On the Marchenko system and the long-time behavior of multi-soliton solutions of the one-dimensional Gross-Pitaevskii equation

0.29 MB·English

by  Haidar Mohamad

5
1 On the Marchenko system and the long-time behavior of
0
2 multi-soliton solutions of the one-dimensional Gross-Pitaevskii

n
a equation
J

4
2
Haidar Mohamad 1

]
P
January 27, 2015
A
.
h
t Abstract
a
m We establish a rigorous well-posedness results for the Marchenko system associated to the
scatteringtheoryoftheonedimensionalGross-Pitaevskiiequation(GP).Undersomeassumptions
[
on the scattering data, these well-posedness results provide regular solutions for (GP). We also
1 construct particular solutions, called N−soliton solutions as an approximate superposition of
v traveling waves. A study for the asymptotic behaviors of such solutions when t → ±∞ is also
3 made.
3
0
6
1 Introduction
0
.
1
0 In any space dimension N, it is usual to call nonlinear Schr¨odinger equation (NLS) the evolution
5 equation
1 i∂ u+∆u+f(u2)u=0, x RN, t R (1)
t
: | | ∈ ∈
v where f : R+ R is some smooth function. We often complete this equation by the boundary
i →
X condition at infinity
lim u(t,x)2 =ρ ,
r 0
a x + | |
| |→ ∞
where the constant ρ 0 is such that f(ρ )=0. Equation (NLS) is called focusing when f (ρ )>0
0 0 ′ 0
≥
and defocusing when f (ρ )<0.
′ 0
Theseequationsarewidelyusedasmodelsinmanydomainsofmathematicalphysics,especiallyin
non-linear optic, superfluidity, superconductivity and condensation of Bose-Einstein (see for example
[1, 2, 3]). In non-linear optic case, the quantity u2 corresponds to a light intensity, in that of the
| |
Bose-Einstein condensation, it corresponds to a density of condensed particles.
Inthefollowing,weextendbyGross-Pitaevskiiequationtheone-dimensionalnon-linearSchr¨odinger
equation with cubic nonlinearity, namely
i∂ u+∂2u+(1 u2)u=0, (GP)
t x −| |
and for which the data at infinity satisfies 0=ρ =1.
0
6
Multi-soliton solutions
In the early seventies, Shabat and Zakharov discovered an integrable system for the one-dimensional
cubic Schr¨odinger equation. They also proposed a semi-explicit method of resolution through the
1D´epartementdeMath´ematiques,Universit´eParis-Sud11,F-91405OrsayCedex. [email protected]
theoryofinversescattering. Theirresultswerepresentedintwoshortarticles,the firstoneis devoted
to the focusing case [4] and the second one is devoted to the defocusing case (i.e. to the Gross-
Pitaevskii equation (GP)) [5]. A similar structure was discovered a few years earlier by, Green,
Kruskal and Miura [6] for Korteweg - de Vries equation
∂ v+∂3v 6v∂ v =0. (KdV)
t x − x
These equations have in common special solutions called solitons, which play an important role in
dynamics in the long time. In the following, we extend by soliton a progressive wave solution, i.e. a
solution of the form
u(t,x)=U(x ct)
−
where U is the wave profile and c its speed. For the Gross-Pitaevskii equation (GP), such solutions
(of finite Ginzburg-Landau energy except for trivial constant solutions) exist if and only if the speed
c satisfies c ( √2,√2). In this case, for a fixedspeed, the profile is unique modulo the invariantsof
∈ −
the equation: specifically
u(t,x)=eiθ0Uc(x x0 ct)
− −
where U is explicitly given by
c
c2 c2 x c
U (x)= 1 tanh 1 +i (2)
c
r − 2 r − 2 √2! √2
andθ ,x Rarearbitraryandreflecttheinvariancebyrenormalizationofthephaseandtranslation.
