Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

On the resonant reflection of weak,nonlinear sound waves off an entropy wave

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one‐dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual type of nonlinear dynamics. On an appropriate long time scale, we show that a complex amplitude function for the spatial profile of the sound waves satisfies a degenerate quasilinear Schrödinger equation. We present some numerical solutions of this equation that illustrate the generation of small spatial scales by a resonant four‐wave cascade and front propagation in compactly supported solutions.     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

Electrostatic wave solutions for a nonlinear coupled system in an inhomogeneous collisional dusty magnetoplasma

In an inhomogeneous collisional dusty magnetoplasma, a new coupled (3 + 1)-dimensional nonlinear system is derived for the low-frequency electrostatic waves considering the collision between ions and neutrals. It is demonstrated that due to the collision, the scaling symmetry of the system is destroyed. By means of the classical Lie group approach, two types of exact similarity waves are obtained which show three important features. First, various waves can be constructed from these solutions, such as solitary waves, shock waves and periodic waves. Second, these waves may have shears that are time-dependent, linear and nonlinear. Third, the electrostatic potential and the parallel electron velocity possess more freedoms in the sense that they can have either same or different wave forms.     

Rotational waves generated by current‐topography interaction

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.     

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Characteristics of rogue waves on a soliton background in a coupled nonlinear Schrödinger equation

In this paper, a coupled nonlinear Schrödinger (CNLS) equation, which can describe evolution of localized waves in a two‐mode nonlinear fiber, is under investigation. By using the Darboux‐dressing transformation, the new localized wave solutions of the equation are well constructed with a detailed derivation. These solutions reveal rogue waves on a soliton background. Moreover, the main characteristics of the solutions are discussed with some graphics. Our results would be of much importance in predicting and enriching rogue wave phenomena in nonlinear wave fields.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively.     

Multi-solitonic solutions for the variable-coefficient variant Boussinesq model of the nonlinear water waves

For the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water, we consider a variable-coefficient variant Boussinesq (vcvB) model with symbolic computation. We construct the connection between the vcvB model and a variable-coefficient Ablowitz-Kaup-Newell-Segur (vcAKNS) system under certain constraints. Using the N-fold Darboux transformation of the vcAKNS system, we present two sets of multi-solitonic solutions for the vcvB model, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Dynamics of those solutions are analyzed and graphically discussed, such as the parallel solitonic waves, shape-changing collision, head-on collision, fusion-fission behavior and elastic-fusion coupled interaction.     

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.     

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

We consider the fully parity‐time (PT) symmetric nonlocal (2 + 1)‐dimensional nonlinear Schrödinger (NLS) equation with respect to x and y. By using Hirota's bilinear method, we derive the N‐soliton solutions of the nonlocal NLS equation. By using the resulting N‐soliton solutions and employing long wave limit method, we derive its nonsingular rational solutions and semi‐rational solutions. The rational solutions act as the line rogue waves. The semi‐rational solutions mean different types of combinations in rogue waves, breathers, and periodic line waves. Furthermore, in order to easily understand the dynamic behaviors of the nonlocal NLS equation, we display some graphics to analyze the characteristics of these solutions.     

The long‐wave limit for the water wave problem I. The case of zero surface tension

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.     

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.     

Linear Waves and Nonlinear Wave Interactions in a Bounded Three‐Layer Fluid System

We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior.     

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.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.     

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.