BOLETÍN ELECTRÓNICO CIENTÍFICO
DEL NODO BRASILERO
DE INVESTIGADORES COLOMBIANOS
Número 3(Artículo 4), 2001
TÍTULO
SCALING, STABILITY AND SINGULARITIES FOR NONLINEAR, DISPERSIVE WAVE EQUATIONS: THE CRITICAL CASE
(ON 28(PM)/03/2001)
TIPO: Proyecto Bi-nacional
AUTORES:
J. Angulo1 angulo@ime.unicamp.br,
J. L. Bona2,
F. Linares3
and M. Scialom1
IDIOMA: Inglés
DIRECCIÓN PARA CONTACTO
1IMECC-UNICAMP, Caixa-Postal 6065, 13081-970,
Campinas, SP, Brazil.
2Department of Mathematics and Texas Institute for Computational and
Applied Mathematics, The University of Texas at Austin, Austin, TX 78712
USA.
3IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320,
Brazil.
ENTIDADES QUE FINANCIARON LA INVESTIGACIÓN: partially suported by Grant CNPq # 300654/96-0, JA and # 300964/88-9, MA.
KEYWORDS: nonlinear dispersive waves,
singularity formation, stability, similarity structure, Korteweg-de
Vries-type equations, Schrödinger-type equations
ABSTRACT Download gziped postscript of the complete work (160KB)
For a class of generalized
Korteweg-de Vries equations of the form
posed in
and for the focusing nonlinear Schrödinger equations
posed on
, it is well known that the initial-value
problem is globally in time well posed provided the exponent
is less than a critical power
. For
, it is known for equation (**) and
suspected for equation (*) that large initial data need not
lead to globally defined solutions. It is our purpose here
to investigate the critical case
in
more detail than heretofore. Building on previous work of Weinstein,
Laedke, Spatschek and their collaborators,
earlier work of the present authors and others, a stability
result is formulated for small perturbations of ground-state
solutions of (**) and solitary-wave solutions of (*). This
theorem features a scaling that is natural in the critical
case. When interpreted in the contexts in view, our general
result provides information about singularity formation in
case the solution blows up in finite time and about
large-time asymptotics in case the solution is globally
defined.
BECNBIC,3(4)2001
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