On Schroedinger Equations with Concentrated Nonlinearities

24 Pages·0.35 MB·English

by  Giovanni Jona-Lasinio

On Schr(cid:127)odinger equations
with concentrated nonlinearities
Giovanni Jona-Lasinio
Dipartimento di Fisica, Universit(cid:18)a di Roma \La Sapienza",
Piazzale A. Moro 2, Roma, Italy 00185
and Centro Linceo Interdisciplinare, via della Lungara 10, Roma, Italy 00165
Carlo Presilla
Dipartimento di Fisica, Universita(cid:18) di Roma \La Sapienza",
Piazzale A. Moro 2, Roma, Italy 00185
5
9
9
1 1
Johannes Sj(cid:127)ostrand

n
D(cid:19)epartement de Math(cid:19)ematiques, B^at. 425, Universit(cid:19)e de Paris-Sud,
a
J Orsay, France 91405

0 and U.R.A. 760, C.N.R.S.
1
(January 10, 1995)

7
3
0
1 Abstract
0
5
9
Schro(cid:127)dinger equations with nonlinearities concentrated in some regions of
/
t
a space are good models of various physical situations and have interesting
m
mathematical properties. We show that in the semiclassical limit it is pos-
- sible to separate the relevant degrees of freedom by noticing that in the re-
d
n gions where the nonlinearities are e(cid:11)ective all states are suppressed but the
o
metastable ones (resonances). In this way the description of the nonlinear
c
regions is reduced to ordinary di(cid:11)erential equations weakly coupled to stan-
dard Schro(cid:127)dinger equations valid in the linear regions. The idea is illustrated
through the study of a prototype equation recently proposed for resonant
tunneling of electrons through a double barrier heterostructure and for which
nonlinear oscillations have been numerically predicted.
03.65.-w, 73.40.Gk
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1
I. INTRODUCTION
In a recent paper [2] a nonlinear Schro(cid:127)dinger equation was proposed to model the e(cid:11)ect
of the accumulation of electric charge in the resonant tunneling of electrons through a dou-
ble barrier heterostructure [3]. The main feature of this equation is that the nonlinearity is
concentrated only in the region where the resonant state is localised, that is within the two
barriers. The equation of Ref.[2] is a prototype of a class of nonlinear Schro(cid:127)dinger equations
which can be used to model several physical situations common in mesoscopic physics, e.g.
superlattices. In this paper we show that as far as the evolution of the solutions over a
(cid:12)nite time interval is concerned, it is possible to carry out an adiabatic theory which sim-
pli(cid:12)es considerably the problem. In fact in the semiclassical limit it is possible to separate
the relevant degrees of freedom by noticing that in the regions where the nonlinearities are
e(cid:11)ective all states are suppressed but the metastable ones (resonances). In this way the
problem can be reformulated in terms of ordinary di(cid:11)erential equations describing the evo-
lution in the nonlinear regions weakly coupled to standard Schro(cid:127)dinger equations describing
the propagation in the linear regions. We shall illustrate this idea, which we think is novel,
in the prototype case of resonant tunneling through a double barrier. Our approach, beside
providing an important mathematicaladvance, allows a better understanding of the physics
involved. In particular we explain why the oscillations discovered in [2] disappear in the
limitof verysmalland verylarge spacial width of the incomingcloud of electronswhile they
are maximal in intermediate situations. From the mathematical standpoint our discussion
will be informal and computer assisted. However we think that a substantial part of it can
be made completely rigorous. It is interesting to remark that the computer time needed
for the numerical solution of the problem is reduced by several order of magnitudes with
excellent agreement with the exact results.
Now we brie(cid:13)y recall the physical origin of our model problem. Interaction among the
electrons can play a crucial role in electrical transport properties of mesoscopic systems [4].
Asanexample,letusconsideracloudofelectronsmovinginadoublebarrierheterostructure
in which the well region con(cid:12)ned between two potential barriers acts like a capacitor whose
energy changes according to the electron charge trapped in it. We emphasize that the
localization of the interaction is justi(cid:12)ed due to the existence of a resonance state which
permits a long sojourn time inside the double barrier and therefore an accumulation of
charge.
