8_189_zhao.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 2, 2010, 227-234

ISSN 1307-5543 – www.ejpam.com

Local Solvability for the 2-Coupled System of Nonlinear Schrödinger

Equations in a Banach Algebra E0
2,1

Zaile He and Xiangqing Zhao∗

School of Mathematics physics & Information Science,Zhejiang Ocean University, Zhoushan, Zhe-

jiang 316000. P. R. China

Abstract. This paper is concerned with initial value problem of the nonlinear coupled Schrödinger

equations. We study local well posedness in the Banach algebra E0
2,1
(Rn) which is the extension of

H s(Rn) when s ≥ n

2
. The method we use is similar to the method of semigroup.

2000 Mathematics Subject Classifications: 35Q53, 35Q35.

Key Words and Phrases: Nonlinear, coupled Schrödinger equations, Banach algebra, local well-

posedness

1. Introduction

It is well-known that Hs(Rn) is an algebra when s > n

2
and the Schrödinger operator

generate an unitary group in Hs(Rn). The well-posedness in Hs(s ≥ n

2
) for the Cauchy problem

of the cubic semi-linear Schrödinger equation

iut +∆u= a|u|2u, x ∈ Rn, t ∈ R.

were treated by using the method of semigroup [we refer to 6]. Recently, Wang et al in [10]

introduce a new Banach algebra E0
2,1(R

n) which is the extension of Hs(Rn) when s ≥ n

2
and

investigated the Cauchy problem of semi-linear Schrödinger equation with nonlinear term

|u|2ku, k ∈ N . We shall study a coupled system by Wang’s approach in this paper.

As a natural extension of the single cubic nonlinear Schrödinger equation, the 2-coupled

nonlinear Schrödinger equations:

¨

iut +∆u= a|u|2u+ |v|2u, x ∈ R, t ∈ R,

ivt +∆v = |u|2v + a|v|2v, x ∈ R, t ∈ R,
(1)

∗Corresponding author.

Email address: zhao-xiangqing�163.
om (X. Zhao)

http://www.ejpam.com 227 c© 2010 EJPAM All rights reserved.



Z. He and X. Zhao / Eur. J. Pure Appl. Math, 3 (2010), 227-234 228

have many applications including, for example, nonlinear optics [cf. 2, 4, 5, 9] and geophysi-

cal fluid dynamics [cf. 7, 8]. In the above equations, a ∈ R, the unknowns u(x , t), v(x , t) are

the envelopes of wave packets in two different degrees of freedom of the underlying physical

systems which we shall call ’modes’. The system is derived as an approximation to a more

complex set of equations by singular perturbation theory.

Instead of studying (1), this paper is concerned with the following general Schrödinger

system:






iut +∆u= a|u|αu+ |v|αu, x ∈ Rn, t ∈ R,

ivt +∆v = |u|αv + a|v|αv, x ∈ Rn, t ∈ R,

u(0, x) = φ(x), x ∈ Rn,

v(0, x) =ψ(x), x ∈ Rn.

(2)

As in [10], for technical reason, the restriction α = 2k, k ∈ N or |u|α = uα( or ūα) and

|v|α = vα( or v̄α), α ∈ N is required in the nonlinear coupled terms. By denoting U =
�u

v

�

,

F(U) =
�a|u|αu+|v|αu

u|αv+a|v|αv

�

and Φ =
�φ

ψ

�

, we see readily that (2) take the following form:

¨

∂t U +∆U = F(U) x ∈ Rn, t ∈ R.

U(0, x) = Φ(x) x ∈ Rn.
(3)

Let

Λ(t) =

�

S(t) 0

0 S(t)

�

,

where S(t) = ei t∆ is the fundamental solution operator of the Schrödinger equation and is

given by

S(t)φ =

∫

Rn

e−i t|ξ|2+i xξφ̂dξ, ∀ φ ∈ S(Rn).

Then by the Duhanmel principle we see that the Cauchy problem (3) is equivalent to the

following integral equation:

U(t) = Λ(t)Φ− i

∫ t

0

Λ(t −τ)F(U(τ))dτ.

Thus, in the sequel we shall solve this integral equation.

We shall use the notation ‖| · ‖| to denote the norm of 2-dimensional vector functions, and

use ‖ · ‖ to denote the norm of scale functions, so that ‖|U‖| = ‖u‖+ ‖v‖ if U =
�u

v

�

.

