Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations ()
1. Introduction
The quantum mechanics theory and application in more field in nature science. The non-linear Schrodinger equation is the basic equation in nonlinear science and widely applied in natural science such as the physics, chemistry, biology, communication and nonlinear optics etc. (See [1-9]) We study this equation to extend them are with important meaning (See[10-12]).
As we all know, the nonlinear Schrodinger equation be description quantum state of microcosmic grain by wave, it is variable for dependent time, and that is most essential equation, which position and action similarly Newton equation in position and action classics mechanics, it is apply to field as optics, plasma physics, laser gather, cohesions etc, particular on that action of power and trap, search analytical solution for Schrodinger equation is also difficult, and more so difficult for complicated power.
Now, we may extend some results in [4] by using Eigen-function method in through paper.
As we all know the solution of initial problem for Schrodinger equation bellow
(1.1)
Assume that real part and imaginary part of
are real analytical function for 
then this solution of the problem may expresses in form:
2. Several Theorems
In this section, we consider the blow-up of solutions to the mixed problems for higher-order nonlinear Schrodinger equation with as bellow.
It is well known the higher order equation:
where
that with new results for higher-order case. Now, we consider the blow-up of solutions to the mixed problems for six-order general Schrodinger equation to extend some results [4] that as bellow form:
(2.1)
Assume that
not identical zero.
Where 
holds complex value function with selfvariable for complex. 
is also complex value
Theorem 2.1. Suppose that nonlinear term 
of problem (2.1) satisfy
,
and 
not identical zero then the classical solution of (2.1) must be for blow-up in finite time in
Proof. Let
(2.2)
Then
(2.3)
By the first Green’s formula, we have
Substituting it into (2.3), then
We may assume 
then we have
Obviously, from 
and 
Therefore, we have
.
By Schwartz inequality:
So,
Inductively, we have
etc.,
Then 
increasing function similar in [4] from
and then there exists 
such that 
that is
So, we complete the proof of this Theorem 2.1.
(As positive integer 
we get it is theorem 3.1 in [4])
Remark. Then we consider that important case is always for the Schrodinger equation may as bellow form
.
Now, we shall consider also in this similar case:
(2.4)
Therefore, we shall obtain the following theorem.
Theorem 2.2. Suppose that non-linear term 
of problem (2.1) satisfy
,
and
then the classical solution of (2.4) must be for blow-up in finite time in 
(as positive 
then it is theorem 3.2 in [4]).
Proof. Since 
satisfies
then 
Thus, from theorem 2.1, we complete the proof of theorem 2.2.
Now, we shall give out the following theorem form. Here, we shall consider the problem:
(2.5)
Theorem 2.3 Suppose that non-linear term 
of problem (2.5) satisfy
,
and
then the classical solution of (2.5) must be for blow-up in finite time in
.
(As positive integer 
then it is theorem 3.2 in [4])
Proof. Since 
we have that
and
Thus, from theorem 2.1, we complete the proof of theorem 2.3. (As 
it is theorem 3.3 in [4])
Now, we may consider the following problem:
(2.6)
where constant 
Theorem 2.4. Assume that 
and 
then the solution of (2.6) must be for blow-up in finite time in
.
Proof. From
then 
satisfy 
and
It holds the condition of theorem 2.1, then by theorem 2.1 that we know the solution of problem (2.6) must be blow-up in finite time. Therefore, we complete the proof of theorem 2.4.
3. Main Results
We consider the initial boundary value of some higher order nonlinear Schrodinger equation. By using of eigenfunction method, we can get new results bellow.
Let
Furthermore, we will consider eight-order nonlinear Schrodinger equation. In first, stating that lemma 3.1.
Lemma 3.1. This Eigen-value problem (see [4])
(*)
As we all know the first Eigen valu1e 
of (*), the corresponding Eigen-function 
assume it with
Let 
be bounded closed domain in 
and by suite smooth conditions of function 
and 
that from Green’s second formula, we easy get following results.
Now, we consider nonlinear Schrodinger equation with eight-order case 
(3.1)
(3.2)
(3.3)
Clearly, 
that is theorem 2.1 in [5].
Theorem 3.1. Assume that problem (3.1)-(3.3) satisfy (where 
out normal direction):
be continuous, convex and even function, here
Then the classical solution of (4.1)-(4.3) must be blowup in finite time.
