1. Introduction
In this paper, we study the stability theory of solutions to the nonlinear Schrödinger equation (NLS).
We consider the Cauchy problem for the nonlinear Schrödinger equation
(1.1)
where
, 
the solution 
is a complex-valued function in 
The Equation (1.1) is called mass-critical or 
critical if
, and it is called mass-supercritical and energy-subcritical when
.
The solutions to (1.1) have the invariant scaling
(1.2)
Definition 1.1 (Solution) Let 
such that
. A function 
is a strong solution to (1.1)
if and only if it belongs to
, and for all 
satisfies the integral equation
(1.3)
A function 
is a weak solution to (1.1)
if and only if
, and for all 
satisfies the integral Equation (1.3).
The solutions to (1.1) have the mass
where
Energy 
where,
Definition 1.2 The problem (1.1) is locally wellposed in 
if for any 
there exist a time 
and an open ball 
in 
such that
, and a subset 
of
, such that for each 
there exists a unique solution 
to the Equation (1.3), and furthermore, the map 
is continuous from
. If 
can be taken arbitrarily large
, the problem is globally wellposed.
Definition 1.3 A global solution 
to (1.1) is scattering in 
as 
if there exists 
such that
Similarly, we can define scattering in 
for
.
For more definition of critical case see [1-3].
In this paper we discuss stability theory of the mass critical, mass-supercritical and energy-subcritical of solution to the nonlinear Schrödinger equation. In section three we discuss the stability of the mass critical solutions and in section four mass-supercritical and energysub-critical solutions are discussed.
Theorem 1.1 Let 
and
. Then there exists a unique maximal-lifespan solution 
to (1.1) with 
and initial data
. Moreover:
1) The interval 
is an open subset of
.
2) For all
, we have 
so, we define
.
3) If the solution 
does not blow up forward in time, then
, and moreover 
scatters forward in time to 
for some
. Converselyif 
then there exists a unique maximallifespan solution 
which scatters forward in time to
.
4) If the solution u does not blow up backward in time, then 
and moreover 
scatters backward in time to 
for some
. Conversely, if
then there exists a unique maximallifespan solution 
which scatters backward in time to
.
5) If 
where a constant 
depending only on 
then
.
In particular, no blowup occurs and we have global existence and scattering both ways.
6) For every 
and 
there exists 
With property: if 
is a solution (not necessarily maximal-lifespan) such that
and
, 
are such that
, then there exists a solution
with 
such that 
and 
for all
.
For proof: See [4-6].
Now in the following we will discuss Standard local well-posedness theorem.
Theorem 1.2 Let
, 
and let
Assume that 
if 
is not an even integer. Then there exists 
such that if between 
and 
there is a compact interval containing zero such that
(1.4)
then there exists a unique solution u to (1.1) on
. Furthermore, we have the bounds
(1.5)
(1.6)
(1.7)
where 
for the closure of all test functions under this norm.
2. Strichartz Estimate
In this section we discus some notation and Strichartz estimate.
2.1. Some Notation
We write 
anywhere in this work whenever there exists a constant 
independent of the parameters, so that
. The shortcut 
denotes a finite linear gathering of terms that “look like” X, but possibly with some factors changed by their complex conjugates.
We start by the definition of space-time norms
The inhomogeneous Sobolev norm 
(when 
is an integer) is defined by:
When s is any real number as
The homogeneous Sobolev norm 
defines as:
For any space time slab
We use 
to denote the Banach space of function 
whose norm is
With the usual adjustments when 
or 
is equal to infinity. When 
we abbreviate 
as
.
A Gagliardo-Nirenberg type inequality for Schrö- dinger equation the generator of the pseudo conformal transformation 
plays the role of partial differentiation.
2.2. Strichartz Estimate
Definition 2.1 The exponent pair 
is says the Schrödinger-admissible if
, and 
Definition 2.2 The exponent pair 
is says the Schrödinger-acceptable if
Let 
be the free Schrödinger evolution. From the explicit formula
we obtain the standard dispersive inequality
(2.1)
for all
.