0 0
∈
We will prove the following result :
Theorem 1.1 Let N 1, c < < c ( √2,√2), θ R and x , ,x R be arbitrary and
1 N 0 1 N
fixed. There exists a sm≥ooth solu·ti·o·n u for∈th−e Gross-Pitaev∈skii equation··(·GP) o∈n whole R R such
×
that
tlim u(t,x+ckt)=ei(θ0+θk)A±kUck(x−xk−ηk±),
→±∞
for all x R and k 1, ,N , with
∈ ∈{ ··· }
N 2 c c + 2 c2 2 c2
η+ = 1 log − j k − j − k , A+ =e2iPNj=k+1θj,
k − 2−c2k j=Xk+1 2−cjck−q2−c2jp2−c2k k
p  q p 
k 1 2 c c + 2 c2 2 c2
ηk− =− 21−c2k Xj=−1log2−−cjjckk−q2−−cj2jp2−−ck2k , A−k =e2iPjk=−11θj,
and p  q p 
c
j
θ =arccos( ), j =1, ,N.
j
√2 ···
In other words, in each coordinate system in translation with speed c , the solution u behaves
k
asymptoticallywhent likethesolitonU translatedappropriately,andthetranslationfactors
when t = ±∞, given b→y±ηk+∞and ηk−, can be icnkterpreted as a shift of binary collisions between the
solitons. Asimilarresultforthefocusingequationwasdescribedin[4],buttheequivalenceofTheorem
1.1 has not been proved by [5].
The solutions of Theorem 1.1 are obtained via an integrable system 1 called Marchenko system.
1Moreexactly, throughafamilyofintegrablesystemdependingontworealvariables.
Let β :R C and N N . For ξ R, we set
∗
→ ∈ ∈
1
λ=λ(ξ)= ξ2+ ,
2
r
c (ξ)=β(λ)+β( λ),
1
−
β(λ) β( λ)
c (ξ)= − − .
2
λ
Assume that the function ξ β(λ(ξ)) belongs to the space L1(R) L2(R). Consider:
7→ ∩
1. N real numbers 1 <λ <...<λ < 1 .
−√2 1 N √2
2. N real strictly negative numbers µ ,...,µ .
1 N
Finally, we define the functions of real variable
F11(z)= N µkλke−νkz, F12(z)= 1 +∞c1(ξ)eiξzdξ,
2π
kX=1 Z−∞
F21(z)= N µke−νkz, F22(z)= 1 +∞c2(ξ)eiξzdξ,
2π
Xk=1 Z−∞
F =F F , F =F F ,
1 12 11 2 22 21
− −
with ν = 1 λ2,k 1,...,N.
k 2 − k ∈
q
Proposition 1.2 Let n 0 be such that
≥
1
λ λn+2β(λ) L ([ ,+ [). (3)
∞
7→ ∈ √2 ∞
There exists a constant ǫ >0, depending on N, n, λ and µ , such that if
0 i i
λnβ(.) ǫ
k kL∞([√12,+∞[) ≤ 0
then for every x R the Marchenko system (with parameter x)
∈
2√2Ψ (x,y)=F (x+y)
1 2
 − x+∞[√2Ψ2(x,s)(F1(s+y)−iF2′(s+y))+Ψ1(x,s)F2(s+y)]ds, (4)
 2√2Ψ2(x,y)=−√Rx2+(∞F1[√(x2+Ψ1y()x+,si)F(F2′(1x(s++yy)))+iF2′(s+y))+Ψ2(x,s)F2(s+y)]ds,
has a unique solution Ψ(x,.R) (Hn(y x))2. Moreover, the function
∈ y ≥
Ψ:R R+ C2 :Ψ(x,p)=Ψ(x,x+p),
× →
belongs to the space k(R,(Hn k(R+))2), k 0,...,n 1 .
−
C ∈{ − }
The well-posedness in the previous proposition are obtained by perturbation of the case β 0,
≡
which corresponds to the exact multi-solitons solutions, and for which the result is completely of
algebraic nature.
