﻿Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations

Applied Mathematics
Vol.3 No.12(2012), Article ID:25463,12 pages DOI:10.4236/am.2012.312263

Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations

Ning Chen*, Jiqian Chen

School of Science, Southwest University of Science and Technology, Mianyang, China

Email: *chenning783@163.com

Received September 18, 2012; revised October 18, 2012; accepted October 26, 2012

Keywords: Nonlinear Schrodinger Equation; Eigen-Function Method; Fractional Order; Blow-Up; Glabal Attractor

ABSTRACT

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.

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 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.

6. Some Notes for Shake Power and Light Power

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 [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).

REFERENCES

1. B. L. Guo, “Initial Boundary Value Problem for One Class of System of Multi-Dimension Inhomogeneous GBBM Equation,” Chinese Annals of Mathematics, Vol. 8, No. 2, 1987, pp. 226-238.
2. B. L. Guo and C. X. Miao, “On Inhomogeneous GBBM Equation,” Journal of Partial Differential Equations, Vol. 8, No. 3, 1995, pp. 193-204.
3. G. G. Cheng and J. Zhang, “Remark on Global Existence for the Superetitical Nonlinear Shrodinger Equation with a Harmonic Potential,” Journal of Mathematical Analysis and Applications, Vol. 320, No. 2, 2006, pp. 591-598. doi:10.1016/j.jmaa.2005.07.008
4. J. Zhang, “Blow-Up of Solutions to the Mixed Problems for Nonlinear Schrodinger Equations,” Journal of Sichuan Normal University, Vol. 3, 1989, pp. 1-8.
5. J. S. Zhao, Q. D. Guo, H. O. Yang and R. Z. Xu, “Blow-Up of Solutions for Initial Value Boundary Problem of a Class of Generalized Non-Linear Schrodinger Equation,” Journal of Nature of Science of Heilongjiang University, Vol. 25, No. 2, 2008, pp. 170-172.
6. N. H. Sweilam and R. F. AI-Bar, “Variational Iteration Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068
7. J. Li and J. Zhan, “Blow-Up for the Stochastic Nonlinear Schrodinger Equation with a Harmonic Potential,” Advances in Mathematics, Vol. 39, No. 4, 2010, pp. 491- 499.
8. J. Zhang, “Sharp Conditions of Global Existence for Nonlinear Schrodinger and Klein-Golden Equations,” Nonlinear Analysis, Vol. 48, No. 1, 2002, pp. 191-207. doi:10.1016/S0362-546X(00)00180-2
9. F. Genoud and C. A. Stuart, “Schrodinger Equations with a Spatially Decaying Non-Linear Existence and Stability of Standing Waves,” Discrete and Continuous Dynamical Systems, Vol. 21, No. 1, 2008, pp. 137-186.
10. S. L. Xu, J. C. Liang and L. Yi, “Exact Solution to a Generalized Nonlinear Schrodinger Equation,” Communications in Theoretical Physics, Vol. 53, No. 1, 2010, pp. 159-165.
11. H. Zhu, Y. Han and J. Zhang, “Blow-Up of Rough Solutions to the Fourth-Order Nonlinear Schrodinger Equation,” Nonlinear Analysts, Vol. 74, No. 17, 2011, pp. 6186- 6201. doi:10.1016/j.na.2011.05.096
12. H. Meng, B. Tian, T. Xu and H. Q. Zhang, “Backland Transformation and Conservation Laws for the VariableCoefficient N-Coupled Schrodinger Equations with Symbolic Computation,” Acta Mathematica Sinica, Vol. 28, No. 5, 2012, pp. 969-974. doi:10.1007/s10114-011-0531-8
13. G. R. Jia, J. C. Zhang, X. Z. Hang and Z. Z. Ren, “Coherent Control of Population Transfer in Li Atoms via Chirped Microwave Pulses,” Chinese Physics Letters, Vol. 26, No. 10, 2009, Article ID: 103201-1-4.
14. N. H. Sweilam and R. F. Ai-Bar, “Variational Iterative Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068
15. C. S. Zhu, “An Estimate of the Global Attractor for the Non-Linear Schrodinger Equation with Harmonic Potential,” Journal of Southwest Normal university, Vol. 30, No. 5, 2005, pp. 788-791.
16. B. L. Guo, T. Q. Han and X. N. Jie, “Existence of Global Smooth Solution to the Periodic Boundary of Fractional Non-Linear Schrodinger Equation,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 468-477. doi:10.1016/j.amc.2008.07.003
17. G. G. Lin and H. J. Gao, “Asymptotic Dynamical Difference between the Nonlocal and Local Swift-Hohenberg Models,” Journal Mathematical Physics, Vol. 41, No. 4, 2000, pp. 2077-2089. doi:10.1063/1.533228
18. L. Wang. J. B. Dang and G. G. Ling, “The Global Attractor of the Fractional Nonlinear Schrodinger Equation and the Estimate of Its Dimension,” Journal of Yunnan University, Vol. 32, No. 2, 2010, pp. 130-135.
19. V. G. Makhankov, “On Stationary Solutions of Schrodinger Equation with a Self-Consistent Satisfying Boussinesq’s Equations,” Physics Letters A, Vol. 500, 1974, pp. 42-44.
20. S. Zhang and F. Wang, “Inter Effects between There Coupled Bose-Einstein Condensates,” Physics Letters A, Vol. 279, 2001, pp. 231-238.
21. A. Aftalion and Q. Do, “Vortices in A Rotating BoseEinstein Condensate: Critical Angular Velocities and Energy Diagrams in the Thomas-Fermi regime,” Physics Letters A, Vol. 64, No. 6, 2001, Article ID: 063603. doi:10.1103/PhysRevA.64.063603
22. C. Tozzo, M. Kramer and F. Dalfovo, “Stability Diagram and Growth Rate of Parametric Resonances in Bose-Einstein Condensates in One Dimensional Optical Lattices,” Physics Letters A, Vol. 72, No. 2, 2005, Article ID: 023613. doi:10.1103/PhysRevA.72.023613
23. J. Y. Zeng, “Quantum Mechanics,” Science Press, Beijing, 2004, pp. 33-144.
24. R. Teman, “Infinite Dimensional Dynamical Systems in Mechanics and Physics,” Springer Verlag, New York, 1988. doi:10.1007/978-1-4684-0313-8

NOTES

*Corresponding author.