Jerry L. Bona

Model Equations for Long Waves in Nonlinear Dispersive Systems

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation ut+ux+uux−uxxt=0,(a), whose solution u(x,t) is considered in a class of real nonperiodic functions defined for ࢤ∞ <x< ∞,t≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation ut+ux+uux−uxxx=0,(b) with which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of (a) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.

The initial-value problem for the Korteweg-de Vries equation

For the Korteweg-de Vries equation ut+ux+uux+uxxx=0, existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞<x<∞) and the periodic initial-value problem (posed on 0 ⩽x⩽l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L2-derivatives as the initial data and these derivatives depend continuously on time, as elements of L2. The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.

Stability and Instability of Solitary Waves of Korteweg-de Vries Type

Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear, dispersive long waves. Special cases for which our analysis is decisive include equations of the Korteweg-de Vries and Benjamin-Ono type. Necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable.

Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Summary. Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.

On the stability theory of solitary waves

Improvements are made on the theory for the stability of solitary waves developed by T. B. Benjamin. The results apply equally to the Kortewegde Vries equation and to an alternative model equation for the propagation of long waves in nonlinear dispersive media.

Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation

AbstractCertain generalizations of one of the classical Boussinesq-type equations,

Nonlocal models for nonlinear, dispersive waves

u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx}

Solitary‐wave interaction

are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.