We refer the reader to [7] and [8] for more details about the derivation of Marchenko system via
the scattering theory associated to (GP). We focus in this paper on the other direction where the
link with the Gross-Pitaevskii equation is obtained by the following proposition which introduces an
additional time parameter t in Marchenko system:
Proposition 1.3 Assume that
β(t,λ)=c(λ)exp( 4iλξt), µ (t)=µ0exp(4λ ν t), 1 k N,
− k k k k ≤ ≤
where the µ0 are assumed to be strictly negative and the function c satisfies assumptions (3) above.
k
Then if the quantity λnc(.) is small enough, for every (t,x) R2 the Marchenko system
k kL∞([√12,+∞[) ∈
(with fixed parameters x and t) has a unique solution Ψ(t,x,.) (Hn(y x))2 such that Ψ
∈ y ≥ ∈
k(R2,(Hn 2k(R+))2) for all k N with k < n. Moreover, if n 3, the function
C − ∈ 2 ≥
u(t,x)=1+2√2iΨ (t,x,0)
12
is solution of the Gross-Pitaevskii equation in 1(R2) such that u(t,.) 2(R) for all t R. Finally,
if the ν are pairwise distinct and c is a real-vaClued function, for t R∈fiCxed, we have ∈
k
∈
lim (u(t,x) u (t,x))=0,
N
x −
→±∞
where u is the solution corresponding to β 0 (the other coefficients λ and µ0 are left equal).
N ≡ k k
ThesolutionsmentionedinTheorem1.1areexactlythe abovesolutionsu . Thelink betweenthe
N
speeds c of Theorem 1.1 and the coefficients λ of Marchenko system is given by the relationships
k k
c =2λ .
k k
2 N-soliton solutions
The purpose of this section is the resolution of Marchenko system when β(λ)=0. We set
1. N real numbers 1 <λ <λ <...<λ < 1 .
−√2 1 2 N √2
2. N negative real numbers µ ,µ ,...,µ .
1 2 N
In this case, the Marchenko system is rewritten as follows
N
2√2Υ1(x,y) = µke−νkxe−νky
−
k=1
X
N +
+ e−νky ∞[√2Υ2(x,s)µk(λk+iνk)e−νks+Υ1(x,s)µke−νks]ds,
k=1 Zx
X
(5)
N
2√2Υ2(x,y) = √2 µk( λk+iνk)e−νkxe−νky
−
k=1
X
N +
+ e−νky ∞[√2Υ1(x,s)µk(λk iνk)e−νks+Υ2(x,s)µke−νks]ds.
−
k=1 Zx
X
(6)
This means that Υ and Υ take the forms
1 2
N N
Υ1(x,y)= fj(x)e−νjy, Υ2(x,y)= gj(x)e−νjy.
j=1 j=1
X X
Denote α =µ (λ iν ), hence
k k k k
−
+ + N
∞Υ2(x,s)e−νksds = ∞ gj(x)e−(νj+νk)sds
Zx Zx j=1
X
N e−(νj+νk)x
= g (x) .
j
ν +ν
j k
j=1
X
Similarly, we have
+∞Υ1(x,s)e−νksds= N fj(x)e−(νj+νk)x,
ν +ν
Zx j=1 j k
X
and the two equations (5) and (6) become
N
2√2Υ1(x,y) = µke−νkxe−νky
−
k=1
X
+ N e−νky N [√2α¯kgj(x)+µkfj(x)]e−(νj+νk)x,
ν +ν
j k
k=1 j=1
X X
N
2√2Υ2(x,y) = √2 αke−νkxe−νky
−
k=1
X
+ N e−νky N [√2αkfj(x)+µkgj(x)]e−(νj+νk)x.
ν +ν
j k
k=1 j=1
X X
Identifying the terms with factors exp( ν y), we get
k
−
2√2fk(x) N [µkfj(x)+√2α¯kgj(x)]e−(νj+νk)x = µke−νkx, k 1,2,...,N . (7)
− ν +ν − ∈{ }
j k
j=1
X
2√2gk(x) N [√2αkfj(x)+µkgj(x)]e−(νj+νk)x = √2αke−νkx, k 1,2,...,N (8)
− ν +ν − ∈{ }
j k
j=1
X
Assuming x as a parameter, the 2N equations (7) and (8) correspond to linear system with 2N
equations and 2N unknowns. To analyze its resolution, we set
1 A B
T = ,
2√2 B A
(cid:18) (cid:19)
1 C1 F1
C = , F = ,
2√2 C2 F2
(cid:18) (cid:19) (cid:18) (cid:19)
with
e−(µi+µj)x
i,j 1,2,..,N (A) = µ i,j 1,2,..,N ,
ij i
∀ ∈{ } − µ +µ ∀ ∈{ }
i j
(B) = √2α e−(µi+µj)x , β e−(µi+µj)x i,j 1,2,..,N ,
ij i i
− µ +µ − µ +µ ∀ ∈{ }
i j i j
(C1)k = µke−νkx, (C2)k = √2αke−νkx k 1,2,..,N ,
− − ∀ ∈{ }
(F1) =f , (F2) =g k 1,2,..,N ,
k k k k
∀ ∈{ }
so that (7) and (8) can be rewritten in the matrix form:
(I +T)F =C. (9)
Proposition 2.1 The matrix (I+T)(x) is invertible for every x R.
