BOLETÍN ELECTRÓNICO CIENTÍFICO
DEL NODO BRASILERO
DE INVESTIGADORES COLOMBIANOS
Número 1, 1999

TÍTULO

EXISTENCE AND STABILITY OF SOLITARY WAVE SOLUTIONS OF THE BENJAMIN EQUATION

TIPO

Trabalho completo publicado em Journal of Differential Equations 152:136-159 (1999)

AUTOR

Jaime Angulo Pava
e-mail: angulo@ime.unicamp.br

DIRECCIÓN PARA CONTACTO

IMECC-UNICAMP-C.P. 6065
CEP 13083-970-Campinas
São Paulo, Brazil

RESUMEN
In this work we study certain properties of the solutions of Benjamin's equation ([1], [2])

$\displaystyle \eta_t + 2\eta \eta_{x} -$l$\displaystyle \; L\eta_x- \eta_{xxx} = 0,\;\;x,t \in \Bbb R, \tag 1 $

for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. Here, $ \eta=\eta(x,t)$ is the vertical displacement of the interface between the two fluids at the spatial point $ x$ at time $ t$, and the equation is written in a non-dimensional, scaled form (see Albert, Bona and Restrepo in [3]). The operator $ L=\Cal H\partial_x$ is the composition of the Hilbert transform $ \Cal H$ and the spatial derivative, that is, its Fourier transform is $ \widehat {L\eta}(\xi)=\vert\xi\vert\widehat \eta(\xi)$, and l is a positive constant.

The approach in this paper is focused on the existence and stability of solitary-wave solutions of Eq. (1) using the concentration compactness method introduced by P.L. Lions ([4], [5]). The solitary-wave solutions of (1) are of the form $ \eta(x,t)=\varphi (x+Ct)$, where $ C>0$ is a dimensionless wave speed, and $ \varphi(\xi)$ and its derivatives tend to zero as the variable $ \xi=x+Ct$ approaches $ \pm \infty$. Substituting this form of $ \eta$ into (1) and integrating with respect to $ \xi$ we obtain that $ \varphi$ satisfies the equation

$\displaystyle C\varphi(\xi)-$l$\displaystyle \;L\varphi(\xi)-\varphi''(\xi)+\varphi^2(\xi)=0. $


References
[1] T.B. Benjamin, A new kind of solitary waves J. Fluid Mechanics, 245:401-411, 1992
[2] T.B. Benjamin, Solitary and periodic waves of a new kind Phil. Trans. R. Soc. lond. A, 354:1775-1806, 1996
[3] J.P. Albert, J.L. Bona and J.M. Restrepo, Solitary-wave solutions of the Benjamin equation, Preprint 1996
[4] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré, Anal. Non linéare, 1:109-145, 1984
[5] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincaré, Anal. Non linéare, 4:223-283, 1984