The exact time evolution of the depicted system can be written in terms of the anticom-
muting (cid:12)eld operator associated to the electron cloud
" #
2 Z 2
@ h(cid:22) e
^ 2 ^y 0 ^ 0 0 ^
ih(cid:22) (cid:9)(~r;t) = (cid:0) r +V(x)+ (cid:9) (~r ;t)(cid:9)(~r ;t) d~r (cid:9)(~r;t) (1)
0
@t 2m "j~r(cid:0)~r j
(cid:16) (cid:17)
where the external potential V(x) = V0 1[a;b] +1[c;d] with a < b < c < d depends only on
the coordinate x orthogonal to the interfaces. As argued in [2] we can assume a decoupling
between the degrees of freedom parallel and orthogonal to the interfaces and set up an
approximate description in terms of a Hartree equation for a one particle wave function
which depends only on the orthogonal coordinate. More precisely we factorize the single
particle mean (cid:12)eld in the following way
2
i
(cid:9)(~r;t) (cid:17) h(cid:9)^(~r;t)i ' u(x;t)(cid:30)(y;z)e(cid:0)h(cid:22)Ekt : (2)
where h:::i means expectation in the many body state at time t; (cid:30)(y;z) is a solution of the
free particle Schro(cid:127)dinger equation in the plane parallel to the interfacesnormalized to unity.
We consider the interaction term e(cid:11)ective only inside the well [b;c] and we approximate it
by the expression
Z 2
e
0 2 0 0 2 0
0 ju(x;t)j j(cid:30)(y ;z )j d~r ' (cid:11)Q[u]1[b;c] : (3)
"j~r(cid:0)~r j
2
The constant (cid:11) is e =C where C summarizes the result of the integration on the plane yz
and an averaging of the potential over the width of the well. Dimensionally C is a length
and de(cid:12)nes the capacitance of the double barrier;
Z
c
2
Q[u] = ju(x;t)j dx (4)
b
is the adimensional charge trapped in the well [b;c] at timet. Within the above approxima-
tions we get the following one-dimensional nonlinear Schro(cid:127)dinger equation
" #
2 2
@ h(cid:22) @
ih(cid:22) u(x;t) = (cid:0) 2 +V(x)+(cid:11)Q[u]1[b;c] u(x;t) (5)
@t 2m@x
which describes the dynamics of our quantum capacitor. An initial condition for u(x;0) has
to be given. We suppose u(x;0) to be a Gaussian wave packet moving toward the double
barrier and initially centered far from the well [b;c] so that Q ' 0 at t = 0.
Equation (5) has two conserved quantities, namely the number of particles
Z
+1
N = u(x;t)u(x;t) dx (6)
(cid:0)1
and the energy
" #
Z +1 h(cid:22)2 @2 1
2
E = u(x;t) (cid:0) +V(x) u(x;t) dx+ (cid:11)Q[u] : (7)
2
(cid:0)1 2m@x 2
Some further comments are in order. The possibility of nonlinear e(cid:11)ects in resonant
tunneling was (cid:12)rst envisaged in a paper by Ricco and Azbel [5] on the basis of a simple
reaction-dissipation argument. The picture which emerges from [2] and the present study,
as well as from a recent study of Malomed and Azbel [6] based on a di(cid:11)erent approach, is
considerably more complex.
The time scale of these e(cid:11)ects is in principle within the reach of present experimental
techniques. An experimental veri(cid:12)cation would be of special interest because as shown
in [7] this type of concentrated nonlinearities can induce a genuine chaotic behavior in
heterostructures which does not have an obvious classical counterpart.
Equation (5) belongs to a class of equations which have other remarkable mathematical
properties [8] beside those discussed in the present paper.
3
II. THE ONE-MODE APPROXIMATION
In the following we will use the atomic unit system with h(cid:22) = 2m = 1. In that case
(cid:0)10
lengths are given in Bohr radii, 1 a0 = 0:529(cid:1)10 m,and energies in Rydbergs, 1 Ry= 13:6
(cid:0)17
eV. The unit of time is 4:83(cid:1)10 s.
