Comparison of quarter-plane and two-point boundary value problems: the BBM-equation
Jerry Bona Hongqiu Chen Shu Ming Sun B.-Y. Zhang
The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.
keywords: two-point boundary value problems BBM equation comparison principles. quarter-plane problems Nonlinear dispersive wave equations
Sharp well-posedness results for the BBM equation
Jerry Bona Nikolay Tzvetkov
The regularized long-wave or BBM equation

$ u_{t}+u_{x}+u u_{x}-u_{x x t} = 0 $

was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg-de Vries equation. Considered here is the initial-value problem wherein $u$ is specified everywhere at a given time $t = 0$, say, and inquiry is then made into its further development for $t>0$. It is proven that this initial-value problem is globally well posed in the $L^2$-based Sobolev class $H^s$ if $s \geq 0$. Moreover, the map that associates the relevant solution to given initial data is shown to be smooth. On the other hand, if $s < 0$, it is demonstrated that the correspondence between initial data and putative solutions cannot be even of class $C^2$. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in $H^s$-based spaces for $s < 0$. One is thus led to surmise that the initial-value problem for the BBM equation is not even locally well posed in $H^s$ for negative values of $s$.

keywords: BBM-equation global well-posedness sharp well-posedness
Well-posedness for regularized nonlinear dispersive wave equations
Jerry Bona Hongqiu Chen
In this essay, we study the initial-value problem

$ u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,$
$u(x,0)=u_0(x), \qquad x\in\mathbb R, (0.1)$

where $u=u(x,t)$ is a real-valued function, $L$ is a Fourier multiplier operator with real symbol $\alpha(\xi),$ say, and $g$ is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for $L_p$, $H^s$ and bore-like or kink-like initial data. In the special case where $\alpha(\xi)=|\xi|^{r}$ wherein $r>1$ and $g(u)=1/2u^2,$ (0.1) is globally well-posed in time if $s$ and $r$ satisfy a simple algebraic relation.

keywords: regularized equation nonlinear dispersive wave equations
Solitary waves in nonlinear dispersive systems
Jerry Bona Hongqiu Chen
Evolution equations that feature both nonlinear and dispersive effects often possess solitary-wave solutions. Exact theory for such waves has been developed and applied to single equations of Korteweg-de Vries type, Schrödinger-type and regularized long-wave-type for example. Much less common has been the analysis of solitary-wave solutions for systems of equations. The present paper is concerned with solitary travelling-wave solutions to systems of equations arising in fluid mechanics and other areas of science and engineering. The aim is to show that appropriate modification of the methods coming to the fore for single equations may be effectively applied to systems as well. This contention is demonstrated explicitly for the Gear- Grimshaw system modeling the interaction of internal waves and for the Boussinesq systems that arise in describing the two-way propagation of long-crested surface water waves.
keywords: solitary wave internal waves Fréchet space Concentration compactness dispersion positive operator travelling waves.
Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane
Jerry Bona Jiahong Wu
Studied here is the large-time behavior and eventual periodicity of solutions of initial-boundary-value problems for the BBM equation and the KdV equation, with and without a Burgers-type dissipation appended. It is shown that the total energy of a solution of these problems grows at an algebraic rate which is in fact sharp for solutions of the associated linear equations. We also establish that solutions of the linear problems are eventually periodic if the boundary data are periodic.
keywords: Large-time behavior BBM-Burgers equation KdV equation BBM equation KdV-Burgers equation eventual periodicity
Singularity formation in the generalized Benjamin-Ono equation
Jerry Bona H. Kalisch
A Fourier-collocation scheme is used to approximate solutions to the generalized Benjamin-Ono equation $u_t + u^pu_x - H u_{x x} = 0$. The numerical simulation suggests that the equation features smooth solutions that become unbounded in finite time.
keywords: integro-differential equations singularity formation. Nonlinear waves Fourier-collocation
Jerry Bona T. Colin Mathieu Colin David Lannes
This special issue of Discrete and Continuous Dynamical Systems grows out of a focused session at the AIMS 6 meeting held in Poitiers, France, in 2006. The session struck its organizers, who were perhaps not unbiased, as very successful. As a consequence, when Shouchuan Hu approached us about editing a special issue based on the papers delivered in the session, we were enthusiastic at the prospect.
The focus of the session was asymptotic models of physical phenomena. This is a large subject, and one special session cannot hope to do it justice. Consequently, some focal point was required, and in the event, most of the lectures in the session were centered upon nonlinear wave equations arising in plasma physics and fluid mechanics. This is also the subject around which most of this special issue turns.

For more information please click the "Full Text" above.
A KdV-type Boussinesq system: From the energy level to analytic spaces
Jerry L. Bona Zoran Grujić Henrik Kalisch
Considered here is the well-posedness of a KdV-type Boussinesq system modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid when the motion is sensibly two dimensional. Solutions are obtained in a range of Sobolev-type spaces, from the energy level to the analytic Gevrey spaces. In addition, a criterion for detecting the possibility of blow-up in finite time in terms of loss of analyticity is derived.
keywords: Boussinesq systems Gevrey-space analysis. two-way propagation of water waves analyticity well-posedness for nonlinear dispersive equations
Comparison of quarter-plane and two-point boundary value problems: The KdV-equation
Jerry L. Bona Hongqiu Chen Shu-Ming Sun Bing-Yu Zhang
This paper is concerned with the Korteweg-de Vries equation which models unidirectional propagation of small amplitude long waves in dispersive media. The two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of length $L$ of the media of propagation is considered. It is shown that the solution of the two-point boundary value problem converges as $L\rightarrow +\infty$ to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. In addition to its intrinsic interest, our result provides justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem for the KdV equation.
keywords: Two-point boundary value problem. Quarter-plane problem KdV equation
Stability of solitary-wave solutions to the Hirota-Satsuma equation
Jerry L. Bona Didier Pilod
The evolution equation

$ u_t- $ uxxt$ +u_x-$uut$ +u_x\int_x^{+\infty}u_tdx'=0, $ (1)

was developed by Hirota and Satsuma as an approximate model for unidirectional propagation of long-crested water waves. It possesses solitary-wave solutions just as do the related Korteweg-de Vries and Benjamin-Bona-Mahony equations. Using the recently developed theory for the initial-value problem for (1) and an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

keywords: Solitary waves nonlinear dispersive wave equations stability Korteweg-de Vries-type equations.

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