∈
Proof. The idea of this proof is to demonstrate that 1 is not an eigenvalue for T. To this end,
−
define
I 0
Q= N ,
D I
N
(cid:18) (cid:19)
withD (C)isadiagonalmatrixdefinedasfollows(D) = βi i 1,...,N .Usingthefact
∈MN ii −µi ∀ ∈{ }
that β 2 =µ2, we get
| i| i
1 A DB B
QTQ−1 = − ,
2√2 0 0
(cid:18) (cid:19)
hence we have the following equivalence
( 1 Sp(T)) ( 1 Sp(A DB)) det(A DB+I )(x)=0, x R,
N
¬ − ∈ ⇔¬ − ∈ − ⇔ − 6 ∀ ∈
where
(µ +β¯ β1)e−2ν1x (µ +β¯ β2)e−(ν1+ν2)x ... (µ +β¯ βN)e−(ν1+νN)x
1 1µ1 2ν1 1 1µ2 ν1+ν2 1 1µN ν1+νN
(µ +β¯ β1)e−(ν1+ν2)x (µ +β¯ β2)e−2ν2x ... (µ +β¯ βN)e−(ν2+νN)x
A DB =  2 2µ1 ν1+ν2 2 2µ2 2ν2 2 2µN ν2+νN .
− − ... ... ...
(µ +β¯ β1)e−(ν1+νN)x (µ +β¯ β2)e−(ν2+νN)x ... (µ +β¯ βN)e−2νNx 
 N Nµ1 ν1+νN N Nµ2 ν1+ν2 N NµN νN 
 
Denote now Dˆ = diag(µ ,µ ,...,µ ). Then we have
1 2 N
−
det(A DB+I )=0 det((A DB+I )Dˆ)=0,
N N
− 6 ⇔ − 6
since det(Dˆ)= N ( µ )>0, and
k=1 − k
Q
(µ2+β¯ β )e−2ν1x (µ µ +β¯ β )e−(ν1+ν2)x ... (µ µ +β¯ β )e−(ν1+νN)x
1 1 1 2ν1 1 2 1 2 ν1+ν2 1 N 1 N ν1+νN
(A DB)Dˆ = (µ2µ1+β¯2β1)e−ν(ν11++νν22)x (µ22+β¯2β2)e−22νν22x ... (µ2µN +β¯2βN)e−ν(ν22++ννNN)x.
− ... ... ...
(µ µ +β¯ β )e−(ν1+νN)x (µ µ +β¯ β )e−(ν2+νN)x ... (µ2 +β¯ β )e−2νNx 
 N 1 N 1 ν1+νN N 2 N 2 ν1+ν2 N N N νN 
 
It is clear that the matrix (A DB)Dˆ is Hermitian and that
−
x
(A DB)Dˆ =(µiµj +β¯iβj) − e(νi+νj)tdt i,j 1,2,..,N .
− ij ∀ ∈{ }
(cid:16) (cid:17) Z−∞
Let now Z CN 0 . Then
∈ \{ }
<(A DB+I )DˆZ,Z > = <(A DB)DˆZ,Z >+<DˆZ,Z >
N
− −
x N x N N
− µ z eνit 2dt+ − β z eνit 2dt µ z 2
i i i i i i
| | | | − | |
Z−∞ Xi=1 Z−∞ Xi=1 Xi=1
>0,
which means that the matrix (A DB+I )Dˆ is positive definite, hence det(A DB+I )Dˆ = 0.
N N
− − 6
Thus, the proof is completed.
