No abstract

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 u t + u x + u u x − u x x t = 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 u t + u x + u u x − u x x x = 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.

In part I of this work (Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory J. Nonlinear Sci. 12 283-318), a four-parameter family of Boussinesq systems was derived to describe the propagation of surface water waves. Similar systems are expected to arise in other physical settings where the dominant aspects of propagation are a balance between the nonlinear effects of convection and the linear effects of frequency dispersion. In addition to deriving these systems, we determined in part I exactly which of them are linearly well posed in various natural function classes. It was argued that linear well-posedness is a natural necessary requirement for the possible physical relevance of the model in question.In this paper, it is shown that the first-order correct models that are linearly well posed are in fact locally nonlinearly well posed. Moreover, in certain specific cases, global well-posedness is established for physically relevant initial data.In part I, higher-order correct models were also derived. A preliminary analysis of a promising subclass of these models shows them to be well posed.

In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three spatial dimensions. These systems are symmetric and conservative. Rigorous convergence results are provided showing that solutions of the complete free-surface Euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large. Using this result as a tool, a rigorous justification of all the two-dimensional, approximate systems recently put forward and analysed by Bona, Chen and Saut is obtained. In particular, this remark applies to the original system derived by Boussinesq. The estimates for the difference between the Euler variables and the system variables is better than that obtained in the two-dimensional context by Schneider and Wayne who approximated with a decoupled pair of Korteweg -de Vries equations. Indeed, the limitations inherent in approximating by a decoupled system are clarified in our analysis. Results are obtained both on an unbounded domain with solutions that evanesce at infinity as well as for solutions that are spatially periodic.

We derived here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on R d , d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators with respect to suitable small parameters that depend variously on the amplitude, wave-lengths and depth ratio of the two layers. We rigorously derive classical models and also some model systems that appear to be new. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established.Nousétablissons ici de manière systématique, et pour une grande classe de régimes, des modèles asymptotiques pour la propagation d'ondes internesà l'interface de deux couches de fluides immiscibles de densité différente, sous l'hypothèse de toit rigide et de fond plat. Leséquations complètes pour cette situation (Euler) sont réduitesà un système d'équations d'évolution posé dans le domaine spatial R d , d = 1, 2, et qui comprend deux opérateurs non locaux. Les divers modèles asymptotiques sont obtenus en développant les opérateurs non locaux par rapportà des petits paramètres convenables (dépendant de l'amplitude, de la longueur d'onde et du rapport de hauteur des deux couches). Nousétablissons rigoureusement des modèles classiques ainsi que d'autres qui semblent nouveaux. De plus, on montre la consistance de ces systèmes asymptotiques avec leséquations d'Euler.