waves. This general approach has enabled the existence of all earlier predicted structures to be verified has enabled new types of structures (three-wave structures) to be revealed and has enabled all the necessary conditions at the discontinuities to be found. All the previously known types of solitary waves are found and new types of solitary waves are revealed (generalized ordinary and 1:1 multisolitons). Methods of finding generalized solitary waves, including those with a finite amplitude of the periodic component, are determined. Examples of the solution of the following problems are given for a fourth-order system: generalized solitary waves as the limiting solutions of two-wave resonance solutions, generalized solitary waves and the structure of a discontinuity with three waves, a 1:1 soliton and the structure of a discontinuity with a single radiated wave, a solitary wave with fixed propagation velocity, and the structure of a discontinuity in the form of a kink with radiation. A generalized 1:1 soliton and the structure of a discontinuity with two radiated waves is considered in the case of sixth-order systems. The discussion is mainly based on the example of travelling waves described by the generalized Korteweg-de Vries equations. Other models with complex dispersion (a plasma and a stratified fluid) are also considered.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).     

Evolution of Higher-Order Gray Hirota Solitary Waves

The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to determine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Convective Linear Stability of Solitary Waves for Boussinesq Equations

Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small-amplitude, long-wave regime. We study the linearized stability of solitary waves for three linearly well-posed Boussinesq models. These are problems for which well-developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small-amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half-plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This "convective linear stability" property has played a central role in the proof of nonlinear asymptotic stability of solitary-wave-like solutions in other systems.     

Explicit series solutions to nonlinear evolution equations: The sine-cosine method

In this paper, a sine-cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems: the coupled quadratic nonlinear equations and the coupled Klein-Gordon-Schrödinger equations. Under different parameter conditions, explicit formulas for some new periodic and solitary wave solutions are successfully obtained. The proposed solutions are found to be important for the explanation of some practical physical problems.     

Steady solution of solitary wave and linear shear current interaction

The steady solution of a solitary wave propagating in the presence of a linear shear background current is investigated by the Green–Naghdi (GN) equations. The steady solution is obtained by use of the Newton–Raphson method. Three aspects are investigated; they are the wave speed, wave profile and velocity field. The converged GN results are compared with results from the literature. It is found that for the opposing-current case of the solitary wave with a small amplitude, the results of the GN equations match results from the literature well, while for the solitary wave with a large amplitude, results from the literature are seen to be not as accurate. In the following-current case, though the amplitude of the solitary wave is small, the GN results are shown to be accurate. The velocity along the water column at the wave crest and the velocity field for different cases are calculated by the GN equations. The results of the GN equations show obvious differences when compared with the results obtained by superposing the no-current results and linear shear current linearly. We find that for the same current strength, the vortex is stronger for the steep solitary-wave case than that for the small solitary-wave case.     

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.     

Study of magnetosonic solitary waves for the electron magnetohydrodynamics equations

Electron magnetohydrodynamics equations are derived with allowance for nonlinearity, dispersion, and dissipation caused by friction between the ions and electrons. These equations are transformed into a form convenient for the construction of a numerical scheme. The interaction of codirectional and oppositely directed magnetosonic solitary waves with no dissipation is computed. In the first case, the solitary waves are found to behave as solitons (i.e., their amplitudes after the interaction remain the same), while, in the second case, waves are emitted that lead to decreased amplitudes. The decay of a solitary wave due to dissipation is computed. In the case of weak dissipation, the solution is similar to that of the Riemann problem with a structure combining a discontinuity and a solitary wave. The decay of a solitary wave due to dispersion is also computed, in which case the solution can also be interpreted as one with a discontinuity. The decay of a solitary wave caused by the combined effect of dissipation and dispersion is analyzed.     

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.     

