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.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

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.     

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

We study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.     

Multi-Solitary Waves for the Nonlinear Klein-Gordon Equation

We consider the nonlinear Klein-Gordon equation in ? ^d . We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.     

Two-dimensional rotating waves in a functional-differential diffusion equation with rotation of spatial arguments and time delay

We consider the Neumann boundary value problem for a parabolic functional-differential equation in a disk. We describe spatially inhomogeneous solutions in the form of rotating waves branching from the homogeneous stationary solution in the case of an Andronov-Hopf bifurcation. By passing to a moving coordinate system and by reducing the original problem to a stationary boundary value problem for a partial differential equation with a deviating argument, we prove the existence of rotating waves appearing in the disk under the Andronov-Hopf bifurcation.     

Anomalous waves as an object of statistical topography: Problem statement

Based on ideas of statistical topography, we analyze the boundary-value problem of the appearance of anomalous large waves (rogue waves) on the sea surface. The boundary condition for the sea surface is regarded as a closed stochastic quasilinear equation in the kinematic approximation. We obtain the stochastic Liouville equation, which underlies the derivation of an equation describing the joint probability density of fields of sea surface displacement and its gradient. We formulate the statistical problem with the stochastic topographic inhomogeneities of the sea bottom taken into account. It describes diffusion in the phase space, and its solution must answer the question whether information about the existence of anomalous large waves is contained in the quasilinear equation under consideration.     

Stochastic wave equation simulating the behavior of random variables whose mean values obey a system of ordinary first-order differential equations

We suggest a theory of stochastic waves that describe the behavior of random vectors satisfying a set of ordinary first-order differential equations. The equation for the stochastic waves is given for the case where the mean values are described by a differential model. The relationship between this equation and the Liouville equation is considered and analogue of the Ehrenfest theorem is proved. For the covariance of a component of a random vector, we obtain an ordinary first-order differential equation. Interpretation of Planck’s constant is discussed. Conditions are formulated under which wave packets propagate with increasing or decreasing covariance. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 356–368, June, 1997.     

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.     

On Periodic Wave Solutions to (1+1)-Dimensional Nonlinear Physical Models Using the Sine-Cosine Method

We investigate two interesting (1+1)-dimensional nonlinear partial differential evolution equations (NLPDEEs), namely the nonlinear dispersion equation with compact structures and the generalized Camassa–Holm (CH) equation describing the propagation of unidirectional shallow water waves on a flat bottom, and arising in the study of a certain non-Newtonian fluid. Using an interesting technique known as the sine-cosine method for investigating travelling wave solutions to NLPDEEs, we construct many new families of wave solutions to the previous NLPDEEs, amongst which the periodic waves, enriching the wide class of solutions to the above equations.     

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.     

Travelling waves for a non-local Korteweg–de Vries–Burgers equation

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.     

Rogue-wave solutions of the Zakharov equation

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these Nth-order rogue-wave solutions explicitly in terms of Nth-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane (x, y) arising from a constant background at t ? 0 and then gradually tending to the constant background for t ? 0. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey–Stewartson equation analytically and graphically.     

Remark on peakon solution to the nonlinear surface wind waves equation

We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

Oscillatory traveling waves of polytropic filtration equation with generalized Fisher–KPP sources

The polytropic filtration equation with generalized Fisher–KPP sources is considered. We will show that the equation may have finite times oscillatory traveling waves, and try to give a complete classification by virtue of a singular exponent in the source according to the finiteness of the oscillatory times of traveling waves.