We are interested in the initial value problem
(cid:16) (cid:17)
2
Dt +Dx +V(x)+(cid:11)Q[u] 1[b;c] u(x;t)= 0 (8a)
Z
c
2 2
Q[u] = kuk[b;c] = ju(x;t)j dx (8b)
b
u(x;0) = (cid:12)e(cid:0)21(x(cid:0)x0)2(cid:27)(cid:0)2+ik0x (8c)
(cid:16) (cid:17)
@ @
where Dt (cid:17) (cid:0)i@t, Dx (cid:17) (cid:0)i@x and with V(x) = V0 1[a;b] +1[c;d] We consider (cid:11) (cid:21) 0, (cid:12) > 0,
2
(cid:27) > 0, k0 > 0, a(cid:0)x0 (cid:29) (cid:27) and k0 close to the realpart of a resonance (cid:21)(0) = ER(0)(cid:0)i(cid:0)(0)=2
2
with 0 < ER(0) < V0 and (cid:0)(0) > 0, of the stationary Schro(cid:127)dinger operator Dx+V(x) on R.
In this section we shall describe a one-mode approximation to problem (8) which hope-
fully can be rigorously justi(cid:12)ed in the limit (cid:27), d (cid:0) c ! 1 when V0 and c (cid:0) b are (cid:12)xed
and (cid:11), (cid:12) are not too large, and provided that (cid:21)(0) i(cid:16)s a resonance s(cid:17)o that ER(0) tends to
some eigenvalue smaller than V0 of the potential V0 1](cid:0)1;b] +1[c;1[ and (cid:0)(0) ! 0 when
d (cid:0) c ! 1. For simplicity we shall also assume that ER(0) is the only eigevalue of this
potential at energy smaller than V0.
We observe that the problem (8) is formally a Schro(cid:127)dinger equation with a time de-
pendent potential. The standard way to study this situation is to work with the in-
stantaneous eigenstates and eigenvalues for the associated time dependent Hamiltonian
2
H = Dx + V(x) + (cid:11)Q1[b;c] [9]. One says that the k-th instantaneous eigenstate evolves
adiabatically when transitions from it to any other m-th eigenstate cannot occur. This
happens when [9]
(cid:12) (cid:12)
(cid:12) _ (cid:12)
(cid:12)Hkm(cid:12)
(cid:28) 1 (9)
2
(Ek (cid:0)Em)
_
where Hkm is the matrix elementof the operator @tH between the k-th and the m-th eigen-
states which have eigenvalues Ek and Em. In the problem (8) the instantaneous spectrum
consists of a resonant state below the potential barrier V0 and a continuum band above V0.
We are interested to make a one-mode adiabatic approximation by keepeng only the in-
stantaneous resonant state in the description of the wave function inside the double barrier.
Since the distance between the resonant state and the rest of the spectrum is of the order
of V0, we get the adiabatic condition
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:11)(cid:12)Q_(cid:12)
(cid:28) 1 : (10)
2
V0
We shall verifya posteriori that this condition is satis(cid:12)ed in the range of parameters we use.
4
e
We now derive the equation for the one-mode approximation. Let V(x) = V0 1[a;d] be
the potential obtained from V(x) by (cid:12)lling the potential well [b;c]. We start by solving the
initial value problem
(cid:16) (cid:17)
2 e
Dt +Dx +V(x) (cid:22)e(x;t) = 0 (11a)
(cid:22)e(x;0) = u(x;0) (11b)
and look for a solution of Eq. (8) of the form u = (cid:22)e +v. Then Eq. (8) becomes
(cid:16) (cid:17) (cid:16) (cid:17)
Dt +Dx2 +V(x)+(cid:11)Q[(cid:22)e +v] 1[b;c] v(x;t)= Ve(x)(cid:0)V(x)(cid:0)(cid:11)Q[(cid:22)e +v] 1[b;c] (cid:22)e(x;t)
(12a)
Z
c
2
Q[(cid:22)e +v]= j(cid:22)e(x;t)+v(x;t)j dx (12b)
b
v(x;0) = 0 : (12c)
2
If 0 (cid:20) s (cid:20) V0, we let (cid:21)(s) = ER(s) (cid:0) i(cid:0)(s)=2 be the resonance of Dx + Vs(x) where
Vs(x) = V(x)+s 1[b;c]. Let e(s;x) be the corresponding resonance state
(cid:16) (cid:17)
2
Dx +Vs(x)(cid:0)(cid:21)(s) e(s;x) = 0 : (13)
(cid:16) q (cid:17)
We recall that e(s;x) is a function on R such that e(s;x) / exp (cid:6)i (cid:21)(s)x , for x ! (cid:6)1.
