Vortex helices for the Gross Pitaevskii equation David CHIRON

David Chiron

Description

Niveau: Supérieur, Doctorat, Bac+8
Vortex helices for the Gross-Pitaevskii equation David CHIRON Laboratoire Jacques-Louis LIONS, Universite Pierre et Marie Curie Paris VI, 4, place Jussieu BC 187, 75252 Paris, France E-Mail : Abstract : We prove the existence of travelling vortex helices to the Gross-Pitaevskii equation in R3. These solutions have an infinite energy, are periodic in the direction of the axis of the helix and have a degree one at infinity in the orthogonal direction. Resume : Nous prouvons l'existence d'ondes progressives a vorticite sur une helice pour l'equation de Gross- Pitaevskii dans R3. Ces solutions sont d'energie infinie, periodiques dans la direction de l'axe de l'helice et ont un degre un dans la direction orthogonale. Keywords : Travelling wave; non linear Schrodinger equation; helix; Ginzburg-Landau; vorticity. 1 Introduction 1.1 Statement of the result In this paper, we are interested in the existence of travelling waves solutions to the Gross-Pitaevskii equation in space dimension 3 i∂?∂t + ∆? + (1 ? |?| 2)? = 0, (1) where ? : R3 ? R ? C. This equation is used as a model for Bose-Einstein condensates, nonlinear optics and superfluidity. On a formal level, it possesses two important quantities constant in time • the energy E(?) = 12 ∫ R3 |??(.

• travelling waves


• u?


• t? ?


