In this paper, the authors study the blow-up of solution for a class of nonlinear Schrodinger equation for some initial boundary problem. On the other hand, the authors give out some analyses and that new conclusion by Eigen-function method. In last section, the authors check the nonlinear parameter for light rule power by using of parameter method to get ground state and excite state correspond case, and discuss the global attractor of some fraction order case, and combine numerical test. To illustrate this physics meaning in dimension d = 1, 2 case. So, by numerable solution to give out these wave expression.
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 [
As we all know the solution of initial problem for Schrodinger equation bellow
Assume that real part and imaginary part of
are real analytical function for then this solution of the problem may expresses in form:
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 [
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
Then
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 [
and then there exists such that that is
So, we complete the proof of this Theorem 2.1.
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:
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 [
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:
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.
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 [
Now, we may consider the following problem:
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.
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 [
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
Clearly, that is theorem 2.1 in [
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 [
Multiplying by the both sides of (3.4) and integral on for, it is form:
Taking then
and that
By in (I) and Green’s second formula:
Substituting (3.6) into (3.5), we get
Hence,
From
Therefore, we have
Combing (3.7)-(3.8), and Jensen’s inequality, we obtain
Here, So,
there exist, such that
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
That is also. From Jensen inequality and is even function, we have
then
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 [
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 [
In the same way, we can consider the higher-order case (integer):
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 [
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.
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 [
The author search the Cauchy problem for fractional order non-linear Schrodinger equation in [
Physics background of (1) is arise the main part of nonlinear interaction for laser and plasma, express the field of electricity [
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
Proof. Multiply for the both sides of (**) act as inner product, we have
and take real part,
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 andfor 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.
We consider some meaning of physic and Energy for nonlinear Schrodinger equation.The numerical test for solution of nonlinear Schrodinger equation with ground state and excite state.
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 [
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:
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:
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 [
So, by check parameter method in [
We consider two class powers (shake power and light power) in (6.3), Setting shake power
taking initial wave
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).
Consider shake power in [14,24]
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)
We consider three-dimension case,
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]).
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).