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In this paper, we examine the following Schrödinger equation:
{iψt=(−Δ)sψ+|x|2ψ−|ψ|2σψinRN×[0,∞),ψ(0,x)=ψ0(x)∈Hs(RN), | (1) |
where
Hs(RN):={u∈L2(RN):∫RN∫RN|u(x)−u(y)|2|x−y|N+2sdydx<∞}. |
with
‖u‖Hs(RN)=√∫RN∫RN|u(x)−u(y)|2|x−y|N+2sdydx+∫RN|u(x)|2dx. |
The fractional Laplacian
(−Δ)su(x)=F−1[|ξ|2s[u]], s>0. | (2) |
For the Cauchy problem, formally we have two conserved quantities [17] by multiplying the conjugate of
M(t)=||ψ(⋅,t)||2:=∫RN|ψ(x,t)|2dx≡M(0) | (3) |
and the total energy:
E(t)=∫RN[Re(ψ∗(x,t)(−Δ)sψ)+|x|2|ψ|2−1σ+1|ψ(x,t)|2(σ+1)]dx≡E(0). | (4) |
In recent years, a great attention has been focused on the study of problems involving the fractional Laplacian, which naturally appears in obstacle problems, phase transition, conservation laws, financial market. Nonlinear fractional Schrödinger equations (FNLS) have been proposed by [20,19,9] in order to expand the Feynman path integral, from the Brownian like to the Lévy like quantum mechanical paths. The stationary solutions of fractional nonlinear Schrödinger equations have also been intensively studied due to their huge importance in nonlinear optics and quantum mechanics [20,19,14,12]. The most interesting solutions have the special form:
ψ(x,t)=e−iλtu(x),λ∈R,u(x) isasmoothfunction. | (5) |
They are called the standing waves. These solutions reduce (1) to a semilinear elliptic equation. In fact, after plugging (5) into (1), we need to solve the following equation
(−Δ)su(x)+|x|2u(x)−|u(x)|2σu(x)=λu(x)inRN. | (6) |
The case
In this paper, we mainly focus on the solutions to (6). To the best of our knowledge, our results are new and will open the way to solve other classes of fractional Schrödinger equations. This paper has two main parts: In the first part, we address the existence of standing waves through a particular variational form, whose solutions are called ground state solutions. We prove the existence of ground state solutions (Theorem 2.1), and show some qualitative properties like monotonicity and radiality (Lemma 4). We also show that the ground state solutions are orbitally stable (Def 4.2, Theorem 2.2) if we have the uniqueness of the solutions for the Cauchy problem (1) (Theorem 4.3). We have also addressed the critical case
The main difficulty of constructing ground state solutions comes from the lack of compactness of the Sobolev embeddings for the unbounded domain
The paper is organized as follows. In section 2, we give our main results about the existence of ground state solutions and the orbital stability of standing waves. In section 3, we provide proof of existence. Then, in section 4, we discuss the orbital stability. In section 5, we use Split-Step Fourier Spectral method to solve (1) numerically. In section 6, instead of using common iterative Newton's method, we use the NGF method to find ground states when
We use a variational formulation to examine the solution to (6). First, note that if
J(u)=12‖∇su‖2L2(RN)+12∫RN|x|2|u|2dx−12σ+2∫RN|u|2σ+2dx, | (7) |
where
‖∇su‖2L2(RN)=CN,s∫RN∫RN|u(x)−u(y)|2|x−y|N+2sdxdy, |
with some normalization constant
We can derive (7) by multiplying smooth enough test function
Ic=inf{J(u):u∈Sc}, | (8) |
with
Sc={u∈Σs(RN):∫RN|u|2dx=c2}, | (9) |
where
Σs(RN)={u∈Hs(RN):‖u‖Σs(RN):=√‖u‖2L2(RN)+‖∇su‖2L2(RN)+‖xu‖2L2(RN)<∞}. | (10) |
is a Hilbert space, with corresponding natural inner product.
We claim that for each minimizer
J∗(u)=J(u)+λ(‖u‖2L2(RN)−c2). | (11) |
The minimizer to problem (8) must be the critical point of (11), satisfying:
∂J∗(u)∂u=0 | (12) |
and
∂J∗(u)∂λ=0, | (13) |
where (12) implies (6) and (13) implies (9). In this paper, we will mainly focus on the minimizers of problem (8). The following theorem discusses the existence of such minimizers.