The main result of this paper is

Theorem 1. Let Φ ∈ E0
2,1(R

n). Then there exists T ∗ ≡ T ∗(‖|Φ‖|E0
2,1(R

n) > 0 such that the Cauchy

problem (3) has a unique solution

U ∈ C([0, T ∗), E0
2,1(R

n)).



Z. He and X. Zhao / Eur. J. Pure Appl. Math, 3 (2010), 227-234 229

Moreover, if T ∗ <∞ then

lim
t→T∗

sup‖|U(t)‖|E0
2,1(R

n) =∞.

E0
2,1(R

n) will be introduced in the next section.

In the sequel, C will denote a constant which may differ at each appearance, possibly

depending on the dimension or other parameters. For p ≥ 1 we set p′ = p

p−1
.

2. Preliminaries

2.1. The Banach Algebra E0
2,1

We denote by S(Rn) and S′(Rn) the Schwartz space and its dual space, respectively. Let ρ ∈
S(Rn) and ρ : Rn → [0,1] be a smooth radial bump function adapted to the ball B(0,

p
2n),

say ρ(ξ) = 1 as 0≤ |ξ| ≤p n

2
, and ρ(ξ) = 0 as |ξ| ≥ p2n. Let ρk be a translation of ρ :

ρk(ξ) = ρ(ξ− k), k ∈ Zn,

where k ∈ Zn means that k = (k1, k2, · · · , kn), and k1, k2, · · · , kn are all integers. Since ρ(ξ) =

1 in the unit closed cube Qk with center k and {Qk}k∈Zn is a covering of Rn, one has that
∑

k∈Zn ρk(ξ)≥ 1 for all ξ ∈ Rn. We write

σk(ξ) = ρk(ξ)
� ∑

k∈Zn

ρk(ξ)
�−1

, k ∈ Zn.

It is easy to see that











|σk(ξ)| ≥ C , ∀ ξ ∈ Qk;

suupσk(ξ)⊂ {ξ : |ξ− k| ≤ p2n};
∑

k∈Zn σk(ξ) = 1, ∀ ξ ∈ Rn;

|σ(m)
k
(ξ)| ≤ Cm, ∀ ξ ∈ Rn.

(4)

Hence, the set

Υ = {{σk}k∈Zn : {σk}k∈Zn satisfies (4)}
is non-void. Let {σk}k∈Zn ∈ Υ be a function sequence. Define operator:

�k ≡F−1σkF , k ∈ Zn,

where the operator F means Fourier transformation.

For any k ∈ Zn, we write |k| = |k1|+ |k2|+ · · ·+ |kn|. Let 0 ≤ λ <∞, 0 < p, q ≤ ∞, we

introduce the following function space

Eλp,q(R
n) =
n

f ∈ S′(Rn) : ‖ f ‖Eλp,q
≡
� ∑

k∈Zn

[2λ|k|‖�k f ‖Lp(Rn)]
q
�

1

q
<∞
o

.



Z. He and X. Zhao / Eur. J. Pure Appl. Math, 3 (2010), 227-234 230

Obviously, the function space Eλp,q(R
n) is modified from the Besov space Bs

p,q(R
n) [see 1]. Since

the relation between Eλp,q(R
n) and the Besov space Bs

p,q(R
n) have nothing to do with our result,

we omit it here [for the details, we refer to 10].

The algebra property of E0
2,1 may deduce from the following embedding property and

bilinear estimate.

Lemma 1. Let 0≤ λ <∞, 0< p1 ≤ p2 ≤∞, 0< q1 ≤ q2 ≤∞. Then we have

Eλp1,q1
(Rn)⊂ Eλp2,q2

(Rn).

Proof. See the proof of Proposition 3.5 in [10].

Lemma 2. Let 0≤ λ <∞, 0< p ≤ p1, p2 ≤∞, 0< q ≤∞. If 1

p
= 1

p1
+ 1

p2
, then we have

‖uv‖Eλp,q
≤ C2Cqλ‖u‖Eλ

p1,q∧1
‖v‖Eλ

p2,q∧1
,

where a∧ b =min{a, b}. C is independent of λ, q and if p is fixed, then C is also independent of

p1, p2.

Proof. See the proof of Lemma 4.1 in [10].