Proof. (I) step, when 
and 
In the similar way by [5] from that (4.1) first we take the real part of both sides for (4.1), we get that
(3.4)
Multiplying by 
the both sides of (3.4) and integral on 
for
, it is form:
Taking 
then 
and that
(3.5)
By 
in (I) and Green’s second formula:
(3.6)
Substituting (3.6) into (3.5), we get
Hence,
(3.7)
From 
Therefore, we have
(3.8)
Combing (3.7)-(3.8), and Jensen’s inequality, we obtain
(3.9)
Here, 
So, 
there exist
, such that
(3.10)
From 
and Holder inequality, we get 
that is 
Therefore,
Hence,
(II) step, when 
taking that
then 
Therefore, let
we have
Combine (4.1)-(4.8) and
, we obtain that
(3.11)
That is also
. From Jensen inequality and 
is even function, we have
then
(3.12)
From (3.12) and similar (I)-step, we can get
Combine (I)-(II) we complete the proof of theorem 3.1.
Clearly, 
that is theorem 2.1 in [5].
Theorem 3.2. Assume that problem (3.1)-(3.3) satisfy:
and
where 
is continuous, convex and even function;
Then the classical solution for this problem (3.1)-(3.3) is blow-up in finite time.
Proof. From 
we discuss two case:
then 
Taking the imaginary part for both sides of (3.1), similar the method of proof for Theorem 3.1, we can easy have
So, we get that
(II)
we may let 
then 
So, 
Taking the imaginary part for both sides of (1), by (II) and similar the method of proof for theorem 3.1, we can easy have
We get that
Combine (I)-(II), we complete the proof of theorem 3.2.
Corollary 3.3. Clearly 
that is theorem 2.2 in [5]. By ([13] )looking it for some applications.
4. Some Higher-Order Case
In the same way, we can consider the higher-order case (integer
):
(4.1)
(4.2)
(4.3)
Clearly, 
that is problem of eight order case.
Theorem 4.1. Assume that problem (4.1)-(4.3) satisfy
and
where 
is continuous, convex and even function;
and 
Then the classical solution for this problem (4.1)-(4.3) is blow-up in finite time.(omit this similar proof )
Remark 4.2. Assume that (here
)
then we will obtain similar results of theorem 3.2 with more case.
Remark 4.3. (See [6,14]) According to the direction of [6], we may consider that coupled nonlinear Schrodinger equation as in the following iterative formulas in an algorithmic form by VIM:
The solution procedure with initial approximations (omit the details ):
The other components can be obtained directly:
Furthermore, the conserved quantities:
and
where 
This numerical results is with higher accuracy.
5. The Global Attractor of the Fractional NSE
Recently, they also showed that dynamic behavior of large time action to investigate for [15,16], they are deepgoing study global attractor and dimension estimate of integer order non-linear Schrodinger equation in [16].
The author search the Cauchy problem for fractional order non-linear Schrodinger equation in [17]. The author search the global attractor problem for a class of fractional order non-linear Schrodinger equation in [17] and we based on [16-18], and combine [19] obtained the condition of existence of solution for following fractional order non-linear Schrodinger equation:
(5.1)
Physics background of (1) is arise the main part of nonlinear interaction for laser and plasma, 
express the field of electricity [20], where
is with standard perpendicular base, i is imaginary unit, the function 
is with one order derivative 
where 
with some consume effect, and as 
express the integral system with soliton solution.
As 
for (3.1), and (4.1) thirdly section case, we will obtain global attractor of initial value problem (5.1) that first give out Lemma as follows.
Lemma 5.1. Let
is the solution of problem (5.1), and
(5.2)
Proof. Multiply 
for the both sides of (**) act as inner product, we have
(**)
and take real part,
(5.3)
From (5.3) and by use of Gronwall inequality, we obtain
Lemma 5.2. Let
is the solution of problem (1), then 
with uniform bounded.
Proof. To establish inner product for both sides of equation (5.1) with for
, and take real part, we have that
easy get that by (5.1),
where
by use of Jensen’s inequality, we have
So,
by use of Gronwall inequality, we obtain
uniform boundary.
Lemma 5.3. Let
is the solution of problem (5.1), then 
with uniform bounded.
Proof. To derivative both sides of Equation (5.1) for 
and take inner product for
, and taking also imaginary part, we have
Then
By Lemma 5.2 and Young inequality, the (5.4) with form
by use of Gronwall inequality, we obtain
Because hold these inequality bellow
Hence 
are uniform boundary. Similar method of [19,20], we give out that condition of yield global attractor of problem (5.1).