Theorem 2.1. If
Remark 1. There were some works considering standing waves for fractional nonlinear Schrödinger equation with different potential enery and nonlinearity terms. In [6], the authors focus on the case where the nonlinearity term is asymptotically linear with
Remark 2. The condition
After we construct the ground state solutions, we further investigate their stability. By the definition of (5), the ground state solution moves around a circle when the time changes. Therefore we consider and prove the orbital stability of ground state solution (Def 4.2).
Theorem 2.2. Suppose that
In this section, we will establish the existence of ground state solutions of (6), the main difficulty comes from the lack of compactness of the Sobolev embeddings. Usually, at least when the potential in (1) is radially symmetric and radially increasing, such difficulty is overcome by considering the appropriate function space. More precisely, we have
Lemma 3.1. Let
Proof. First, when
∫|x|≥R|u|2dx≤|R|−2∫|x|≥R|x|2|u|2dx<|R|−2‖u‖2Σs(RN) | (14) |
By the classical Sobolev embedding theorem, for any fixed
Second, for
‖u‖pLp(RN)≤K‖u‖1−θL2(RN)‖∇su‖θL2(RN)≤K‖u‖1−θL2(RN)‖u‖θΣs(RN), | (15) |
for some positive constant
Then we have a lemma showing the existence of
Lemma 3.2. If
Proof. First, we prove that
‖u‖L2σ+2(RN)≤K‖u‖1−θL2(RN)‖∇su‖θL2(RN), | (16) |
for some positive constant
On the other hand, let
‖u‖(2σ+2)(1−θ)L2(RN)‖∇su‖θ(2σ+2)L2(RN)≤1pϵp‖∇su‖pθ(2σ+2)L2(RN)+1qϵq‖u‖q(1−θ)(2σ+2)L2(RN). | (17) |
Combining (16) and (17), we obtain that for any
∫RN|u(x)|2σ+2dx≤ϵpK2σ+2p‖∇su‖2L2(RN)+K2σ+2qϵqc2q(1−θ)(1+σ), | (18) |
where
Hence, from (18) we get:
J(u) | (19) |
=12‖∇su‖2L2(RN)+12∫RN|x|2|u|2dx−12σ+2∫RN|u|2σ+2dx | (20) |
≥12‖∇su‖2L2(RN)+12∫RN|x|2|u|2dx | (21) |
−12σ+2(ϵpK2σ+2p‖∇su‖2L2(RN)+K2σ+2qϵqc2q(1−θ)(1+σ)) | (22) |
≥(12−ϵpK2σ+22p(σ+1))‖∇su‖2L2(RN)+12∫RN|x|2|u|2dx−K2σ+2c2q(1−θ)(1+σ)2q(1+σ)ϵq. | (23) |
Then we choose
Now, we can use compactness (Lemma 3.1) and boundedness (Lemma 3.2) to prove our existence Theorem 2.1.
Proof. Let
Since
On the other hand, thanks to the lower semi-continuity, we have
‖xu‖L2(RN)+‖∇su‖L2(RN)≤lim infn→∞‖∇sun‖L2(RN)+‖xun‖L2(RN). |
Therefore
Ic≤J(u)≤lim infn→∞J(un)=Ic, | (24) |
which yields that
The second step consists of constructing a nonnegative, radial, and radially decreasing minimizer. First, we note that:
‖∇s|u|‖L2(RN)≤‖∇su‖L2(RN), | (25) |
which implies that
meas{x∈RN:|u(x)|>μ}=meas{x∈RN:u∗(x)>μ}foranyμ>0. |
It is well-known [15] that
{∫RN|u|2σ+2dx=∫RN|u∗|2σ+2dx,∫RN|u|2dx=∫RN|u∗|2dx,∫RN|x|2|u∗|2dx≤∫RN|x|2|u|2dx. | (26) |
Besides, from [3] Lemma 6.3, we also have
‖∇su∗‖L2(RN)≤‖∇su‖L2(RN). | (27) |