3 Resolution of Marchenko system: proofs of Propositions 1.2
and 1.3
3.1 The case with no time dependence
The solution corresponding to β = 0 will be obtained by perturbation of the N-soliton solution
6
Υ
constructed in the previous section. Denote Υ = 1 the solution of Marchenko system (4) corre-
Υ
2
(cid:18) (cid:19)
sponding to β 0 and the same values of the other scattering coefficients, i.e. λ and µ . Define
i i
≡
Ψr =Ψ Υ , Ψr =Ψ Υ .
1 1− 1 2 2− 2
Ψr
The Marchenko system (4) is thus transformed into a system for Ψr = 1 :
Ψr
(cid:18) 2 (cid:19)
(2√2I +Ω )Ψr(x,y)= Ψr(x,y), (10)
d x
F
where the two components (Ω Ψr) (x,.) and (Ω Ψr) (x,.) of Ω Ψr(x,y) are given by
x 1 x 2 x
+ +
(ΩxΨr)1(x,y)=− ∞√2Ψr2(x,s)(F11−iF2′1)(s+y)ds− ∞Ψr1(x,s)F21(s+y)ds,
Zx Zx
+ +
(Ω Ψr) (x,y)= ∞√2Ψr(x,s)(F +iF )(s+y)ds ∞Ψr(x,s)F (s+y)ds,
x 2 − 1 1 2′1 − 2 21
Zx Zx
and the two components ( Ψr) (x,.) and ( Ψr) (x,.) of Ψr(x,.) are given by
1 2
F F F
+
( Ψr) (x,y) = ∞√2(Υ +Ψr)(x,s)(F iF )(s+y)ds
F 1 − 2 2 12− 2′2
Zx
+
+ ∞(Υ +Ψr)(x,s)F (s+y)ds+F (x+y), (11)
1 1 22 22
Zx
+
(FΨr)2(x,y) = − ∞√2(Υ1+Ψr1)(x,s)(F12+iF2′2)(s+y)ds
Zx
+
+ ∞(Υ +Ψr)(x,s)F (s+y)ds+√2(F +iF )(x+y). (12)
2 2 22 12 2′2
Zx
The operator Ω (L2(y x)) satisfies the following property:
x ∈L y ≥
Lemma 3.1 The operator Ω is positive independently of x. Namely,
x
+
∞Ω Φ Φ¯ 0, Φ (L2(y x))2.
x · ≥ ∀ ∈ y ≥
Zx
Proof. In view of the definition of F and F , we have
11 21
+ N +
∞(ΩxΦ)1Φ¯1 = µk ∞e−νkyΦ1(y)dy 2
− | |
Zx k=1 Zx
X
N + +
√2 µk(λk+iνk) ∞e−νksΦ2(s)ds ∞e−νkyΦ¯1(y)ds,
−
k=1 Zx Zx
X
(13)
3.1 The case with no time dependence
and
+ N +
∞(ΩxΦ)2Φ¯2 = µk ∞e−νkyΦ2(y)dy 2
− | |
Zx k=1 Zx
X
N + +
√2 µk(λk iνk) ∞e−νksΦ1(s)ds ∞e−νkyΦ¯2(y)ds.
− −
k=1 Zx Zx
X
(14)
Combining (13) and (14), we obtain
+ N + N +
∞ΩxΦ Φ¯ = µk ∞e−νkyΦ1(y)dy 2+ µk ∞e−νkyΦ2(y)dy 2
− · | | | |
Zx k=1 Zx k=1 Zx
X X
N + +
+2√2 µkλkRe ∞e−νksΦ2(s)ds ∞e−νkyΦ¯1(y)ds
k=1 (cid:18)Zx Zx (cid:19)
X
N + +
2√2 µkνkIm ∞e−νksΦ2(s)ds ∞e−νkyΦ¯1(y)ds . (15)
−
k=1 (cid:18)Zx Zx (cid:19)
X
Since µ are assumed negative, the sum of the last two terms of the right-hand side of (15) is upper-
k
bounded by
N + +
−2√2 µk λ2k+νk2 ∞e−νksΦ1(s)ds ∞e−νkyΦ¯2(y)dy
kX=1 q (cid:12)(cid:12)Zx Zx (cid:12)(cid:12)
N + (cid:12) + (cid:12)
µk ∞e−(cid:12)νkyΦ1(y)dy 2+ ∞e−νkyΦ2(y)dy 2 (cid:12).