As a matter of fact, by Lemma 1 and Lemma 2, we have

‖uv‖E0
2,1
≤ C‖uv‖E0

1,1
≤ C‖u‖E0

2,1
‖v‖E0

2,1
. (5)

Which suggest that E0
2,1 is a Banach algebra.

From the comparison between E0
2,q(R

n) and Hs(Rn), we find that E0
2,1(R

n) is the extension

of Hs(Rn):

Hs(Rn)⊂ E0
2,1(R

n) for s >
n

2
, and Hs(Rn)⊂ E0

2,1 fails, for s ≤ n

2
.

Indeed, we have

Lemma 3. We have

Hs(Rn)⊂ E0
2,q(R

n), s > n(
1

q
− 1

2
), 0< q < 2,

L2(Rn) = E0
2,2(R

n) (equivalent norm),

E0
2,q(R

n)⊂ Hs(Rn), s < n(
1

q
− 1

2
), 2< q ≤∞.

Furthermore, E0
2,1 is the intermediate space of Hs(Rn) and L∞(Rn), that is

Hs(Rn)⊂ E0
2,1 ⊂ L∞(Rn), s > n/2.

[(3.43) in 10].

Proof. See the proof of Proposition 3.8 in [10].



Z. He and X. Zhao / Eur. J. Pure Appl. Math, 3 (2010), 227-234 231

2.2. Some Preliminary Lemmas

Estimate for the Schrödinger group

Lemma 4. Let 0 < r ≤ 2 ≤ p ≤∞, 0 < q ≤∞. Then for the Schrödinger group S(t) = ei t∆ we

have the estimate

‖S(t)φ‖E0
p,q
≤ C‖φ‖E0

r,q
.

In particular,

‖S(t)φ‖E0
2,1
≤ C‖φ‖E0

2,1
.

Proof. See the proof of Proposition 5.5 in [10].

From Lemma 4, we deduce that

Lemma 5. Let 0< r ≤ 2≤ p ≤∞, 0< q ≤∞. Then for the group Λ(t) we have the estimate

‖|Λ(t)Φ‖|E0
p,q
≤ C‖|Φ‖|E0

r,q
.

In particular,

‖|Λ(t)Φ‖|E0
2,1
≤ C‖|Φ‖|E0

2,1
.

With the algebra property, we have the Estimates for the nonlinear coupled terms

Lemma 6.

‖|F(U)‖|E0
2,1
≤ C‖|U‖|α+1

E0
2,1

,

and

‖|F(U1)− F(U2)‖|E0
2,1
≤ C‖|U1 − U2‖|E0

2,1

h

‖|U1‖|αE0
2,1

+ ‖|U2‖|αE0
2,1

i

,

where U =
�u

v

�

, U1 =
�u1

v1

�

, U2 =
�u2

v2

�

.

Proof. By (5), we have

‖|F(U)‖|E0
2,1
= ‖a|u|αu+ |v|αu‖E0

2,1
+ ‖|u|αv + a|v|αv‖E0

2,1

≤ |a|‖|u|αu‖E0
2,1
+ ‖|v|αu‖E0

2,1
+ ‖|u|αv‖E0

2,1
+ |a|‖|v|αv‖E0

2,1

≤ |a|‖u‖α+1

E0
2,1

+ ‖|v|‖α
E0

2,1

‖u‖E0
2,1
+ ‖u‖α

E0
2,1

‖v‖E0
2,1
+ |a|‖v‖α+1

E0
2,1

≤ ‖u‖α
E0

2,1

�

‖u‖E0
2,1
+ ‖v‖E0

2,1

�

+ C‖v‖α
E0

2,1

�

‖u‖E0
2,1
+ ‖v‖E0

2,1

�

≤ C‖|U‖|α+1

E0
2,1

By mean value theorem we obtain xα − yα = α(x − y)(x − ηy)α−1, (0 ≤ η ≤ 1). Using this

