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.     

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.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

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.     

The KdV Equation in Stratified Fluid Flow

This paper describes a method for the formal derivation of the equation governing long waves on the surface of a shallow fluid which is stably and continuously stratified. By using a three variable expansion procedure involving the normal time scale together with two slow time scales suggested as a result of the formulation of the problem the governing equation is shown to be the ordinary Korteweg-de Vries equation.

AMS(MOS) CLASSIFICATION: 76D30, 35Q20, 35C20.     

Optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities

In this paper, optical rogue waves for the coherently coupled nonlinear Schrödinger equation with alternate signs of nonlinearities are investigated via Darboux transformation. We derive family of structures of rogue wave, including rogue waves with one peak and two valleys, bright rogue waves without valleys while with one peak or two peaks.     

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation.     

Traveling waves for a dissipative modified KdV equation

In this paper we consider a dispersive–dissipative nonlinear equation which can be regarded as a dissipation perturbed modified KdV equation, governing the evolution of long waves in an elastic rod immersed inside a viscoelastic medium. Using geometric singular perturbation theory, a construction of traveling waves for the equation is shown. This also is illustrated by presenting some numerical calculations.     

A new highly nonlinear shallow water wave equation

We derive a quasilinear shallow water equation directly from the governing equations for gravity water waves within a certain regime for large-amplitude waves which has not been studied so far. Furthermore, we demonstrate local well-posedness of the corresponding Cauchy problem and finally discuss some aspects of the blowup behavior of solutions.     

Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

Surfactant Influence on Water Wave Packets

The influence of surfactant on water wave packets is investigated. An envelope equation for a slowly varying wave packet in the potential flow equations with variable Bond number is derived. The properties of this equation depend on the relative phases of the wave packet and the distribution of surface tension. We observe that small variations in the Bond number may change the focusing nature of the envelope equation from that of the constant Bond number problem. Variations in Bond number can thus suppress, or incite, the Benjamin‐Feir instability. The existence of envelope solitary waves depends in a similar way on the Bond number variation. The envelope equation is also derived in a larger class of models.     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

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.     

Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.     

On the Equation of Motion of a Thin Layer of Uniform Vorticity

A higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equation is used to examine the growth of disturbances on a straight, steady layer of uniform vorticity. The growth rate for long waves is in good agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be suppressed by making this approximate allowance for finite thickness. However, it is found that very short waves, which are quite outside the range of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the equation will require use of smoothing techniques to suppress this spurious growth of short wave disturbances.     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen-condition on an eigen-value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.     

Analytic rogue wave solutions for a generalized fourth‐order Boussinesq equation in fluid mechanics

A bilinear transformation method is proposed to find the rogue wave solutions for a generalized fourth‐order Boussinesq equation, which describes the wave motion in fluid mechanics. The one‐ and two‐order rogue wave solutions are explicitly constructed via choosing polynomial functions in the bilinear form of the equation. The existence conditions for these solutions are also derived. Furthermore, the system parameter controls on the rogue waves are discussed. The three parameters involved in the equation can strongly impact the wave shapes, amplitudes, and distances between the wave peaks. The results can be used to deeply understand the nonlinear dynamical behaviors of the rogue waves in fluid mechanics.     

Navier Stokes model of solitary wave collision

Wave collision and its interaction characteristics is one of the important challenges in coastal engineering. This article concerns the collision of solitary waves over a horizontal bottom considering unsteady, incompressible viscous flow with free surface. The method solves the two dimensional Naiver–Stokes equations for conservation of momentum, continuity equation, and full nonlinear kinematic free-surface equation for Newtonian fluids, as the governing equations in a vertical plan. A mapping was developed to trace the deformed free surface encountered during wave propagation, transforms and interaction by transferring the governing equations from the physical domain to a computational domain. Also a numerical scheme is developed using finite element modeling technique in order to predict the solitary wave collision. Consequently results compared with other researches and show the inelastic behavior of solitary wave collision.