Combining (26) and (27), we obtain that
J(|u|∗)≤J(|u|)≤J(u),forany u∈Σs(RN). |
Remark 3. By (24) and the weakly convergence, we can also see that:
‖xu‖L2(RN)+‖u‖L2(RN)+‖∇su‖L2(RN)=limn→∞‖∇sun‖L2(RN)+‖un‖L2(RN)+‖xun‖L2(RN), |
which implies that there is a minimizing subseqence
Remark 4. If
Ic=J(u)=J(|u|)=J(|u|∗). | (28) |
By (27) and (28), we obtain that
‖∇su‖L2(RN)=‖∇s|u|‖L2(RN), | (29) |
which implies that
∫RN|x|2(|u|∗)2dx=∫RN|x|2|u|2dx. | (30) |
which further implies
In this section, we will deal with the orbital stability of the ground state solutions. Let us introduce the appropriate Hilbert space:
˜Σs(RN):={ω=u+iv:(u,v)∈Σs(RN)×Σs(RN)}, |
equipped with the norm
In term of the new coordinates, the energy functional reads
˜J(ω)=12‖∇sω‖2L2(RN)+12∫RN|x|2|ω(x)|2dx−12σ+2∫RN|ω|2σ+2dx, |
where
Then, for all
˜Ic=inf{˜J(ω),ω∈˜Sc}, |
where
˜Sc={ω∈˜Σs(RN),∫RN|ω(x)|2dx=c2}. |
We also introduce the following sets
Oc={u∈Sc:Ic=J(u)},˜Oc={ω∈˜Sc:˜Ic=˜J(ω)}. |
Proceeding as in [4,16], we have the following lemma:
Lemma 4.1. If
(i) The energy functional
(ii) There exists a constant
‖J′(u)‖Σ−1s(RN)≤C(‖u‖Σs(RN)+‖u‖2σ+1Σs(RN)),‖˜J′(ω)‖Σ−1s(RN)≤C(‖ω‖Σs(RN)+‖ω‖2σ+1Σs(RN)). |
(iii) All minimizing sequences for
(iv) The mappings
(v) Any minimizing sequence of
(vi) For any
Ic=˜Ic. |
Proof. (i) We follow the steps of Proposition 2.3 [16] by choosing
−∫RN|u(x)|2σu(x)v(x)dx |
is of class
(ii) From (i),
⟨J′(u),v⟩=CN,S∫RN∫RN|u(x)−u(y)||v(x)−v(y)||x−y|N+2sdxdy+∫RN|x|2u(x)v(x)dx−∫RN|u(x)|2σu(x)v(x)dx. |
For the last term, by Hölder's inequality, we have
∫RN|u(x)|2σu(x)v(x)dx≤‖u‖2σ+1L2σ+2(RN)‖v‖L2σ+2(RN). |
Therefore, there exists
‖J′(u)‖Σ−1s(RN)≤C(‖u‖Σs(RN)+‖u‖2σ+1Σs(RN)). |
(iii) This is a direct result of Lemma 3.2.
(iv) Let
Icn≤J(un)<Icn+1n. | (31) |
From (iii), there exists a constant
‖un‖Σs(RN)≤C1,∀n∈N. |
Setting
vn∈Scand‖un−vn‖Σs(RN)=|1−ccn|‖un‖Σs(RN)≤C1|1−ccn|, |
which implies that
‖un−vn‖Σs(RN)≤C1+1fornlargeenough. | (32) |
We deduce by part (ii) that there exists a positive constant
‖J′(u)‖Σ−1s(RN)≤K,forallu∈Σs(RN)with‖u‖Σs(RN)≤2C1+1. | (33) |
By (32) and (33), we get that
|J(vn)−J(un)|=|∫10ddtJ(tvn+(1−t)un)dt|≤sup‖u‖Σs(RN)≤2C1+1‖J′(u)‖Σ−1s(RN)‖vn−un‖Σs(RN)≤KC1|1−ccn|. | (34) |
Then, from (31) and (34), we obtain that
Icn≥J(un)−1n≥J(vn)−KC1|1−ccn|−1n≥Ic−KC1|1−ccn|−1n |
Combining this with the fact that
lim infn→∞Icn≥Ic. | (35) |
Now, from Lemma 3.2 and by the definition of
‖un‖Σs(RN)≤C2andlimn→∞J(un)=Ic. |
Set
‖vn−un‖Σs(RN)≤C2|1−cnc|and|J(vn)−J(un)|≤LC2|1−cnc|. |
Combining this with (31), we obtain that
Icn≤J(vn)≤J(un)+LC2|1−cnc|. |
Since
lim supn→∞Icn≤Ic. | (36) |
It follows from (35) and (36) that
limn→∞Icn=Ic. |
(v) This is a direct result of Remark 3.
