Affiliation:
1. Laboratory of Mathematics, School of Engineering, Graduate School of Advanced Science and Engineering, Hiroshima University , Higashihiroshima , 739-8527 , Japan
Abstract
Abstract
We study the global structure of the oscillatory perturbed bifurcation problem which comes from the stationary logarithmic Schrödinger equation
−
u
″
(
t
)
=
λ
(
log
(
1
+
u
(
t
)
)
+
sin
u
(
t
)
)
,
u
(
t
)
>
0
,
t
∈
I
≔
(
−
1
,
1
)
,
u
(
±
1
)
=
0
,
-{u}^{^{\prime\prime} }\left(t)=\lambda (\log \left(1+u\left(t))+\sin u\left(t)),\hspace{1.0em}u\left(t)\gt 0,\hspace{1.0em}t\in I:= \left(-1,1),\hspace{1.0em}u\left(\pm 1)=0,
where
λ
>
0
\lambda \gt 0
is a parameter. It is known that
λ
\lambda
is a continuous function of
α
>
0
\alpha \gt 0
, written as
λ
(
α
)
\lambda \left(\alpha )
, where
α
\alpha
is the maximum norm
α
=
‖
u
λ
‖
∞
\alpha =\Vert {u}_{\lambda }{\Vert }_{\infty }
of the solution
u
λ
{u}_{\lambda }
associated with
λ
\lambda
. In the field of bifurcation theory, the study of global structures of bifurcation curves is one of the main subjects of research, and it is important to investigate the influence of the oscillatory term on the global structure of bifurcation curve. Due to the effect of
sin
u
\sin u
, it is reasonable to expect that an oscillatory term appears in the second term of the asymptotic formula for
λ
(
α
)
\lambda \left(\alpha )
as
α
→
∞
\alpha \to \infty
(cf. [1]). Contrary to expectation, we show that the asymptotic formula for
λ
(
α
)
\sqrt{\lambda \left(\alpha )}
as
α
→
∞
\alpha \to \infty
does not contain oscillatory terms by the third term of
λ
(
α
)
\sqrt{\lambda \left(\alpha )}
. This result implies that the oscillatory term has almost no influence on the global structure of
λ
(
α
)
\lambda \left(\alpha )
. The result is, therefore, unexpected, new and novel, since such phenomenon as this is not known so far. For the proof, the involved time map method and stationary phase method are used.