Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

Are Solitary Internal Waves Solitons?

Results of fully nonlinear numerical simulations of the interaction of two mode-1 solitary internal waves, both propagating in the same direction, are presented. After the interaction, two solitary internal waves emerge. The large wave is slightly larger than the initial large solitary wave, while the small one is slightly smaller than the initial small solitary wave. Some small-amplitude trailing mode-1 and mode-3 waves are generated by the interaction.     

Solitary wave solutions of the high-order KdV models for bi-directional water waves

An approach, which allows us to construct specific closed-form solitary wave solutions for the KdV-like water-wave models obtained through the Boussinesq perturbation expansion for the two-dimensional water wave problem in the limit of long wavelength/small amplitude waves, is developed. The models are relevant to the case of the bi-directional waves with the amplitude of the left-moving wave of O(ϵ) (ϵ is the amplitude parameter) as compared with that of the right-moving wave. We show that, in such a case, the Boussinesq system can be decomposed into a system of coupled equations for the right- and left-moving waves in which, to any order of the expansion, one of the equations is dependent only on the (main) right-wave elevation and takes the form of the high-order KdV equation with arbitrary coefficients whereas the second equation includes both elevations. Then the explicit solitary wave solutions constructed via our approach may be treated as the exact solutions of the infinite-order perturbed KdV equations for the right-moving wave with the properly specified high-order coefficients. Such solutions include, in a sense, contributions of all orders of the asymptotic expansion and therefore may be considered to a certain degree as modelling the solutions of the original water wave problem under proper initial conditions. Those solitary waves, although stemming from the KdV solitary waves, possess features found neither in the KdV solitons nor in the solutions of the first order perturbed KdV equations.     

Time-linearized,compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.     

Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.     

Nonlinear Wave Equations for Oceanic Internal Solitary Waves

In the coastal ocean, the interaction of barotropic tidal currents with topographic features such as the continental shelf, sills in narrow straits, and bottom ridges are often observed to generate large amplitude, horizontally propagating internal solitary waves. These are long nonlinear waves and hence can be modeled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography, with the consequence that the appropriate nonlinear evolution equation has variable coefficients. Further, as these waves can be long‐lived it is necessary to take account of the effects of the Earth's background rotation. We review this family of model evolution equations and some of their pertinent solutions, obtained both asymptotically and numerically.     

The effects of interplay between the rotation and shoaling for a solitary wave on variable topography

This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modeling. One of them is a constant‐slope bottom and the other is a specific bottom profile providing a constant amplitude solitary wave. Estimates with real oceanic parameters show that the predicted effects of stable soliton dynamics in a coastal zone can occur, in particular, for internal waves.     

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.     

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.     

Exact solutions of nonlinear generalizations of the wave equation

Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.     

缓变的任意截面渠道中的孤立波

本文研究了在流动方向可以有缓慢变化的任意截面渠道中的孤立波,导出了缓变系数KdV方程,并求出了此方程的首项近似解,导出了孤立波的速度的表示式,以及孤立波的波幅与渠道几何尺寸的关系,并把它们应用于三角形渠道、矩形渠道,对于变深度、变宽度矩形渠道的情况,本文的结果与Johnson、Shuto及Mile等人所得的结果一致.     

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.     

The Equivalence of Bernoulli's Equation and a Set of Integral Relations for Periodic Waves

In a recent paper Longuet-Higgins (1978) discovered some newrelations between Stokes' coefficients in the theory of periodicgravity waves. These were shown to give rise to a set of integralrelations. In this paper we show that this set is complete inthe sense that it is equivalent to Bernoulli's equation. Wealso show that a suitably redefined set exists in the theoryof the solitary wave.     

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.     

Solitary waves of the EW and RLW equations

Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long–wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank–Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the L₂-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.     

A Model Equation for Wavepacket Solitary Waves Arising from Capillary-Gravity Flows

A model equation governing the primitive dynamics of wave packets near an extremum of the linear dispersion relation at finite wavenumber is derived. In two spatial dimensions, we include the effects of weak variation of the wave in the direction transverse to the direction of propagation. The resulting equation is contrasted with the Kadomtsev–Petviashvilli and Nonlinear Schrödinger (NLS) equations. The model is derived as an approximation to the equations for deep water gravity-capillary waves, but has wider applications. Both line solitary waves and solitary waves which decay in both the transverse and propagating directions—lump solitary waves—are computed. The stability of these waves is investigated and their dynamics are studied via numerical time evolution of the equation.     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

两层流体界面上的孤立波

本文讨论两水平固壁间两层不可压无粘流体界面上的孤立波,计及界面上的表面张力效应.首先建立了适用于这种模型的基本方程组,并在弱色散近似下应用约化摄动法,导得了一阶界面升高所满足的Korteweg-de Vries方程,指出了按该方程系数α和μ的符号的异同,KdV孤立波可能凸向上或凸向下.然后详细讨论了原有近似下非线性效应与色散效应不能平衡的两种临界情形.在采用了适当的近似之后,对第一种临界情形(α=0)得到了修正的KdV方程,并指出,在所考虑的情形中,当μ>0时孤立波不存在,当μ<0时,孤立波仍可能存在,其形式与KdV孤立波不同;对第二种临界情形(μ=0),导得了推广的KdV方程,这时存在振荡型孤立波.文中还对近临界情形作了讨论.本文结果与一些经典结果完全一致,并把它们作了拓广.     

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.