(vi) First, we can see
˜J(ω)=J(ω), |
which implies that
Ic≥˜Ic. | (37) |
Second, for any
‖∇sω‖2L2(RN)≥‖∇s|ω|‖2L2(RN), |
which implies that
˜J(ω)≥˜J(|ω|)=J(ω)≥Ic,∀ω∈˜Σs(RN), |
and
˜Ic≥Ic. | (38) |
Combining (37) and (38), we finally have
Now, for a fixed
Definition 4.2 We say that
●
● For all
‖ω0−ψ0‖˜Σs(RN)<δ⟹infω∈˜Oc‖ω−ψ‖˜Σs(RN)<ε, |
where
If
Theorem 4.3. Suppose that
‖ψ(t,.)‖L2(RN)=‖ψ0(t,.)‖L2(RN)and˜J(ψ(t,.))=˜J(ψ0(t,.)), | (39) |
then
Proof. The proof is by contradiction: Suppose that
infZ∈˜Oc‖Φn(tn,.)−Z‖˜Σs(RN)≥ε | (40) |
for some sequence
Let
‖Φn0‖L2(RN)→cand˜J(Φn0)→˜Ic, n→∞. |
Thus, we deduce from (39) that
‖ωn‖L2(RN)=‖Φn0‖L2(RN)→cand˜J(ωn)=˜J(Φn0)→˜Ic, n→∞. | (41) |
Since
un⇀u,vn⇀vandlim infn→∞‖∇sun‖L2(RN)+‖∇svn‖L2(RN)exists. | (42) |
Now, by a straightforward computation, we obtain that
˜J(ωn)−˜J(|ωn|)=12‖∇sωn‖2L2(RN)−12‖∇s|ωn|‖2L2(RN)≥0, | (43) |
which implies that
˜Ic=limn→∞˜J(ωn)≥lim supn→∞J(|ωn|). |
Besides, by (41),
‖ωn‖2L2(RN)=‖|ωn|‖2L2(RN)=c2n→c2. |
It follows from Lemma 4.1 that we have
lim infn→∞J(|ωn|)≥lim infn→∞Icn=Ic. |
Hence
˜Ic=limn→∞˜J(ωn)=limn→∞J(|ωn|)=Ic. | (44) |
It follows from (42), (43) and (44) that
limn→∞‖∇sun‖2L2(RN)+‖∇svn‖2L2(RN)−‖∇s(u2n+v2n)12‖2L2(RN)=0, |
which is equivalent to say that
limn→∞‖∇swn‖2L2(RN)=limn→∞‖∇s|wn|‖2L2(RN). | (45) |
The boundedness of
|ωn|→φinΣs(RN)and‖φ‖L2(RN)=cwithJ(φ)=Ic. | (46) |
Next, let us prove
un⟶uandvn⟶vinL2(B(0,R))forallR>0. |
Since
(u2n+v2n)12⟶(u2+v2)12inL2(B(0,R)). |
But
(u2+v2)12=|ω|=φ. |
This further implies that
‖ω‖L2(RN)=‖φ‖L2(RN)=c,‖ω‖L2σ+2(RN)=‖φ‖L2σ+2(RN). | (47) |
and
12∫RN|x|2|ω|2dx−12σ+2∫RN|ω|2σ+2dx=12∫RN|x|2|φ|2dx−12σ+2∫RN|φ|2σ+2dx=limn→∞12∫RN|x|2|ωn|2dx−12σ+2∫RN|ωn|2σ+2dx. | (48) |
Additionally, by the lower semi-continuity, we further have
‖∇ω‖2L2(RN)≤lim infn→∞‖∇ωn‖2L2(RN). | (49) |
Combining (48) with (49) and
˜Ic≤˜J(ω)≤limn→∞˜J(ωn)=˜Ic, |
which implies that
ω∈˜Ocand‖∇sω‖2L2(RN)=limn→∞‖∇sωn‖2L2(RN). | (50) |
Finally, by (46), (47), and (50), we obtain that
ωn→ωin˜Σs(RN), |
which contradicts to (40).
In this section, we consider numerical methods to solve (1) and introduce the Split-Step Fourier Spectral method.
