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.     

On the stability of solitary-wave solutions of model equations for long waves

Summary After a review of the existing state of affairs, an improvement is made in the stability theory for solitary-wave solutions of evolution equations of Korteweg-de Vries-type modelling the propagation of small-amplitude long waves. It is shown that the bulk of the solution emerging from initial data that is a small perturbation of an exact solitary wave travels at a speed close to that of the unperturbed solitary wave. This not unexpected result lends credibility to the presumption that the solution emanating from a perturbed solitary wave consists mainly of a nearby solitary wave. The result makes use of the existing stability theory together with certain small refinements, coupled with a new expression for the speed of propagation of the disturbance. The idea behind our result is also shown to be effective in the context of one-dimensional regularized long-wave equations and multidimensional nonlinear Schr?dinger equations.     

Weakly Nonlocal Solitary Waves in a Singularly Perturbed Nonlinear Schrödinger Equation

We consider the nonlinear Schrödinger equation perturbed by the addition of a third-derivative term whose coefficient constitutes a small parameter. It is known from the work of Wai et al. [1] that this singular perturbation causes the solitary wave solution of the nonlinear Schrödinger equation to become nonlocal by the radiation of small-amplitude oscillatory waves. The calculation of the amplitude of these oscillatory waves requires the techniques of exponential asymptotics. This problem is re-examined here and the amplitude of the oscillatory waves calculated using the method of Borel summation. The results of Wai et al. [1] are modified and extended.     

Oleinik type estimates for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model of small-amplitude long waves in a rotating fluid of finite depth. This is a nonlinear evolution equation. In this study, we consider the well-posedness of the Cauchy problem associated with this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky–Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

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.     

Forced Waves Near Resonance at a Phase-Speed Minimum

When a dispersive wave system is subject to forcing by a moving external disturbance, a maximum or minimum of the phase speed is associated with a critical forcing speed at which the linear response is resonant. Nonlinear effects can play an important part near such resonances, and the salient characteristics of the nonlinear response depend on whether the maximum or minimum of the phase speed is realized in the long-wave limit (zero wavenumber) or at a finite wavenumber. The focus here is on the latter case that, among other physical systems, applies to gravity–capillary waves on water of finite or infinite depth. The analysis, for simplicity, is based on a forced–damped fifth-order Korteweg–de Vries equation, a model problem that features a phase-speed minimum at a finite wavenumber. When damping is not too strong compared with forcing, multiple subcritical finite-amplitude steady-solution branches coexist with the small-amplitude response predicted by linear theory. For forcing speed well below critical, the transient response from rest approaches the small-amplitude state, but at speeds close to critical, jump phenomena can occur, and reaching a time-periodic state that involves shedding of wavepacket solitary waves is also possible.     

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.     

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.     

Generation of Secondary Solitary Waves in the Variable-Coefficient Korteweg–de Vries Equation

We consider the solitary wave solutions of a Korteweg–de Vries equation, where the coefficients in the equation vary with time over a certain region. When these coefficients vary rapidly compared with the solitary wave, then it is well known that the solitary wave may fission into two or more solitary waves. On the other hand, when these coefficients vary slowly, the solitary wave deforms adiabatically with the production of a trailing shelf. In this paper we re-examine this latter case, and show that the trailing shelf, on a very long time-scale, can lead to the generation of small secondary solitary waves. This result thus provides a connection between the adiabatic deformation regime and the fission regime.     

Three-Dimensional Water Waves

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.     

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.     

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).     

Numerical Solution of Some Nonlocal, Nonlinear Dispersive Wave Equations

Summary. We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory. Received October 28, 1997; revised February 11, 1999; accepted April 7, 1999     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

Solitary wave families of a generalized microstructure PDE

Wave propagation in a generalized microstructure PDE, under the Mindlin relations, is considered. Limited analytic results exist for the occurrence of one family of solitary wave solutions of these equations. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work are also mentioned.     

B‐spline collocation algorithms for numerical solution of the RLW equation

Both sextic and septic B‐spline collocation algorithms are presented for the numerical solutions of the RLW equation. Numerical results resolve the fine structure of the single solitary wave propagation, two and three solitary waves interaction, and evolution of solitary waves. Comparison of the numerical results is done by the results of some earlier schemes mentioned in the article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 581–607, 2011     

Change of Polarity for Periodic Waves in the Variable‐Coefficient Korteweg‐de Vries Equation

We examine the variable‐coefficient Kortweg‐de Vries equation for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here, we examine the same case but for a modulated periodic wave train. Using an asymptotic analysis, we show that in contrast a periodic wave is preserved with a finite amplitude as it passes through the critical point, but a phase change is generated causing the wave to reverse its polarity.     

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.