We also observe that, as a special case of the method of complex scaling [10], e(s;x) is of
S S
2 (cid:0)i(cid:18) i(cid:18)
class L on the contour (cid:13) = e ](cid:0)1;a] [a;d] e [d;+1[ if (cid:18) > 0 is conveniently
chosen. We normalize the resonance state to unity
Z
2
e(s;x) dx = 1 : (14)
(cid:13)
In the spirit of the one-mode approximation, we assume v(x;t) = ze(t)e(s;x) with
s = (cid:11)Q[(cid:22)e +zee(s)] : (15)
The problem (12) then becomes
(cid:16) (cid:17)
2
Dt +Dx +V(x)+(cid:11)Q[(cid:22)e +zee] 1[b;c] ze(t)e(s;x)= (V0 (cid:0)(cid:11)Q[(cid:22)e +zee])1[b;c] (cid:22)e(x;t) (16)
with initial condition ze(0) = 0. If Eq. (15) de(cid:12)nes a unique s = s[(cid:22)e;ze], Eq. (16) reduces to
(Dt +(cid:21)(s))ze(t)e(s;x) = (V0 (cid:0)s) 1[b;c] (cid:22)e(x;t) : (17)
We multiply the above equation by e(s;x) and integrate with respect to x on the contour
R
(cid:13). We also use the normalization condition for e implying (cid:13) Dt[e(s;x)]e(s;x) dx = 0 and
we obtain
5
Z
c
(Dt +(cid:21)(s))ze(t) = (V0 (cid:0)s) (cid:22)e(x;t)e(s;x) dx : (18)
b
Introducing (cid:22)(x;t) = eiE0t(cid:22)e(x;t) and z(t) = eiE0tze(t), where E0 = k02, we can rewrite Eq.
(15) and Eq. (18) as
Z
dz(t) c
= (cid:0)i((cid:21)(s)(cid:0)E0)z(t)+i(V0 (cid:0)s) (cid:22)(x;t)e(s;x) dx (19a)
dt b
s = (cid:11)Q[(cid:22)+ze] (19b)
z(0) = 0 : (19c)
The above three equations represent our one-mode approximation. Since s depends on z(t)
when (cid:11) 6= 0, Eq. (19) is clearly a nonlinear ordinary di(cid:11)erential equation with a driving
term.
A. The driving term
In principlea completeintegral representation of the driving term(cid:22)e(x;t) or (cid:22)(x;t) could
be given, but we prefer to use various approximations which seem justi(cid:12)ed in the limit (cid:27),
d(cid:0)a ! 1. (cid:16) (cid:17)
Westart bylooking fora solution ofthe stationary problem Dx2 +Ve(x)(cid:0)E ee(k;x) = 0
of the form
8
ikx (cid:0)ikx
><e +r(k)e x < a
1
+ (cid:20)x (cid:0) (cid:0)(cid:20)x
ee(k;x) = p g (k)e +g (k)e a < x < d (20)
>
2(cid:25) : ikx
t(k)e d < x
p p
2
where k = E and (cid:20) = V0 (cid:0)k . The normalization constant is chosen in such a way that
R
0 0 (cid:6)
ee(k;x)ee(k ;x) dx = (cid:14)(k(cid:0)k ). The functions r(k), t(k) and g (k) can be evaluated exactly
1
by requiring eeto be of class C . In the limitof (cid:20)(d(cid:0)a) (cid:29) 1 we get the simplesemiclassical
result
(cid:16) (cid:17)
ik +(cid:20)
2ika (cid:0)(cid:20)(d(cid:0)a)
r(k) = e +O e (21)
ik (cid:0)(cid:20)
(cid:16) (cid:17)
2ik
(cid:0) (ik+(cid:20))a (cid:0)(cid:20)(d(cid:0)a)
g (k) = e +O e (22)
ik(cid:0)(cid:20)
(cid:16) (cid:17)
+ (cid:0)(cid:20)(d(cid:0)a)
t(k); g (k) = O e : (23)
Now we have the solution of Eq. (11) of the form
Z
1
(cid:0)ik2t
(cid:22)e(x;t) = f(k)e ee(k;x) dk (24)
0
6
where, in the limit k0(cid:27) (cid:29) 1, f(k) is the Fourier transform of (cid:22)e(x;0)
f(k) = (cid:12)(cid:27)e(cid:0)21(k0(cid:0)k)2(cid:27)2+i(k0(cid:0)k)x0 : (25)
The integral in Eq. (24) can be evaluated analytically by extending the lower limit to (cid:0)1
and approximating the smooth functions of k with their value at k = k0. The integration of