First, we truncate (1) into a finite computational domain
{iψt=(−Δ)sψ+|x|2ψ−|ψ|2σψ, t>0,ψ(0,x)=ψ0(x), | (51) |
for
Let
(xj)n=−L+(j)nh, 1≤n≤N, | (52) |
where
Denote
ψ(xj,t)=∑k∈K^ψk(t)exp(iμkxj), | (53) |
where
Now, we introduce the Split-step Fourier Spectral method. The main idea of this method is to solve (51) in two splitting steps from
iψt=|x|2ψ−|ψ|2σψ, | (54) |
iψt=(−Δ)sψ. | (55) |
First, by multiplying
iψt=|x|2ψ−|ψ(x,tn)|2σψ. | (56) |
Second, taking Fourier transform on both sides of (55), we obtain that
id^ψk(t)dt=|μk|2s^ψk(t). | (57) |
We use the second order Strang splitting method with (56) and (57) as follows:
ψn,1j=ψnjexp(−i(|xj|2−|ψnj|2σ)τ/2) | (58) |
ψn,2j=∑k∈K^ψn,1kexp(−i|μk|2sτ)exp(iμkxj) | (59) |
ψn+1j=ψn,2jexp(−i(|xj|2−|ψn,2j|2σ)τ/2) | (60) |
where
ψ0j=ψ0(xj) | (61) |
This method has spectral order accuracy in space and second order in time. Similar to [8], this method preserves discrete mass corresponding to (3) defined as
Mn=(hN∑j|ψnj|2)1/2. | (62) |
To find ground state solutions, we have to solve the following equation corresponding to
(−Δ)su+|x|2u−|u|2σu=λu x∈RN. | (63) |
As discussed previously, for
Thus, for a given sequence of time
∂u∂t=−∂E(u)∂u |
and project it onto
{∂˜u∂t=−(−Δ)s˜u−|x|2˜u+|˜u|2σ˜u,tn<t<tn+1,˜u(x,tn)=u(n)(x),u(n+1)(x)=c˜u(x,tn+1)||˜u(⋅,tn+1)||L2(RN),. |
Here, we use semi-implicity time discretization scheme:
{˜u(n+1)−˜u(n)Δt=−(−Δ)s˜u(n+1)−|x|2˜u(n+1)+|˜u(n)|2σ˜u(n+1),tn<t<tn+1,˜u(x,tn)=u(n)(x),u(n+1)(x)=c˜u(n+1)(x)||˜u(n+1)(⋅)||L2(RN), |
with
δsxu|j=∑k∈K|μk|2sexp(iμkxj) |
to discretize fractional Laplacian, where
In conlusion, in each step, we solve:
{˜u(n+1)j−u(n)jτ=−δsx˜u(n+1)|j−(|xj|2−|u(n)j|2σ)˜u(n+1)ju(n+1)j=c˜u(n+1)jMn, | (64) |
where
We need to notice that we can only solve (8) for
For
Lc=inf{K(u):u∈Tc} | (65) |
with
Tc={u∈Σs(RN):‖u‖L2σ+2(RN)=c}, |
K(u)=12‖∇su‖2L2(RN)+12∫RN|x|2|u(x)|2dx+ω2∫RN|u(x)|2dx, | (66) |
where
For
∫RN|u(x)|2dx≤|R|−2∫|x|>R|x|2|u|2dx+∫|x|<R|u|2dx≤|R|−2∫|x|>R|x|2|u|2dx+CR,σ‖u‖2L2σ+2(RN), |
where
K(u)≥12‖∇su‖2L2+(12+ωR2)∫RN|x|2|u(x)|2dx+ω2CR,σc2>−∞ | (67) |
for
Now we see
K∗(u)=K(u)−λ2σ+2(‖u‖2σ+2L2σ+2(RN)−c2σ+2), |
then the critical points
(−Δ)su∗+|x|2u∗−λ|u∗|2σu∗+ωu∗=0,x∈RN | (68) |
by
K∗(u∗)=K(u∗)=λ2c2σ+2>0, | (69) |
which implies that
(−Δ)suω,c+|x|2uω,c−|uω,c|2σuω,c+ωuω,c=0,x∈RN, | (70) |
which means
Now, for
{˜u(n+1)j−u(n)jτ=−δsx˜u(n+1)|j−(|xj|2+u(n)j)˜u(n+1)ju(n+1)j=c˜u(n+1)jMn2σ+2, | (71) |
where
Mn2σ+2=(hN∑j|unj|2σ+2)1/(2σ+2) |
and
In this section, we show some numerical results, which can help us understand the ground state solution and also illustrate theoretical results. We have mainly investigated: 1. Ground state solutions with different
First, we solve (8) numerically by (64) in one dimension
Second, we use
By Theorem 2.1, we can obtain the existence of ground state solutions with
From figure 3a and figure 3b, it seems ground state solutions change continuously with
Then, we test another two things. The first is the relationship between the constrained minimal energy in (8) and
E(s)=hJ−1∑j=0[J/2−1∑l=−J/2|μl|2s|ˆul|2+|xj|2|uj|2−1σ+1|uj|2(σ+1)]. | (72) |
From figure 5a, we find the energy's dependence
Second, we test the relationship between
limc→0λc=λ0. |
We also test this with
Up to now, we only consider the case with radially symmetrical potential. However, when the potential is not radially symmetric, we can still find a standing wave to (1) using (8). We tried cases where the potential has the form
First, we consider the case
D(s,t)=‖us(x,t)−ψs(x,t)‖˜Σs(R)‖us(x,t)‖˜Σs(R). |
We test initial conditions
Second, we try to obtain some numerical results when we touch the critical point
Finally, we test some simple time dynamics of FNLS, we let
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