Non Scattering of the Solution of the Nonlinear Schrödinger Equation on the Torus

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2017.59162 Oct. 16, 2017 1917 Journal of Applied Mathematics and Physics

Non-Scattering of the Solution of the Nonlinear

Schrödinger Equation on the Torus

Xing Cheng, Qiang Li

College of Science, Hohai University, Nanjing, China

Abstract

In this article, we will show non-scattering of the solution of the nonlinear Schrodinger equation on the torus. The result extends the result of Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T. for the cubic nonlinear Schrödinger equation on 2-dimensional torus.

Keywords

Schrödinger Equation, Torus, Scattering

1. Introduction

In this article, we will consider the nonlinear Schrödinger equation on d-dimensional torus:

� 𝑖𝑖𝜕𝜕𝑡𝑡𝑢𝑢+∆𝑢𝑢 = |𝑢𝑢|𝑝𝑝−1𝑢𝑢.

𝑢𝑢(0,𝑥𝑥) = 𝑢𝑢0(𝑥𝑥)∈ 𝐻𝐻1(𝕋𝕋𝑑𝑑), (1)

where 𝑢𝑢: ℝ×𝕋𝕋𝑑𝑑 _{→ ℂ ,1 < p < ∞. This kind of nonlinear Schrödinger equations} has been studied intensively in the last two decades. See [1] [2] [3] [4].

Many mathematicians believe this equation does not have nontrivial solutions which scatter, i.e. which approach a solution to the linear equation at time t =

∞. Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T. [5] consider the cubic nonlinear Schrödinger equation on two dimensional torus, and prove the solution cannot scatter to free solution in 𝐻𝐻1_(𝕋𝕋𝑑𝑑_).

As in [5], the explicit solution

𝑢𝑢(𝑡𝑡,𝑥𝑥) = Ae𝑖𝑖𝑖𝑖_𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥_𝑒𝑒𝑖𝑖𝑡𝑡|𝑖𝑖|2

𝑒𝑒𝑖𝑖|𝐴𝐴|𝑝𝑝−1_𝑡𝑡

, (2) where 𝐴𝐴 ≥0,𝑖𝑖 ∈₂ℝ_𝜋𝜋ℤ,𝑖𝑖 ∈ ℤ𝑑𝑑 _{cannot converge to a free solution e}𝑖𝑖𝑡𝑡∆_u

+ due to

the presence of the phase rotation e𝑖𝑖|𝐴𝐴|𝑝𝑝−1𝑡𝑡 _{which is caused by the nonlinearity.} We will show this is the only solution that scatters modulo phase rotation in H1

How to cite this paper: Cheng, X. and Li, Q. (2017) Non-Scattering of the Solution of the Nonlinear Schrödinger Equation on the Torus. Journal of Applied Mathematics and Physics, 5, 1917-1922.

https://doi.org/10.4236/jamp.2017.59162

DOI: 10.4236/jamp.2017.59162 1918 Journal of Applied Mathematics and Physics in the sense that there exists 𝑢𝑢+∈ 𝐻𝐻1(𝕋𝕋𝑑𝑑) and function 𝜃𝜃:ℝ → ℝ/2πℤ such

that

�𝑢𝑢(𝑡𝑡)− 𝑒𝑒𝑖𝑖𝜃𝜃(𝑡𝑡)_𝑒𝑒−𝑖𝑖𝑡𝑡∆_𝑢𝑢

+�_𝐻𝐻1_(𝕋𝕋𝑑𝑑₎→0,𝑎𝑎𝑎𝑎𝑡𝑡 → ∞.

is of the form (2), which then reveals that no solution of the nonlinear Schrödinger equation on torus can scatter to free solution.

2. Main Theorem

In this section, we will present the main theorem in this article. We will show the only solution that scatters modulo phase rotation is of the form (2).

Theorem 1 (No non-trivial solution scatters modulo phase rotation). Let

𝑢𝑢: ℝ×𝕋𝕋𝑑𝑑 _{→ ℂ be an H}1 _{solution to (1) which scatters modulo phase in H}1_,

then u is of the form (2) for some 𝐴𝐴 ≥0,𝑖𝑖 ∈ ℝ/2πℤ,𝑖𝑖 ∈ ℤ𝑑𝑑_. To prove this theorem, we first need some lemmas.

