São Paulo Journal of Mathematical Sciences
https://doi.org/10.1007/s40863-019-00144-5
ORIGINAL PAPER
Initial boundary value problems for some nonlinear
dispersive models on the half-line: a review and open
problems
Márcio Cavalcante
1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2019
Abstract
In the last years the study of initial boundary value problems for nonlinear dispersive
equations on the half-lines has given attention of many researchers. This turns out to
be a rather challenging problem, mainly when studied in low Sobolev regularity. In
this note we present a review of the main results about this topic and also introduce
interesting open problems which still requires attention from the mathematical point
of view.
Keywords Initial boundary value problems · Nonlinear dispersive equations · Open
problems
Mathematics Subject Classification 35Q53 · 35Q55 · 35G31
1 Introduction
In this note, we comment some recent results of the initial boundary value problems
(IBVPs) for some nonlinear dispersive models in the form
i ∂
t
u (x , t ) − (i ∂
x
)
j
u (x , t ) = N (u ,∂
x
u (x , t ),...,∂
k
x
u (x , t )), x ∈ I , t ∈ (0, T )
u (x , 0) = u
0
(x ), x ∈ I
(1.1)
where, j and k are integer numbers, the set I is the positive half-line R
+
or the
negative R
−
and N is a nonlinear function. Models in the form (1.1) is considered
with an appropriate number of conditions on the boundary x = 0, which depends of
the dispersion term (i ∂
x
)
j
and it is given by the physical condition of the model. Note
that, in the case, where j is a odd number equation (1.1) changes his behavior when
B Márcio Cavalcante
marciocavalcante@im.ufal.br
1
Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL 50740-545, Brazil
123