the remaining quadratic exponential gives for a < x < d
( )
(cid:0)(cid:20)0(x(cid:0)a) (cid:0)1 2
2ik0e (cid:12) [k0(cid:27)+i(a(cid:0)x0)(cid:27) ] 1 2 2
(cid:22)e(x;t) = ik0(cid:0)(cid:20)0 p1+2it(cid:27)(cid:0)2 exp 2(1+2it(cid:27)(cid:0)2) (cid:0) 2k0(cid:27) +ik0x0 (26)
q
2
where (cid:20)0 = V0 (cid:0)k0. This expression further simpli(cid:12)es for small times or large (cid:27). By
(cid:0)2
expanding with respect to t(cid:27) and using k0(cid:27) (cid:29) 1 we get to the leading order
(cid:22)(x;t) = 2i(cid:12)k0e(cid:0)(cid:20)0(x(cid:0)a)+ik0a e(cid:0)12((cid:0)20)2(t(cid:0)t0)2 (27)
ik0 (cid:0)(cid:20)0
where t0 = (a(cid:0)x0)=(2k0) is the timpe a classical particle with kinetic energy E0 spends to
(cid:0)1=2 2
cover the distance a(cid:0)x0 and (cid:0)0 = 2 2k0=(cid:27) is the full width at height e of jf(k)j , i.e.
the energy spread of the initial wave function u(x;0).
B. Solution of the one-mode approximation in the linear case
We limit ourselves to a qualitative discussion of the limiting behaviors for (cid:0)0 (cid:28) (cid:0)(0)
and (cid:0)0 (cid:29) (cid:0)(0). Since (cid:11) = 0, Eq. (19) becomes
Z
dz(t) c
= (cid:0)i((cid:21)(0)(cid:0)E0)z(t)+iV0 (cid:22)(x;t)e(0;x) dx (28)
dt b
with initial condition z(0) = 0. In the resonant case E0 = ER(0) we get
Z
dz(t) (cid:0)(0) c
+ z(t) = iV0 (cid:22)(x;t)e(0;x) dx : (29)
dt 2 b
The driving term in the right hand side has been evaluated in the(cid:16)previous section. It(cid:17)has
1 2 2
a roughly costant argument and a modulus which behaves like exp (cid:0)2 ((cid:0)0=2) (t(cid:0)t0) at
least for (cid:27) large. In this case if (cid:0)0 (cid:28) (cid:0)(0) the Gaussian right hand side of Eq. (29) varies
very little during the characteristic time period 2=(cid:0)(0) of the left hand side and we get the
approximate solution
Z
2iV0 c
z(t) ' (cid:22)(x;t)e(0;x) dx (30)
(cid:0)(0) b
which has the same shape as the driving term, i.e. a Gaussian shape. On the other hand if
(cid:0)0 (cid:29) (cid:0)(0) we get the approximate solution
Z Z
(cid:0)(0) t c
(cid:0) t 0 0
z(t) ' iV0e 2 (cid:22)(x;t)e(0;x) dx dt (31)
0 b
7
characterized by a linear exponential decay slow with respect to the rise time.
The above results can be generalized to the non resonant case. We get
Z
V0 c
z(t) ' (cid:22)(x;t)e(0;x) dx (32)
ER(0)(cid:0)E(0)(cid:0)i(cid:0)(0)=2 b
when (cid:0)0 (cid:28) (cid:0)(0) and
Z Z
(cid:0)(0) t c
z(t) ' iV0e(cid:0)i(ER(0)(cid:0)E(0)(cid:0)i 2 )t (cid:22)(x;t0)e(0;x) dx dt0 (33)
0 b
when (cid:0)0 (cid:29) (cid:0)(0).
It is worth noting the conceptual simplicity of the one-mode approach in this case if
compared with previous treatments [11].
C. Numerical simulations
We now introduce the numericalsimulationsof the one-mode approximationobtained by
a standard Runge-Kutta integration of Eq. (19). For comparison we give also the numerical
simulations of the Eq. (8) obtained with the algorithm described in Appendix B.
We consider an external potential V(x) with c (cid:0)b = 15, d(cid:0)c = b(cid:0)a = 20 and V0 =
(cid:0)3
0:3=13:6. In this case we have a resonace (cid:21)(0) with ER(0) ' 0:5 V0 and (cid:0)0 ' 5(cid:1)10 =13:6.