Lemma 2 (Pre-compactness). For 𝑢𝑢+∈H1(𝕋𝕋𝑑𝑑) , {𝑒𝑒𝑖𝑖𝜃𝜃𝑒𝑒𝑖𝑖𝑡𝑡∆𝑢𝑢+∶ 𝜃𝜃 ∈

ℝ/2𝜋𝜋ℤ,𝑡𝑡 ∈ ℝ} is pre-compact in 𝐻𝐻1_(𝕋𝕋𝑑𝑑_).

Proof. It is equivalent to show {𝑒𝑒𝑖𝑖𝜃𝜃_𝑒𝑒−𝑖𝑖𝑡𝑡|𝑖𝑖|2

𝑢𝑢�+(𝑖𝑖): 𝜃𝜃 ∈₂ℝ_πℤ,𝑡𝑡 ∈ ℝ} is pre-

compact in l2_{(〈𝑖𝑖〉}2_{𝑑𝑑𝑖𝑖) after taking Fourier transforms. By monotone}

conver-gence, ∀ϵ>0, there exists R > 0 such that ∑ 〈𝑖𝑖〉2_|𝑢𝑢�

+(𝑖𝑖)|2≤ 𝜖𝜖

𝑖𝑖∈ℤ𝑑𝑑_{,𝑖𝑖≥𝑅𝑅} . We can

conclude {𝑒𝑒𝑖𝑖𝜃𝜃_𝑒𝑒𝑖𝑖𝑡𝑡| |2

𝑢𝑢�+:𝜃𝜃 ∈ ℝ/2πℤ,𝑡𝑡 ∈ ℝ} is covered by finitely many balls of

radius 𝑂𝑂(𝜖𝜖), and the claim follows. □

Lemma 3 (Diamagnetic inequality) If 𝑢𝑢 ∈H1_(𝕋𝕋𝑑𝑑_{), then |𝑢𝑢|∈H}1_(𝕋𝕋𝑑𝑑_{) and}

‖|𝑢𝑢|‖_𝐻𝐻1_(𝕋𝕋𝑑𝑑₎≤ ‖𝑢𝑢‖_𝐻𝐻1_�𝕋𝕋𝑑𝑑_�.

Proof. It suffices to verify this when 𝑢𝑢 is smooth. For any ϵ> 0, we have 2��𝜖𝜖2_{+ |𝑢𝑢|}2_{∇�𝜖𝜖}2_{+ |𝑢𝑢|}2_{� = |∇(𝜖𝜖}2_+|𝑢𝑢|2_{)| = 2|𝑅𝑅𝑒𝑒(𝑢𝑢�∇𝑢𝑢)|≤2|𝑢𝑢||∇𝑢𝑢|,}

and hence

�∇�𝜖𝜖2_{+ |𝑢𝑢|}2_{� ≤|∇𝑢𝑢|. Taking distributional limits as ϵ →0, we obtain the}

claim.

□ Lemma 4 (H1 _{has no step functions). Let 𝑢𝑢 ∈H}1_(𝕋𝕋𝑑𝑑_{) be such that 𝑢𝑢(𝑥𝑥)} takes at most two value. Then u is constant.

Proof. We may assume that 𝑢𝑢 takes 0 and 1 only, thus u2_{=𝑢𝑢. On the one}

hand, differentiating this we obtain 2𝑢𝑢∇𝑢𝑢 = ∇𝑢𝑢, thus (1-2𝑢𝑢)∇𝑢𝑢 = 0. On the other hand, since u2_{=𝑢𝑢, (1−2u)}2_{= 1, and thus ∇𝑢𝑢= 0, therefore 𝑢𝑢 is}

constant.

□ Proof of Theorem 1. Let u, u+,θ be as above. We may assume 𝑢𝑢 has

non-zero mass. From Lemma 2, we see that {u(t): t∈[0,∞)} is precompact in

𝐻𝐻1_{. Thus we can find a sequence t}

𝑚𝑚 → ∞ such that 𝑢𝑢(t𝑚𝑚)→ 𝑣𝑣0 in H1(𝕋𝕋𝑑𝑑), as

DOI: 10.4236/jamp.2017.59162 1919 Journal of Applied Mathematics and Physics Since 𝑢𝑢 has non-zero mass, we see 𝑣𝑣+ also has non-zeromass. Let