In particular, as the free propagator conserves the 
- norm,
(2.2)
For all 
If 
solves the inhomogeneous Equation (1.1) for some
and 
in the integral.
Duhamel (1.3). Then we have
(2.3)
for some constant 
depending only on the dimension
.
For some constant 
depending only on 
we have the Holder inequality
We now return to prove Theorem 1.2.
Proof Theorem 1.2 The theorem follows from a contraction mapping argument. More accurate, defined
using the Strichartz estimates, we will show that the map 
is a contraction on the set 
where
under the metric given by
Here 
denotes a constant that changes from line to line. Note that the norm appearing in the metric scales like
. Note also that both 
and 
are closed (and hence complete) in this metric.
Using the Strichartz inequality and Sobolev embedding, we find that for 
And similarly,
Arguing as above and invoking (1.4), we obtain
Thus, choosing 
suciently small, we see that for
, the functional 
maps the set 
back to itself. To see that 
is a contraction, we repeat the above calculations to obtain
Therefore, choosing 
even smaller (if necessary), we can ensure that 
is a contraction on the set
. By the contraction mapping theorem, it follows that 
has a fixed point in
. Furthermore, noting that 
maps into 
(not just
). We now turn our attention to the uniqueness. Since uniqueness is a local property, it enough to study a neighbourhood of 
By Definition of solution (and the Strichartz inequality), any solution to (1.1) belongs to 
on some such neighbourhood. Uniqueness thus follows from uniqueness in the contraction mapping theorem.
The claims (1.6) and (1.7) follow from another application of the Strichartz inequality. □
Remark 2.1 By the Strichartz inequality, we know that
Thus, (1.4) holds with 
for initial data with suciently small norm instead that, by the monotone convergence theorem, (1.4) holds provided 
is chosen suciently small. Note that by scaling, the length of the interval 
depends on the fine properties of
, not only on its norm.
3. Stability of the Mass Critical
In this section we discuss the stability theory at mass critical case. Consider the initial-value problem (1.1)
with 
.An important part of the local well-posedness theory is the study of how the strong solutions built in the past subsection depend upon the initial data. More accurate, we want to know if the small perturbation of the initial data gives small changes in solution. In general, we take care in developing a stability theory for nonlinear Schrödinger Equation (1.1). Even though stability is a local question, it plays an important role in all existing treatments of the global well-posedness problem for nonlinear Schrödinger equation at critical case, for more see [7]. It has also proved useful in the treatment of local and global questions for more exotic nonlinearities [8,9]. In this section, we will only discus the stability theory for the mass-critical NLS.
Lemma 3.1 Let 
be a compact interval and let 
be an approximate solution to (1.1) meaning that
for some function
. Suppose that
(3.1)
for some positive constant
. Let 
and let 
be such that
(3.2)
for some
. Suppose also the smallness conditions
(3.3)
(3.4) 
(3.5)
for some 
where 
is a small constant. Then, there exists a solution 
to (1.1) on 
with initial data 
at time 
satisfying
(3.6)
(3.7)
(3.8)
(3.9)
Proof: By symmetry, we may assume
. Let
. Then 
satisfies the initial value problem
For 
we define
By (3.3),
(3.10)
Furthermore, by Strichartz, (3.4), and (3.5), we get
(3.11)
Combining (3.10) and (3.11), we obtain
A standard continuity argument then shows that if 
is taken sufficiently small,
which implies (3.9). Using (3.9) and (3.11), we obtained (3.6). Furthermore, by Strichartz, (3.2), (3.5), and (3.9),
which establishes (3.7) for 
sufficiently small.
To prove (3.8), we use Strichartz, (3.1), (3.2), (3.9), and (3.3):
Choosing 
sufficiently small, this finishes the proof. □
Based on the previous result, we are now able to prove stability for the mass-critical NLS.