≤− | | | |
k=1 (cid:18) Zx Zx (cid:19)
X
Then
+
∞Ω Φ Φ¯ 0.
x
· ≥
Zx
The operator 2√2I +Ω (L2(y x)) is coercive independently of x. Namely, we have
d x ∈L y ≥
+
∞(2√2I +Ω )Φ Φ¯ 2√2 Φ , Φ (L2(y x))2, x R, (16)
d x · ≥ k kL2 ∀ ∈ y ≥ ∀ ∈
Zx
whichimpliesthatitisanisomorphismon(L2(y x))2 andthat (2√2I +Ω ) 1 1 .This
y ≥ ||| d x − |||L(L2) ≤ 2√2
allows us to establish the following corollary:
Corollary 3.2 For every k N, the operator 2√2I +Ω is an isomorphism of
d x
∈
(Hk(y x)). Moreover, we have
L y ≥
1
(2√2I +Ω ) 1 .
||| d x − |||L(Hk) ≤ 2√2
Proof. Let Φ (Hk(y x))2. We know that there exists a unique Θ (L2(y x))2 such that
∈ y ≥ ∈ y ≥
(2√2I +Ω )Θ=Φ.
d x
3.1 The case with no time dependence
In view of the definition of Ω , we have Ω Θ (Hk(y x))2, hence
x x ∈ y ≥
1
Θ= (Ω Θ Φ) (Hk(y x))2.
2√2 x − ∈ y ≥
This proves that 2√2I +Ω : (Hk(y x))2 (Hk(y x))2 is invertible, hence 2√2 is not an
d x y ≥ → y ≥
eigenvalue for Ω ((Hk(y x))2). This operator is compact, since it is a finite-rank one. Then
− x ∈ L y ≥
in view of Fredholm alternative, 2√2 belongs to the resolvent of Ω . This means that 2√2I +Ω
x d x
−
is an isomorphism on (Hk(y x))2. On the other hand, the point spectrum of 2√2I +Ω relatively
y ≥ d x
to the space (Hk(y x))2 is included in the point spectrum relatively to (L2(y x))2, and similarly
y ≥ y ≥
for (2√2I +Ω ) 1. Since Ω ((Hk(y x))2) is compact, the spectrum of 2√2I +Ω contains
d x − x ∈ L y ≥ d x
only eigenvalues. We deduce therefore that
1
(2√2I +Ω ) 1 (2√2I +Ω ) 1 .
||| d x − |||L(Hk) ≤||| d x − |||L(L2) ≤ 2√2
Equation (10) is equivalent to
Ψr(x,y)=(2√2I +Ω ) 1 Ψr(x,y). (17)
d x −
F
For x R, we use the fixed point argument to prove that (17) has a unique solution Ψr(x,.)
∈ ∈
(Hn(y x))2. To this end, the following lemma will be needed:
y ≥
Lemma 3.3 Let x R. The function defined above is Lipschitz continuous on (Hn(y x))2 and
∈ F y ≥
there exists a constant C > 0 depending on β such that for all Ψr(x,.),Ψl(x,.) (Hn(y x))2, we
∈ y ≥
have
( Ψr Ψl)(x,.) C Ψr(x,.) Ψl(x,.) .
(Hn(y x))2 (Hn(y x))2
k F −F k y ≥ ≤ k − k y ≥
Proof. Let Ψr(x,.),Ψl(x,.) (Hn(y x))2. For every k n, we have
∈ y ≥ ≤
+
∂k( Ψr Ψl) (x,y) = ∞√2(Ψr Ψl)(x,s)(F iF )(k)(s+y)ds
y F −F 1 − 2− 2 12− 2′2
Zx
+
+ ∞(Ψr Ψl)(x,s)F(k)(s+y)ds, (18)
1− 1 22
Zx
+
∂yk(FΨr−FΨl)2(x,y) = − ∞√2(Ψr1−Ψl1)(x,s)(F12+iF2′2)(k)(s+y)ds
Zx
+
+ ∞(Ψr Ψl)(x,s)F(k)(s+y)ds, (19)
2− 2 22
Zx
where
1 ξ ξ
(F12−iF2′2)(k)(s+y)= 2π R(iξ)k (1+ λ)β(λ)+(1− λ)β(−λ) eiξ(s+y)dξ,
Z (cid:18) (cid:19)
1 (iξ)k
F(k)(s+y)= (β(λ) β( λ))eiξ(s+y)dξ.