𝑣𝑣(𝑚𝑚)_{:ℝ×𝕋𝕋}𝑑𝑑 _{→ ℂ be the time-translated solution 𝑣𝑣}(𝑚𝑚)_{(𝑡𝑡) = 𝑢𝑢(𝑡𝑡+𝑡𝑡}

𝑚𝑚), thus

𝑣𝑣(𝑚𝑚)_{(0)→ 𝑣𝑣}

0 in 𝐻𝐻1(𝕋𝕋𝑑𝑑). Let 𝑣𝑣:ℝ×𝕋𝕋𝑑𝑑 → ℂ be solution to NLS with initial

data 𝑣𝑣(0) =𝑣𝑣0. By the local well-posedness theory in 𝐻𝐻1 we conclude that

𝑣𝑣(𝑚𝑚) _{converge uniformly in 𝐻𝐻}1 _{to 𝑣𝑣 on every compact time interval [-T,T].}

On the other hand, by hypothesis, ∀𝑡𝑡 we have

�𝑣𝑣(𝑚𝑚)_{(𝑡𝑡)− 𝑒𝑒}𝑖𝑖𝜃𝜃(𝑡𝑡+𝑡𝑡𝑚𝑚)_𝑒𝑒𝑖𝑖(𝑡𝑡+𝑡𝑡𝑚𝑚)∆_𝑢𝑢+�

𝐻𝐻1→0,𝑎𝑎𝑎𝑎𝑚𝑚 → ∞. Since 𝑒𝑒𝑖𝑖𝑡𝑡𝑚𝑚∆_{𝑢𝑢+→ 𝑣𝑣+, as 𝑚𝑚 → ∞}, we conclude that

�𝑣𝑣(𝑚𝑚)_{(𝑡𝑡)− 𝑒𝑒}𝑖𝑖𝜃𝜃(𝑡𝑡+𝑡𝑡𝑚𝑚)_𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣+�

𝐻𝐻1 →0,𝑎𝑎𝑎𝑎𝑚𝑚 → ∞. By taking limits, we conclude that

𝑣𝑣(𝑡𝑡) = 𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡)_𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣

+

for some α(t)∈ ℝ/2πℤ.

In particular, since 𝑣𝑣+ has non-zero mass, we have

𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡) ₌ 1

‖𝑣𝑣+‖2𝐿𝐿2� 𝑣𝑣(𝑡𝑡)𝑒𝑒 𝑖𝑖𝑡𝑡∆_𝑣𝑣

+d𝑥𝑥.

𝕋𝕋𝑑𝑑

From (1) and Sobolev, we see that 𝑣𝑣(t) is continuously differentiable in

𝐻𝐻−1_(𝕋𝕋𝑑𝑑_{), and so from the above identity we see that α is continuously} differen-tiable in time. Now we apply i∂𝑡𝑡+∆ to both sides of (3). Using the NLS equa-tion, we conclude that

(i∂𝑡𝑡+∆)𝑣𝑣= (−i∂𝑡𝑡+∆)�𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡)𝑒𝑒𝑖𝑖𝑡𝑡∆𝑣𝑣+�

⟹|𝑣𝑣(𝑡𝑡,𝑥𝑥)|𝑝𝑝−1_{𝑣𝑣(𝑡𝑡,𝑥𝑥) =𝑖𝑖}′_{(𝑡𝑡)𝑒𝑒}𝑖𝑖𝑖𝑖(𝑡𝑡)_𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣

+(𝑥𝑥).

And thus by (1.3), we have whenever 𝑣𝑣(𝑡𝑡,𝑥𝑥)≠0, |𝑣𝑣(𝑡𝑡,𝑥𝑥)|𝑝𝑝−1_𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡)_𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣

+(𝑥𝑥) = 𝑖𝑖′𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡)𝑒𝑒𝑖𝑖𝑡𝑡∆𝑣𝑣+(𝑥𝑥)

⟹|𝑣𝑣(𝑡𝑡,𝑥𝑥)|𝑝𝑝−1_{= 𝑖𝑖}′_(𝑡𝑡).

Thus we see that ∀t, |𝑣𝑣(𝑡𝑡,𝑥𝑥)| takes at most two values. By Lemma 3 and Lemma 4, we conclude that |𝑣𝑣(𝑡𝑡,𝑥𝑥)| is constant in 𝑥𝑥, by (3), we see that the same is true for |𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣

+(𝑥𝑥)|. By mass conservation we conclude that |𝑒𝑒𝑖𝑖𝑡𝑡∆𝑣𝑣+(𝑥𝑥)|

is also constant in time.