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 wave solutions for a general Boussinesq type fluid model

The possible solitary wave solutions for a general Boussinesq (GBQ) type fluid model are studied analytically. After proving the non-Painlevé integrability of the model, the first type of exact explicit travelling solitary wave with a special velocity selection is found by the truncated Painlevé expansion. The general solitary waves with different travelling velocities can be studied by casting the problems to the Newtonian quasi-particles moving in some proper one dimensional potential fields. For some special velocity selections, the solitary waves possess different shapes, say, the left moving solitary waves may possess different shapes and/or amplitudes with those of the right moving solitons. For some other velocities, the solitary waves are completely prohibited. There are three types of GBQ systems (GBQSs) according to the different selections of the model parameters. For the first type of GBQS, both the faster moving and lower moving solitary waves allowed but the solitary waves with“middle” velocities are prohibit. For the second type of GBQS all the slower moving solitary waves are completely prohibit while for the third type of GBQS only the slower moving solitary waves are allowed. Only the solitary waves with the almost unit velocities meet the weak non-linearity conditions.     

Bright solitons of the variants of the Novikov–Veselov equation with constant and variable coefficients

In this paper, the existence of the bright soliton solution of four variants of the Novikov–Veselov equation with constant and time varying coefficients will be studied. We analyze the solitary wave solutions of the Novikov–Veselov equation in the cases of constant coefficients, time-dependent coefficients and damping term, generalized form, and in 1 + N dimensions with variable coefficients and forcing term. We use the solitary wave ansatz method to derive these solutions. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Parametric conditions for the existence of the exact solutions are given. The solitary wave ansatz method presents a wider applicability for handling nonlinear wave equations.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

Approximate solutions for magmon propagation from a reservoir

^** Email: tim_marchant{at}uow.edu.au^*** Email: n.smyth{at}ed.ac.uk A 1D partial differential equation (pde) describing the flowof magma in the Earth's mantle is considered, this equationallowing for compaction and distension of the surrounding matrixdue to the magma. The equation has periodic travelling wavesolutions, one limit of which is a solitary wave, called a magmon.Modulation equations for the magma equation are derived andare found to be either hyperbolic or of mixed hyperbolic/elliptictype, depending on the specific values of the wave number, meanheight and amplitude of the underlying modulated wave. The periodicwave train is stable in the hyperbolic case and unstable inthe mixed case. Solutions of the modulation equations are foundfor an initial-boundary value problem on the semi-infinite line,these solutions representing the influx of magma from a largereservoir. The modulation solutions are found to consist ofa full or partial undular bore. Excellent agreement with numericalsolutions of the governing pde is obtained, except in the limitwhere the wave train becomes a train of magmons. An alternativeapproximation based on the assumption that the wave train isa series of uniform magmons is also derived and is found tobe superior to modulation theory in this limit.     

Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Rossby惯性重力孤立波与东、西太平洋副高活动的遥相关

根据季内西太平洋副高活动异常的观测事实，考虑夏季北太平洋地区500hPa层以下的有限纬带区域，应用非线性浅水模式得到了一种孤立波。该波的产生、维持及传播与中、东太平洋副高主体的活动关系密切。 通过对这类孤立波传播特征的讨论和模式大气计算分析，发现它与夏季北太平洋副热带地区存在的位势中心的传播路径和范围比较一致，太平洋中、东部副高主体位置和强度的变化通过这种孤立波的传播可能导致西太平洋副高出现相应的变化。     

New solitary wave and periodic wave solutions for general types of KdV and KdV–Burgers equations

In this paper, we first introduced improved projective Riccati method by means of two simplified Riccati equations. Applying the improved method, we consider the general types of KdV and KdV–Burgers equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions are obtained. Some of them are found for the first time.     

On the solitary wave solutions of the CQNLS

In this paper, coexistence and simplified formulations of the solitary waves of the cubic–quintic non-linear Schrödinger equation (CQNLS) are investigated by analyzing the steady bifurcation and the energy integral of the conservative dynamical system satisfied by the wave packet. It is found that the bright solitary waves can coexist with kinks and anti-kinks in a range of the bifurcation control parameter. There exists a critical parameter value at which the dark solitary waves are distinguished from the bright solitary waves, kinks and anti-kinks. All of the simplified solitary wave solutions, kinks and anti-kinks are obtained by using our previously developed approximate method.