• gross-pitaevskii equation


• direction de l'axe de l'helice

Sujets

• Chiron
• Gross?Pitaevskii equation

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

Vortex helices for the Gross-Pitaevskii equation
David CHIRON
Laboratoire Jacques-Louis LIONS,
Universite Pierre et Marie Curie Paris VI,
4, place Jussieu BC 187, 75252 Paris, France
E-Mail : [e-mail]
3Abstract : We prove the existence of travelling vortex helices to the Gross-Pitaevskii equation in R . These
solutions have an in nite energy, are periodic in the direction of the axis of the helix and have a degree one at
in nit y in the orthogonal direction.
Resume : Nous prouvons l’existence d’ondes progressives a vorticite sur une helice pour l’equation de Gross-
3Pitaevskii dans R . Ces solutions sont d’energie in nie, periodiques dans la direction de l’axe de l’helice et ont
un degre un dans la direction orthogonale.
Keywords : Travelling wave; non linear Schr odinger equation; helix; Ginzburg-Landau; vorticity.
1 Introduction
1.1 Statement of the result
In this paper, we are interested in the existence of travelling waves solutions to the Gross-Pitaevskii
equation in space dimension 3
@ 2i + + (1 j j ) = 0; (1)
@t
3where : R R ! C. This equation is used as a model for Bose-Einstein condensates, nonlinear
optics and super uidit y. On a formal level, it possesses two important quantities constant in time
the energy Z
1 12 2 2E( ) = jr (:;t)j + (1 j (:;t)j ) dx;
2 3 2R
the momentum Z Z
~P( ) =Im r dx = (i ;r ) dx;
3 3R R
R
2 ~where (:;:) is the scalar product in R ’ C. The rst component ofP is denotedP( ) = (i ;@ )dx.3 1R
Travelling waves solutions to (1) are solutions of the form (up to a rotation)
(x;t) =U(x Ct;x ;x ):1 2 3
1The equation on reads now on U
@U 2iC = U + (1 jUj )U: (2)
@x1
The question of the existence of such travelling waves for small speeds has been studied in [BS] in
dimension 2 and in [BOS] and [C1] in dimension larger than 2. We refer to these papers for details
and references about the Gross-Pitaevskii equation. In [BS], travelling waves with a structure of two
vortices of degrees 1 and 1 are exhibited, and in [BOS] and [C1] the travelling wave is a vortex ring
(like a \smoke ring").
We consider a function U de ned in the following way. We use cylindrical coordinates (x ;r;),1L
where (r;)2 R (R=2Z) are the polar coordinates in the (x ;x )-plane. We set T := R=2Z (we+ 2 3
1do not identify T with S to be able to de ne @ for example). We x L 0 and de ne in cylindricalx1
coordinates
2H :=fx2 T R ; r =L; x =g:L 1
p
2 ~This is an helix of axis x , of pitch L, and length M(H ) = 2 1 +L , that we denote H when1 L L
endowed with the orientation given by the natural parametrization
T37!() := (;Lcos;Lsin):
~If L = 0, then H = Tf0g is the x axis. We may then see 2H as a prescribed vorticity and0 1 L
1 2 1consider a mapU 2C (TR nH ;S ), which will be precisely de ned at the end of the subsection,LL
such that its vorticity concentrates on the helix H in the sense thatL
~curl(U rU ) = 2H and div(U rU ) = 0; (3)LL L L L
that is the vector eld U rU , representing the gradient of the phase of U , is given in the gureL L L
1 ~below. The mapU is therefore smooth outsideH , is S -valued and has a degree one aroundH andL LL
x
1
2p
L
Figure 1: The vector eld U rUL L
at in nit y (in the (x ;x )-plane). Our main result states the existence, after rescaling, of solutions to2 3
(2) close toU . Due to the degree one at in nit y, they are of in nite energy. Moreover, theseL
are periodic in the x variable of the axis of the helix.1
2Theorem 1. For every L> 0, there exists " (L)> 0 such that, for every 0<"<" (L), there exists0 0
2a solution U to (2), -periodic in the x variable, with C =C(") verifying, if "! 0," 1"
2C(") 1 L
! p and P(U ) = 2 : (4)"
2"jlog"j "1 +L
Moreover,
jU (x)j! 1 as j(x ;x )j! +1 (5)" 2 3
and, for every k2 N,
x k 2U !U in C (T R nH ): (6)" LL loc"
Remark 1. In the limiting case L = 0, we can nd solutions of (2) independent of x , that is1
2U(x) =V (x ;x ), with V solution of in nite energy (in R ) and with a degree one at in nit y of2 3
2 2V +V (1 jVj ) = 0 in R : (7)
These solutions have been studied in [BMR] and also [Sha], [San1] and [Mi]. The associated functions
U clearly have a vanishing momentum and are solutions of (2) for any speed C 2 R. There exists a
particular radially symmetric solution of (7) of degree one at in nit y of the form
z
V (z) =(jzj) ;0
jzj
where (r) increases from 0 to 1 as r goes from 0 to +1.
2Remark 2. It is important to note that the solution is -periodic in thex variable, and its singular1"
L 2set is an helix of pitch . Therefore, we will work with functions U which are de ned on T R ,""
2with T := R=( Z)." "
Remark 3. We nally emphasize that the momentum in (4) is not exactly the one already introduced.
Indeed, since the solution U is periodic in the x variable, the integral which de nes the momentum" 1
3is clearly not convergent in R . We will instead consider a momentum de ned only on a period, that
2is T R . Even in this case, we clarify just below the de nition."
We clarify the notion of momentum for our problem, and adapt to the situation with a degree one
at in nit y the de nition given in [BOS]. Note that neither the de nition ofP, since (iU;@ U) may not1
1be in L at in nit y, nor an energy space is clear, since the degree one at in nit y makes the energies
2to diverge. We denote D (R> 0) the disk in R of radius R centered at 0. We consider the class ofR
functions
R
1 1 2 2 1 2 2Y :=fU 2H \L (T R ;C); j@ Uj + (1 jUj ) <1, 9R> 0 s.t.2" " 1loc T R 2"
for rR, jU(x)j 1=2 andU has degree one outside T D g." R
Note that if U 2 Y and jU(x)j 1=2 for r R, the degree of U outside T D is well-de ned." " R
0Indeed, from [BLMN], we know that, for every R >R, since
U 1 102H (T (D nD );S );" R R
jUj
then its degree on almost every slice fx g (D 0 nD ) is well-de ned and is independent of x and1 R R 1
0R >R : we will call this integer the degree of U outside T D . For U 2Y , we may then write" R "
U(x) =(x)exp(i’(x) +i );
31 2for r R, where (x) = jU(x)j 1=2 and ’ 2 H (T (R nD );R) is well-de ned modulo a" Rloc
2 2multiple of 2 (note that imposing @ U 2 L (T R ) prevents U from having a degree in the x1 " 1
variable). We de ne then
Z Z Z
2P(U) := (iU;@ U) + (1 )( 1)@ ’ + ’@ (1 ); (8)1 1 1
2 2 2T R T R T R" " "
where is a smooth function compactly supported, such that 0 1 and = 1 on T D . It is" R
easy to verify that this de nition of P(U) does not depend on the exact choice of and’.
For our problem, it is convenient to perform the rescaling
x C(")
u (x) :=U ; c := :" " "
" "jlog"j
2The function u is then de ned in T R and equation (2) reads now on u" "
@u 1" 2ic jlog"j = u + u (1 ju j ): (9)" " " "2@x "1
The expressions of the (diverging) energy and momentum are now
Z Z
1 12 2 2E (u ) ="E(U ) = jru j + (1 ju j ) dx = e (u ) dx" " " " " " "22 2"2 2TR TR
and Z
2p(u ) =" P(U ) = (iu ;@ u ) dx:" " " 1 "
2TR
Finally, we would like to mention why we have been interested in these solutions. In [BOS] (see
Theorem 4), the study of the asymptotic of a general Ginzburg-Landau equation including (9) in a
Ndomain
R ,N 3 under assumption sup jc j<1 for solutionsu satisfying the natural energy" " "
bound
E (u )M jlog"j" " 0
leads to the mean curvature equation for the concentration set
dJ~H(x) =? c~e ^? ;1
d
where (all these limits are for a subsequence" ! 0)c = lim c ,? is Hodge duality,J is a limitingn "!0 "
e (u )dx" " dJmeasure of the jacobian, a limiting measure of , is the Radon-Nikodym derivative and jlog"j d
~H is the generalized mean curvature of the varifoldV ( ; ) ( is the 1-dimensional density of
dkJ kand =f > 0g its geometrical support). If N = 3 and = 1, the singular set is a smooth d
curve and this equation rewrites
~ =c~e ~; (10)1
d~ 3where~ is the unit tangent and~ := is the curvature vector of. The solutions in R are the circlesds
1 3 3a +f0g@D(0;c ) (a2 R ), the straight lines a + R~e (a2 R ), and helices of axis parallel to~e .1 1
The case of a singular circle comes from Theorem 1 in [BOS] (forN = 3). A straight line singular set
comes from a two dimensional solution (independent ofx ) of the classical Ginzburg-Landau equation1
in two dimensions, having a singularity of degree 1 in (x ;x ) = (0;0), as the mapV (see Remark 1),2 3 0
having radial symmetry. We have constructed the last type of solution.
4 De nition of the map U . In order to de ne precisely the mapU , we note that the natural vectorL L
eld ~v verifying (3) is given by Biot-Savart law
Z Z +1 0~1 (x y) (2H (y)) 1 (x ()) ()L
~v(x) := dy = d : (11)
3 34 3 jjx yjj 2 jjx ()jjR 1
0 2 2Note that the integral is convergent since k k = 1 +L and k()k jj for jj ! +1. By
construction, the vector eld ~v is smooth outside H , satis es div~v = 0 and its vorticityL
~curl~v = 2HL
~is concentrated on H . Moreover, ~v h