Since𝑣𝑣+has non-zero mass, we can thus write

𝑒𝑒𝑖𝑖𝑡𝑡∆_𝑣𝑣

+(𝑥𝑥) =𝐴𝐴𝐴𝐴, a.e.,

where 𝐴𝐴:ℝ×𝕋𝕋𝑑𝑑 _{→ 𝕊𝕊}1 _{is some function and 𝐴𝐴> 0. Since 𝑣𝑣}

+ was in 𝐻𝐻1, we

see that 𝐴𝐴(𝑡𝑡) is in 𝐻𝐻1 _{also for every t. Differentiating the identity 𝐴𝐴𝐴𝐴� = 1,}

We see that ∂𝑗𝑗𝐴𝐴𝐴𝐴 is imaginary almost everywhere for j = 1, 2, thus ∂𝑗𝑗𝐴𝐴(𝑡𝑡,𝑥𝑥) is an imaginary multiple of 𝐴𝐴(𝑡𝑡,𝑥𝑥) for almost every (t, x). This implies the im-aginary vector field ∇𝐴𝐴.𝐴𝐴� is curl-free and thus by Hodge theory, we may write

∇𝐴𝐴.𝐴𝐴�=𝑖𝑖∇𝜔𝜔 for some ω:ℝ×ℝ𝑑𝑑 _{→ ℝ which is locally in 𝐻𝐻}1 _{uniformly in t.}

This implies that ∇(𝐴𝐴𝑒𝑒−𝑖𝑖𝜔𝜔_{) = 0, thus by adjustingω by a constant independent} of space. We may assume 𝐴𝐴=𝑒𝑒−𝑖𝑖𝜔𝜔_{, thus 𝑒𝑒}𝑖𝑖𝑡𝑡∆_𝑣𝑣

+=𝐴𝐴𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥).

DOI: 10.4236/jamp.2017.59162 1920 Journal of Applied Mathematics and Physics = (i∂𝑡𝑡+∆)�𝐴𝐴𝑒𝑒−𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)�

=𝐴𝐴(𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)_𝜕𝜕

𝑡𝑡𝜔𝜔+∆𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)) = 𝐴𝐴(𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)_𝜕𝜕

𝑡𝑡𝜔𝜔+∇(𝑖𝑖∇𝜔𝜔(𝑡𝑡,𝑥𝑥)∆𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥))) = 𝐴𝐴(𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)_𝜕𝜕

𝑡𝑡𝜔𝜔+𝑖𝑖∆𝜔𝜔(𝑡𝑡,𝑥𝑥)∆𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)+𝑖𝑖∇𝜔𝜔(𝑡𝑡,𝑥𝑥)𝑖𝑖∇𝜔𝜔(𝑡𝑡,𝑥𝑥)𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)) = 𝐴𝐴(𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)_𝜕𝜕

𝑡𝑡𝜔𝜔+𝑖𝑖∆𝜔𝜔(𝑡𝑡,𝑥𝑥)∆𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)−|∇𝜔𝜔(𝑡𝑡,𝑥𝑥)|2𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)) = 𝐴𝐴𝑒𝑒𝑖𝑖𝜔𝜔(𝑡𝑡,𝑥𝑥)_(𝜕𝜕

𝑡𝑡𝜔𝜔+𝑖𝑖∆𝜔𝜔 −|∇𝜔𝜔|2) in the sense of distributions.

Since A is non-zero, we conclude that

𝜕𝜕𝑡𝑡𝜔𝜔+𝑖𝑖∆𝜔𝜔 −|∇𝜔𝜔|2= 0

Taking imaginary parts, we conclude that ∆ω= 0 and in particular at time t = 0, ω is a harmonic function from ℝ𝑑𝑑 _{to ℝ. On the other hand, from the} identity ∇𝐴𝐴𝐴𝐴�=𝑖𝑖∇𝜔𝜔, we know that ∇𝜔𝜔 is periodic, so 𝜔𝜔 has at most linear growth. Thus 𝜔𝜔 must in fact be linear. Descending back to 𝕋𝕋𝑑𝑑_{, we conclude} that

