John P. Albert

Sufficient conditions for stability of solitary-wave solutions of model equations for long waves

Abstract We consider solitary-wave solutions of model equations for long waves that feature a general form of linear dispersion. Sufficient conditions for the non-linear stability of such solutions are derived. These conditions are shown to obtain for the Korteweg-de Vries equation and certain of its generalizations such as the Benjamin-Ono equation and the intermediate long-wave equation.

Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation

On considere les equations suivantes: u t +u x +(F(u)) x +u xxx =0(1) et u t +u x +(F(u)) x −u xxt =0(2) avec F:R→R est C ∞ . On donne le comportement des solutions ondes a basse energie pour certains F

Solitary-wave solutions of the Benjamin equation

Considered here is a model equation put forward by Benjamin that governs approximately the evolution of waves on the interface of a two-fluid system in which surface-tension effects cannot be ignored. Our principal focus is the traveling-wave solutions called solitary waves, and three aspects will be investigated. A constructive proof of the existence of these waves together with a proof of their stability is developed. Continuation methods are used to generate a scheme capable of numerically approximating these solitary waves. The computer-generated approximations reveal detailed aspects of the structure of these waves. They are symmetric about their crests, but unlike the classical Korteweg--de Vries solitary waves, they feature a finite number of oscillations. The derivation of the equation is also revisited to get an idea of whether or not these oscillatory waves might actually occur in a natural setting.

Positivity Properties and Stability of Solitary–Wave Solutions of Model Equations For Long Waves

Sufficient conditions are given for stability of solitary-wave solutions of model equations for one-dimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of the solitary waves. Their use leads to simplified proofs of existing stability results for the Korteweg-de Vries, BenjaminOno, and Intermediate Long Wave equations; and to new stability results for certain solitary-wave solutions of partial differential equations of Korteweg-de Vries type.

Comparisons between model equations for long waves

SummaryConsidered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.

Model equations for waves in stratified fluids

Model equations for gravity waves in horizontally stratified fluids are considered. The theories to be addressed focus on stratifications featuring either a single pycnocline or neighbouring pairs of pycnoclines. Particular models investigated include the general version of the intermediate long-wave equation derived by Kubota, Ko and Dobbs to simulate waves in a model system consisting of two homogeneous layers separated by a narrow region of variable density, and the related system of equations derived by Liu, Ko and Pereira for the transfer of energy between waves running along neighbouring pycnoclines. Issues given rigorous mathematical treatment herein include the well-posedness of the initial value problem for these models, the question of existence of solitary wave solutions, and theoretical results about the stability of these solitary waves.

Existence and stability of ground-state solutions of a Schrödinger—KdV system

We consider the coupled Schrodinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δ i , c i , we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves

We consider systems of equations which arise in modelling strong interactions of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we prove the existence and stability of localized solutions representing coupled solitary waves travelling at a common speed. Our results apply in particular to the systems derived by Gear and Grimshaw and by Liu, Kubota and Ko as models for interacting gravity waves in a density-stratified fluid. For the latter system, we also prove that any coupled solitary-wave solution must have components which are all symmetric about a common vertical axis.