Theorem 5.4. Assume hat
then the periodic global attractor of initial value problem (4.1-4.3):
where 
for operator semi-group with needing define in prove and
for with the bounded attractor set in following in prove processes.
Proof. We omit the proof (by using of similar proof method in [18,19]).
Remark 5.5. Furthermore, we shall study global attractor of fraction order non-linear Schrodinger type equation, and the estimate for its dimensions, and that blowing-up of solution for some fraction order non-linear Schrodinger type equation.
6. Some Notes for Shake Power and Light Power
Atoms absorb energy from the ground state transition to the excited state, learned through experiments in extreme case, the ground state solution is not controlled solution-Blow-up solution.
Thus, strictly control the number and perturbation for impulsive velocity of the atomic transition, is one of the main methods to produce new material structure. Strict control of the atomic transition to the first, second and third excited state is more practical significance, especially the transition to the first excited state. As we all now, the ultra-low temperatures, the atomic gas in the magnetic potential well Boer-Einstein condensation experiments [21], promotion of scholars study the macroscopic quantum behavior of atoms and kinetic characteristics.
By using of above stating method we consider calculate to the ground state solution and excite state of ddimension BECS (Bose-Einstein condensate) with mix harmonic potential and crystal lattice potential.
The Gross-Pitaevskii equation:
(6.1)
where 
expresses mass of atoms, 
be planck constant, 
be number of atoms in cohesion system, 
be outer power,
describe interaction between the atoms cohesion (
means repel; 
shows attract each other). Thus, by pass appropriate immeasurable process, then the (6.1) may be written:
(6.2)
The parameter 
for positive, or negative, describe that repel or attract corresponding, out power 
be defined by physic system for us to study things. By using of the imaginary time method to calculate it in [22] that let 
substituting it into (6.2), we have
(6.3)
So, by check parameter method in [23] we check nonlinear parameter for light rule power, then we get ground state and excite state correspondingly.
6.1. One Dimension Case (d = 1)
We consider two class powers (shake power and light power) in (6.3), Setting shake power
taking initial wave
(6.4)
to calculate ground state 
For (6.4) we calculate first arouse state 
space field for 
the time step for 
Similar above way, taking
and (6.3) for
and 
and 
On the other hand, by the MATLAB search the solution of Equation (6.3) in case (1) and (2) as follow with 
(See Figures 1 and 2).
6.2. Two-Dimension Case (d = 2)
Consider shake power in [14,24]
Figure 1. Ground state phi0. First excited state phi1. V = x2/2; b = 500, bi = 2.
The grain energy:
We take initial wave function for
To calculate ground state
; For
and
.
By calculating 
along the direction of axe 
and 
in direction of axe y, and calculating first excited of 
along direction for axe x and axe y, and space field for 
time step:
Combine these cases as Fig: (See Figures 3(a) and (b), Figures 4-6)
Figure 2. Ground state phi0. First excited state phi1. V = x2/2 + 25*(sin(pi*x/4))2; b = 500, bi = 2.
(a)
(b)
Figure 3. (a) Ground state phi0. First excited state phi1. a = 5, b = 2. (b) Ground state phi0. First excited state phi1. a = 5, b = 2.
Figure 4. Ground state phi0 a = 5, b = 2.
Figure 5. First excited state phi1-x a = 5, b = 2.
Figure 6. First excited state phi2-y a = 5, b = 2.
Figure 7. First excited state phi3-xy a = 5, b = 2.
We consider three-dimension case, Figure 4 for ground state 
corresponding case, the 
as with express along direction of axe x (wave surface) in Figure 5, the 
as with express along direction of axe y (wave surface) in Figure 6, the 
as for express along direction of axe x and axe y (wave surface) in Figure 7.
7. Concluding Remarks
Recently, the higher-order Schrodinger differential equations is also a very interesting topic, and that application of some physics and mechanics of for some more fields as nonlinear Schrodinger equations and some compute methods etc. In our future work, we may obtain some better results.
The application of some physics and mechanics of for some more fields with some combine equations (look [7, 13]).
8. Acknowledgements
This work is supported by the Nature Science Foundation (No.11ZB192) of Sichuan Education Bureau (No.11zd 1007 of Southwest University of Science and Technology).
NOTES
*Corresponding author.