𝑒𝑒𝑖𝑖𝜔𝜔(0,𝑥𝑥) _=𝑒𝑒𝑖𝑖(𝑖𝑖𝑥𝑥+𝛽𝛽)_{, for some 𝑖𝑖 ∈ ℤ}𝑑𝑑_{,𝛽𝛽 ∈ ℝ/2𝜋𝜋ℤ.} Thus, we have

𝑣𝑣+(𝑥𝑥) =𝑒𝑒𝑖𝑖𝜔𝜔(0,𝑥𝑥)=𝑒𝑒𝑖𝑖(𝑖𝑖𝑥𝑥+𝛽𝛽)

Since

𝑒𝑒𝑖𝑖𝑡𝑡𝑚𝑚∆_{𝑢𝑢+→ 𝑣𝑣+ in 𝐻𝐻}1_{, as m→ ∞,} 𝑒𝑒𝑖𝑖𝑡𝑡𝑚𝑚∆_{𝑣𝑣+→ 𝑢𝑢+ in 𝐻𝐻}1_{, as m→ ∞,}

but 𝑒𝑒−𝑖𝑖𝑡𝑡𝑚𝑚∆_𝑣𝑣+ is a multiple of _{𝐴𝐴𝑒𝑒}𝑖𝑖𝑖𝑖𝑥𝑥 by a phase, thus _𝑢𝑢+(_𝑥𝑥) = A_𝑒𝑒𝑖𝑖𝑖𝑖_𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥.

Ap-plying phase rotation, we may assume 𝑖𝑖= 0, thus 𝑢𝑢+(𝑥𝑥) = A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥, and we

have

�𝑢𝑢(𝑡𝑡)− 𝑒𝑒𝑖𝑖𝜃𝜃(𝑡𝑡)_𝑒𝑒−𝑖𝑖𝑡𝑡∆_A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥_�

𝐻𝐻1(𝕋𝕋𝑑𝑑) →0, as 𝑡𝑡 → ∞.

From mass and energy conservation, we conclude

� |𝑢𝑢(𝑡𝑡,𝑥𝑥)|2_d𝑥𝑥

𝕋𝕋𝑑𝑑 = lim𝑡𝑡→∞� |𝑒𝑒

𝑖𝑖𝜃𝜃(𝑡𝑡)_𝑒𝑒−𝑖𝑖𝑡𝑡∆_A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥_|2_d𝑥𝑥

𝕋𝕋𝑑𝑑

= lim 𝑡𝑡→∞� |𝑒𝑒

−𝑖𝑖𝑡𝑡∆_A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥_|2_d𝑥𝑥

𝕋𝕋𝑑𝑑

= lim 𝑡𝑡→∞� |A𝑒𝑒

𝑖𝑖𝑖𝑖𝑥𝑥_|2_d𝑥𝑥

𝕋𝕋𝑑𝑑 = 𝐴𝐴

2_|𝕋𝕋𝑑𝑑_|,

and

� 1₂|∇𝑢𝑢(𝑡𝑡,𝑥𝑥)|2₊ 1

𝑝𝑝+ 1|𝑢𝑢(𝑡𝑡,𝑥𝑥)|𝑝𝑝+1d𝑥𝑥 𝕋𝕋𝑑𝑑

= lim_{𝑡𝑡→∞}� 1_{2 |}∇(𝑒𝑒𝑖𝑖𝜃𝜃(𝑡𝑡)_𝑒𝑒−𝑖𝑖𝑡𝑡∆_A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥_)|2₊ 1

𝑝𝑝+ 1 |𝑒𝑒𝑖𝑖𝜃𝜃(𝑡𝑡)𝑒𝑒−𝑖𝑖𝑡𝑡∆A𝑒𝑒𝑖𝑖𝑖𝑖𝑥𝑥|𝑝𝑝+1d𝑥𝑥 𝕋𝕋𝑑𝑑

= lim

t→∞� 1

2|∇(Aeinx)|2+ 1

p + 1Ap+1|e−it∆Aeinx|p+1dx 𝕋𝕋d

=� 1₂𝐴𝐴2_{|𝑖𝑖𝑖𝑖𝑒𝑒}𝑖𝑖𝑖𝑖𝑥𝑥_|2₊ 1

DOI: 10.4236/jamp.2017.59162 1921 Journal of Applied Mathematics and Physics =𝑖𝑖₂2𝐴𝐴2_|𝕋𝕋𝑑𝑑_{| +} 𝐴𝐴𝑝𝑝+1