Theorem 3.2 Let 
be a compact interval and let 
be an approximate solution to (1.1) in the sense that
for some function
. Assume that condition (3.1) in Lemma 3.1 holds and
(3.12)
for some positive constant
. Let 
and let 
obey (3.2) for some
. Furthermore, suppose the smallness conditions (3.4), (3.5) in Lemma 3.1. For some 
where
is a small constant. Then, there exists a solution 
to (1.1) on 
with initial data 
at time 
satisfying
(3.13)
(3.14)
(3.15)
Proof: Subdivide 
into 
subintervals 
such that
where 
is as in Lemma 3.1. We replaced 
by 
as the mass of the difference 
might grow slightly in time. By choosing 
sufficiently small depending on 
and
, we can apply Lemma 3.1 to obtain for each 
and all 
Provided, we can prove that their counterparts of (3.2) and (3.4) hold with replace 
by
. To verify this, we use an inductive argument. By Strichartz, (3.2), (3.5), and the inductive hypothesis,
Similarly, by Strichartz, (3.4), (3.4), and the inductive hypothesis,
Choosing 
sufficiently small depending on 
and
, we can ensure that hypotheses of Lemma 3.1 continue to hold as j varies. □
Lemma 3.3 (Stability) Fix 
and
. For every 
and 
there exists 
with the property: if 
is such that 
and that 
approximately solves (1.1) in the sense that
. (3.16)
And
, 
are such that
Then there exists a solution 
to (1.1) with 
such that
.
Note that, the masses of 
and 
do not appear immediately in this lemma, although it is necessary that these masses are finite. Similar stability results for the energy-critical NLS (in
) instead of
, of course) have appeared in [10-14]. The mass-critical case it is actually slightly simpler as one does not need to deal with the existence of a derivative in the regularity class. For more see [15].
Proof: (Sketch) First let prove the claim when 
is suciently small depending on
. Let 
be the maximal-lifespan solution with initial data
. Writing 
on the interval
, we see that
and
.
Thus, if we set
by the triangle inequality, (2.3), and (3.16), we have
hence, by (2.4) and the hypothesis
,
where 
depends only on
. If 
is suciently small depending on
, and 
is suciently small depending on 
and
, then standard continuity arguments give 
as desired. To deal with the case when 
is large, simply iterate the case when 
is small (shrinking
, 
repeatedly) after a subdivision of the time interval
.
4. Stability of the Mass-Supercritical and Energy-Subcritical
In this section we discuss the Stability theory of the mass-supercritical and energy-subcritical to the nonlinear Schrödinger equation. Consider the initial-value problem (1.1) with 
and 
we chose
.
In this case the initial-value problem 
is locally well-posed in
. Now we rewrite (1.1) as
(1.1)*
We discuss the stability by the following proposition. Before beginning we need define the Kato inhomogeneous Strichartz estimate. See [16]
(4.1)
Proposition 4.1 For each 
there exists 
and 
such that the following holds.
Let 
and solve
.
Let 
for all 
and define
If
And
(4.2)
Then
Proof: Let w be defined by 
then 
solves the equation
(4.3)
since
.Can be divided 
into
in intervals 
Such that for all
, the quantity
, is Appropriate small
(δ to be selected below).
Integration (4.3) with initial time 
is
(4.4)
where
.
Applying the Kato Strichartz estimate (4.1) on
, to obtain
. (4.5)
Note that
.
Similarly,
and
Substituting the above estimates in (4.5), to get,
(4.6)
As long as
and
(4.7)
We obtain
(4.8)
Taken now 
in (4.4) and apply 
to both sides to obtain
(4.9)
Since the Duhamel integral is restricted to
by again applying the Kato estimate, similarly to (4.6) we obtain,
By (4.8) and (4.9), we bound the Former of expression to obtain
Start iterates with
, we obtain
To absorb the second part of (4.7) for all intervals 
we require
(4.10)
We review that the dependence of parameters δ is an absolute constant chosen to meet the first part of (4.7). The inequality (4.10) determines how the small 
needs to be taken in terms of 
(and thus, in terms of
). We were given 
which then determined 
□