Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition

Theoretical study of the phenomenon of blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors. It is said that the maximal time interval of existence of the solution blows up in a finite time when this time is finite, and the solution develops a singularity in a finite time. In fact, semilinear Schrhödinger equation models a lot of physical phenomenon such as nonlinear optics, energy transfer in molecular systems, quantum mechanics, seismology, plasma physics. In the past, certain authors have used numerical methods to study the phenomenon of blow-up for semilinear Schrödinger equations. They have considered the same problem and one proves that the energy of the system is conserved, and the method used to show blow-up solutions are based on the energy's method. This paper proposes a method based on a modification of the method of Kaplan using eigenvalues and eigenfunctions to show that the semidiscrete solution blows up in a finite time under some assumptions. The semidiscrete blow-up time is also estimate. Similar results are obtain replacing the reaction term by another form to generalise the result. Finally, this paper propose two schemes for some numerical experiments and a graphics is given to illustrate the analysis.

When / is finite, it is say that the solution . / of (4)--(6) blows up in a finite time and the time / is called the semidiscrete blow-up time of solution . / . It is show that under some assumptions, the solution of the semidiscrete problem defined in (4)--(6) blows up in a finite time and estimate its semidiscrete blow-up time. This paper proposes also some schemes and algorithms for the numerical calculation of the blow-up time. In the past, certain authors have used numerical methods to study the phenomenon of blow-up for semilinear Schrödinger equations but they have considered the problem (1)--(3) in the case where the term − | , | is replaced by − | , | <) , (see for instance [3,24]). In this case, one proves that the energy of the system is conserved, and the method used to show blow-up solutions are based on the energy's method. This paper propose a method based on a modification of the method of Kaplan (see [14]) using eigenvalues and eigenfunctions to show that the solution . / of (4)--(6) blows up in a finite time if ∑ tan C D 6 ℎE 2<) *F) sin +Hℎ I 8 * is large enough. The above result is also extend replacing -ib in (4) by b and also by J − where b>0 and c>0. One integrates the semidiscrete scheme and obtain some discrete schemes where the convergence and stability have been proved (see for instance [8,11,13,25]). We utilize these schemes to compute the numerical blow-up time by means of appropriate algorithms. In [6,19,20], one may find some results about the numerical study of the phenomenon of blow-up and extinction for semilinear parabolic equations.
This paper is written in the following manner. In the next section, the authors give some conditions under which the solution of (4)--(6) blows up in a finite time and estimate its semidiscrete blow-up time. In the last section, we propose some schemes and algorithms to compute the numerical blow-up time. Some numerical values are given.

Semidiscrete Blow-up Solutions
In this section, under some assumptions, we show that the solution of the semidiscrete problem blows up in a finite time and estimate its semidiscrete blow-up time. One need the following Lemma. Then we have ∑ ℎ. * 5 6 U * = ∑ ℎU * 5 6 Proof. A straightforward computation reveals that ∑ ℎ. * 5 6 U * = ∑ ℎU * 5 6  Then the solution . / of (4)-(6) blows up in a finite time / which is estimated as follows Proof. Since 0, / is the maximal time interval on which ‖. / ‖ % is finite, our aim is to show that / is finite and obeys the above inequality. Introduce the functions v and w defined as follows Taking the derivative of v in t and using (4), we get One observes that 5 6 ' +Hℎ = −d / ' +Hℎ . Due to Lemma 2.2, we arrive at which implies that .
We also observe that, taking the derivative of w in t and using (4), we discover that b i = − L tan S Hℎ 2 T sin +Hℎ 5 6  Since the quantity on the right-hand side of the above inequality is positive, we see that the time / * is estimated as follows is positive, we deduce that Consequently / * = / is finite. Use the fact that j 0 = e to complete the rest of the proof. Now, we consider the following initial-boundary value problem − = | | , ∈ 0, 1 , ∈ 0, 0, = 0, 1, = 0, ∈ 0, , 0 = , ∈ [0,1] where ( > 1, 0 = 0 1 = 0. Approximate the solution u of (9)-(11) by the solution . / = . , … , . 2 # of the following semidiscrete equations where (0, / is the maximal time interval on which ‖. / ‖ % is finite. Our second result on blow-up is the following. Then the solution . / of (12)- (14) blows up in a finite time / which is estimated as follows Proof. Since 0, / is the maximal time interval on which ‖. / ‖ % is finite, our aim is to show that / is finite and obeys the above inequality. Introduce the functions v and w defined as follows Taking the derivative of v in t and using (12), we get .
We also observe that, taking the derivative of w in t and using (12) We deduce that Since This implies that / * = / is finite. Therefore / is finite and use the fact that Z(0)=A to complete the rest of the proof.

2<) *F)
is large enough, the solution . * of the above semidiscrete problem blows up in a finite time.

Numerical Results
In this section, one present some numerical approximations of the blow-up time for the solution of the problem (1)  In the tables 1 and 2 in rows, we present the numerical blow-up times, the numbers of iterations, the CPU times and the orders of the approximations corresponding to meshes of 16, 32, 64, and 128. We take the numerical blow-up time which is computed at the first time when ∆ t ƒ | |9) | | -10 <)z . The order (s) of the method is computed from For the numerical values, we take p=2, . * 20 sin H + , a=1, b=1 and € ,ˆ/ 6 . In this graphics, one can see that the norm of the solution u of the problem (1)-(3) is increasing and develops a singularity in a finite time. Also, we see that the blow-up rate occurs at the middle of the solution for the mesh i=I/2. This graphics respect . 0, . 2 0, ∈ 0, / . But this condition doesn't prevent the blow-up of the solution.

Conclusion
Under some assumption, and using a method based on a modification of the method of Kaplan, it is show that the semidiscete solution of the semilinear solution of the problem (1)-(3) blows up in a finite time and the semidiscrete blow-up time is estimate. The result obtains with the problem (1)-(3) is generalize considering a reaction term more complex. At the end, two schemes proposed, permit to illustrate the estimation of the numerical blow-up time which converge to 0,0782 (see Tables 1 and 2). But the convergence of the schemes proposed was not proof and can be the subject of another investigation.