𝑝𝑝+ 1|𝕋𝕋𝑑𝑑| . On the other hand, from Hölder’s inequality, we have

�� |𝑢𝑢(𝑡𝑡,𝑥𝑥)|2_{𝑑𝑑𝑥𝑥}

𝕋𝕋d �

1 2

≤ ‖𝑢𝑢(𝑡𝑡,𝑥𝑥)‖_𝐿𝐿 𝑥𝑥

𝑝𝑝+1_�𝕋𝕋𝑑𝑑_�|𝕋𝕋𝑑𝑑|

1 2−𝑝𝑝1+1

=‖𝑢𝑢(𝑡𝑡,𝑥𝑥)‖_𝐿𝐿 𝑥𝑥

𝑝𝑝+1_�𝕋𝕋𝑑𝑑_�|𝕋𝕋𝑑𝑑| 𝑝𝑝 −1 2(𝑝𝑝+1).

then

� _{𝑝𝑝+ 1}1 |𝑢𝑢(𝑡𝑡,𝑥𝑥)|𝑝𝑝+1_{𝑑𝑑𝑥𝑥}

𝕋𝕋𝑑𝑑 ≥

1

𝑝𝑝+ 1��𝕋𝕋𝑑𝑑|𝑢𝑢(𝑡𝑡,𝑥𝑥)|2𝑑𝑑𝑥𝑥� 𝑝𝑝+1

2

|𝕋𝕋𝑑𝑑_|−𝑝𝑝−₂1

=_{𝑝𝑝+ 1}1 (𝐴𝐴2_|𝕋𝕋𝑑𝑑_|)𝑝𝑝+1_{2 |𝕋𝕋}𝑑𝑑_|−𝑝𝑝 −₂1

=_𝑝𝑝1₊₁𝐴𝐴𝑝𝑝+1_|𝕋𝕋𝑑𝑑_|. Thus, we must have

∫ 1₂|∇𝑢𝑢(𝑡𝑡,𝑥𝑥)|2_{𝑑𝑑𝑥𝑥}

𝕋𝕋𝑑𝑑 = 0.

Thus, 𝑢𝑢 is constant in space, and thus it is of the form

𝑢𝑢(𝑡𝑡,𝑥𝑥 ) =𝐴𝐴𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡)_.

□ Applying (1), we see that

i ∂𝑡𝑡𝑢𝑢+△ 𝑢𝑢= (i∂𝑡𝑡+△)𝐴𝐴e𝑖𝑖𝑖𝑖(𝑡𝑡)= A(𝑖𝑖′(𝑡𝑡)e𝑖𝑖𝑖𝑖(𝑡𝑡)) = A𝑖𝑖′(𝑡𝑡)𝑒𝑒𝑖𝑖𝑖𝑖(𝑡𝑡) | u|p−1u = Ap_eik(t)_,

then

A𝑝𝑝 _{= A𝑖𝑖}′_{(𝑡𝑡)⟹k}′_{(t) =𝐴𝐴}𝑝𝑝−1_,

so

𝑖𝑖(𝑡𝑡) =𝑖𝑖(0) +𝐴𝐴𝑝𝑝−1_𝑡𝑡,

so 𝑢𝑢(𝑡𝑡,𝑥𝑥) =𝐴𝐴e𝑖𝑖𝑖𝑖(0)_𝑒𝑒𝑖𝑖|𝐴𝐴|𝑝𝑝−1_𝑡𝑡

, which is exactly the form of (2).

References

[1] Bourgain, J. (1999) Global Well-Posedness of Defocusing 3D Critical NLS in the Radial Case. J. Amer. Math. Soc., 12, 145-171.

https://doi.org/10.1090/S0894-0347-99-00283-0

[2] Cazenave, T. (2003) Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI.

[3] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T. (2008) Global Well-Posedness and Scattering for the Energy-Critical Nonlinear Schrödinger Equ-ation in 𝑅𝑅3_{, Ann. of Math., 167, 767-865.}

https://doi.org/10.4007/annals.2008.167.767

DOI: 10.4236/jamp.2017.59162 1922 Journal of Applied Mathematics and Physics [5] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T. (2010) Transfer of Energy to High Frequencies in the Cubic Defocusing Nonlinear Schrödinger Equa-tion. Invent. Math, 181, 39-113. https://doi.org/10.1007/s00222-010-0242-2

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