We consider the
critical gKdV equation with a saturated perturbation:
, where
and
. For any
initial data
,
the corresponding solution is always global and bounded in
. This
equation has a family of solutions, and our goal is to classify the dynamics near
solitons. Together with a suitable decay assumption, there are only three
possibilities: (i) the solution converges asymptotically to a solitary wave whose
norm is of
size
as
; (ii) the
solution is always in a small neighborhood of the modulated family of solitary waves, but blows
down at
;
(iii) the solution leaves any small neighborhood of the modulated family of the
solitary waves.
This extends the classification of the rigidity dynamics near the ground state for the unperturbed
critical gKdV
(corresponding to
)
by Martel, Merle and Raphaël. However, the blow-down behavior (ii) is completely new,
and the dynamics of the saturated equation cannot be viewed as a perturbation of the
critical
dynamics of the unperturbed equation. This is the first example of classification of the
dynamics near the ground state for a saturated equation in this context. The cases of
critical
NLS and
supercritical gKdV, where similar classification results are expected, are completely
open.
Keywords
gKdV, $L^2$-critical, saturated perturbation, dynamics near
ground state, blow down