fact and (5), Young’s inequality (since α−1

α
+ 1

α
= 1), we have

‖|F(U1)− F(U2)‖|E0
2,1

= ‖a|u1|αu1 + |v1|αu1 − (a|u2|αu2 + |v2|αu2)‖E0
2,1



Z. He and X. Zhao / Eur. J. Pure Appl. Math, 3 (2010), 227-234 232

+‖|u1|αv1+ a|v1|αv1 − (|u2|αv2 + a|v2|αv2)‖E0
2,1

= ‖(a|u1|α+ |v1|α)(u1− u2) + (a(|u1|α− |u2|α) + (|v1|α− |v2|α))u2‖E0
2,1

+‖(|u1|α+ a|v1|α)(v1− v2) + ((|u1|α− |u2|α) + a(|v1|α− |v2|α))v2‖E0
2,1

≤ ‖(a|u1|α+ |v1|α)(u1− u2)‖E0
2,1
+ ‖(a(|u1|α− |u2|α) + ‖(|v1|α− |v2|α))u2‖E0

2,1

+‖(|u1|α+ a|v1|α)(v1− v2)‖E0
2,1
+ ‖((|u1|α− |u2|α) + a(|v1|α− |v2|α))v2‖E0

2,1

≤ C
h

‖u1‖αE0
2,1

+ ‖u2‖αE0
2,1

+ ‖v1‖αE0
2,1

+ ‖v2‖αE0
2,1

i

(‖u1 − u2‖E0
2,1
+ ‖v1 − v2‖E0

2,1
)

≤ C(‖|U1‖|αE0
2,1

+ ‖|U2‖|αE0
2,1

)‖|U1− U2‖|E0
2,1

.

3. Proof of the Main Result

We shall make use of the fixed point Theorem to solve the integral equation

U = T (U) = Λ(t)Φ− i

∫ t

0

Λ(t −τ)F(U(τ))dτ. (6)

Define a metric space as follows:

D = {U : ‖|U‖|C(0,T ;E0
2,1)
≤ M},

d(U , V ) = ‖|U − V‖|C(0,T ;E0
2,1)

.

By Lemma 5, we have

‖|Λ(t)Φ‖|C(0,T ;E0
2,1)
≤ C‖|Φ‖|E0

2,1
. (7)

By Lemma 5 and the first inequality of Lemma 6, we obtain







�

�

�

∫ t

0

Λ(t −τ)F(U(τ))dτ






�

�

�

C(0,T ;E0
2,1)
≤ C T‖|U‖|α+1

C(0,T ;E0
2,1)

. (8)

Let us consider the mapping T : U → Λ(t)Φ − i
∫ t

0
Λ(t − τ)F(U(τ))dτ. We show that T :

(D, d)→ (D, d) is a contraction mapping. Indeed, for any U ∈ D, by (7) and (8) we have

‖|T (U)‖|C(0,T ;E0
2,1)
≤ C‖|Φ‖|E0

2,1
+ C T‖|U‖|α+1

C(0,T ;E0
2,1)

.

Put M = 2C‖|Φ‖|E0
2,1

, we have

‖|T (U)‖|C(0,T ;E0
2,1)
≤ M

2+ C T Mα+1
. (9)

Let T be small enough to satisfies C T Mα ≤ 1

4
. It follows from (9) that T (U) ∈ D.



REFERENCES 233

Similarly, we have

‖|T (U)−T (V )‖|C(0,T ;E0
2,1)
≤ 1

2
‖|U − V‖|C(0,T ;E0

2,1)
.

Indeed, ∀ U , V ∈ (D, d), by Lemma 5 and the second inequality of Lemma 6, we obtain

‖|T (U)−T (V )‖|C(0,T ;E0
2,1)
≤






�

�

�

∫ t

0

Λ(t −τ)[F(U(τ))− F(V )dτ







�

�

�

C(0,T ;E0
2,1)

≤ C T‖|F(U)− F(V )‖|C(0,T ;E0
2,1)

≤ C T
h

‖|U‖|α
C(0,T ;E0

2,1)
+ ‖|V‖|α

C(0,T ;E0
2,1)

i

‖|U − V‖|C(0,T ;E0
2,1)

≤ 2C T Mα‖|U − V‖|C(0,T ;E0
2,1)

≤ 1

2
‖|U − V‖|C(0,T ;E0

2,1)
.

Hence, by Banach fixed point theorem, we see that T has a fixed point U ∈ D which is a

solution of integral equation (6). We can extend this solution step by step and finally find

a maximal T ∗ > 0 such that U ∈ C([0, T ∗), E0
2,1(R

n)) and lim
t→T∗

sup‖|U(t)‖|E0
2,1(R

n) = ∞. The

uniqueness of such solutions can also be shown in a standard way. This finishes the proof of

the Theorem.

ACKNOWLEDGEMENTS This work is financially supported by the research projects of Zhe-

jiang Ocean University under the grant numbers X08M014 and 21065030608.

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