22 2π R λ − −
Z
Denote
ξ ξ
d ( ξ)=(iξ)k (1+ )β(λ)+(1 )β( λ) ,
k
− λ − λ −
(cid:18) (cid:19)
(iξ)k
e ( ξ)= (β(λ) β( λ)).
k
− λ − −
3.2 Variation for Proposition 1.3 : addition of time dependence
Since λ λn+2β(λ) is assumed to be in L ([ 1 ,+ [), there exists C >0 such that
7→ ∞ √2 ∞
ξkλ2β(λ(ξ)) λk+2 β(λ) C,k 0,...,n .
| |≤ | |≤ ∈{ }
Then d ,e L1(R) L (R) and (18) can be rewritten as follows
k k ∞
∈ ∩
∂k( Ψr Ψl) (x,y) = ω1(s)e (s+y)ds ω2(s)d (s+y)ds
y F −F 1 R x k − R x k
Z Z
= (ωx1⋆ek(−b·))(−y)−(ωx2⋆dk(−c·))(−y),
(20)
b c
with ω1(s)=1 (Ψr Ψl)(x,s), ω2(s)=√21 (Ψr Ψl)(x,s), where d ,e are the Fourier
x ]x,+∞[ 1− 1 x ]x,+∞[ 2− 2 k k
transforms of d ,c respectively. Since the functions d ,e , ω1 and ω2 belongs to L2(R), we have
k k k k x x
c b
ωx1⋆ek(−·)=F−1[ωx1dk], ωcx2⋆ebk(−·)=F−1[ωx2ek],
where F 1 is the inverseFourier transform. This implies that the left hand side members of previous
− c c
equations belong to L2(R) abnd that ∂k( Ψr Ψl)(xb,.) (L2(y x))2. Applying the Fourier
transform on (20) and taking the L2(R)ynForm,−weFobtain ∈ y ≥
k∂yk(FΨr−FΨl)1(x,.)kL2y(y≥x) ≤ √2kdkkL∞(R)k(Ψr2−Ψl2)(x,.)kL2y(y≥x)
+kekkL∞(R)k(Ψr1−Ψl1)(x,.)kL2y(y≥x)
2√2 λkβ(.)
≤ k kL∞([√12,+∞[)
( (Ψr Ψl)(x,.)
× k 1− 1 kL2y(y≥x)
+ (Ψr Ψl)(x,.) ).
k 2− 2 kL2y(y≥x)
A similar argument yields the same inequality from (19). This completes the proof.
Consequently, if λnβ(.) is small enough, the operator (2√2I +Ω ) 1 has a unique
k kL∞[√12,+∞[ d x − F
fixed point Ψr(x,.) (Hn(y x))2 which is the solution of (10).
∈ y ≥
3.2 Variation for Proposition 1.3 : addition of time dependence
Assume that
β(t,λ)=c(λ)exp( 4iλξt), µ (t)=µ0exp(4λ ν t), 1 k N.
− k k k k ≤ ≤
In the case c 0, we deduce, following the same steps followed in Section 2, that there exist two
families f , ≡g of functions in the space (R2) such that for all (t,x) R2, the function
k k ∞
{ } { } C ∈
t
N N
y Υ(t,x,y)= fk(t,x)e−νky, gk(t,x)e−νky
7→ !
k=1 k=1
X X
is the unique solution of the corresponding Marchenko system. The dependence on t of the two
operators Ω and does not change the previous results. More specifically, the operator Ω stays
x x
non-negative indepFendently of (t,x) R2. This allows us to establish a similar corollaryto Corollary
∈
3.2. On the other hand, we have
ξkβ(t,λ(.)) L (R) = ξkc(λ(.)) L (R),k n.
k k ∞ k k ∞ ≤
Hence, if λnc(.) is small enough, for all (t,x) R2, the operator (2√2I +Ω ) 1 has
k kL∞([√12,+∞[) ∈ d x − F
a unique fixed point Ψr(t,x,.) (Hn(y x))2 which is the solution of (10).
∈ y ≥