2
The initial wave function is chosen to be in resonance with k0 = ER(0). We consider
three di(cid:11)erent values for its width: (cid:27) = 5775, (cid:27) = 825 and (cid:27) = 110 corresponding to
p
(cid:0)1=2
(cid:0)0 ' 0:14 (cid:0)(0), (cid:0)0 ' (cid:0)(0) and (cid:0)0 ' 7:3 (cid:0)(0), respectively. A constant (cid:12) = (825 (cid:25))
is used is every case. Finally we put a (cid:0) x0 = 5(cid:27) so that no appreciable charge Q is
present in the well at t = 0. Notice that the semiclassical condition and the condition
k0(cid:27) (cid:29) 1, employed in evaluating the approximate expression (26) for the driving term, are
well satis(cid:12)ed with the above parameters.
Fig. 1, 2 and 3 show the behavior of the charge Q(t) for the three given values of (cid:27) and
for di(cid:11)erent values of the parameter (cid:11). Fig. 4, 5 and 6 show the trajectories of z(t) in the
complex plane for the same cases. We notice that the one-mode approximation reproduces
exceptionally well the exact result. A small di(cid:11)erence appears and grows up when (cid:27) is
decreased and (cid:11) is increased in agreementwith the adiabatic condition (10). This point will
be discussed in detail in Section IV.
The limiting behaviors for (cid:0)0 (cid:28) (cid:0)(0) and (cid:0)0 (cid:29) (cid:0)(0) in the linear case ((cid:11) = 0) as
expressed by Eq. (30) and Eq. (31) are apparent in Fig. 1 and Fig. 3. To understand qual-
itatively the nonlinear behavior we introduce a simpli(cid:12)ed one-mode approximation which
keeps the basic features of Eq. (19) and is analytically more manageable.
III. THE SIMPLIFIED ONE-MODE APPROXIMATION
If we assume that Eq. (19b) has a unique solution s = s[(cid:22) + ze], then the one-mode
approximation (o.m.a.) de(cid:12)ned by Eq. (19) can be written
z_ = V(t;z(t)) (34)
8
where
Z
c
V(t;z) = (cid:0)i((cid:21)(s[(cid:22)+ze])(cid:0)E0)z +i(V0 (cid:0)s[(cid:22)+ze]) (cid:22)(x;t)e(s[(cid:22)+ze];x) dx : (35)
b
It turns out that the qualitativenumericalaspects of the vector (cid:12)eld V are already captured
by a simpli(cid:12)edone-mode approximation (s.o.m.a.). The s.o.m.a. is obtained from the o.m.a.
i) replacing e(s;x) everywhere by e(0;x), ii) replacing V0(cid:0)s by V0, iii) replacing Q[(cid:22)+ze]
0
by Q[ze] and iv) replacing (cid:21)(s)(cid:0)E0 by ER(0)s(cid:0)i(cid:0)(0)=2, where the derivative
hq i
(cid:12)
0 dER(s)(cid:12)(cid:12) ER(0)cos2 ER(0)(c(cid:0)b)=2
E (0) = (cid:12) = 1(cid:0) hq i q (36)
R ds (cid:12) 2
s=0 V0cos ER(0)(c(cid:0)b)=2 +ER(0) V0 (cid:0)ER(0)(c(cid:0)b)=2
is evaluated from Eq. (A4). The condition in Eq. (19b) now simpli(cid:12)es to
Z
c
2 2
s = (cid:11)qjzj q = je(0;x)j dx (37)
b
and the vector (cid:12)eld becomes
!
Z
(cid:0)(0) c
0 2
V(t;z)= (cid:0) +iER(0)(cid:11)qjzj z +iV0 (cid:22)(x;t)e(0;x) dx : (38)
2 b
According to Eq. (27) during the interesting part of the evolution the driving term is well
approximated by
iV0Z c(cid:22)(x;t)e(0;x) dx = F e(cid:0)21((cid:0)20)2(t(cid:0)t0)2 (39)
b
where
2(cid:12)k0V0eik0a Z c (cid:0)(cid:20)0(x(cid:0)a)
F = e e(0;x) dx : (40)
(cid:20)0 (cid:0)ik0 b
A. solution of the simpli(cid:12)ed one-mode approximation
We consider (cid:12)rst two limiting behaviors in analogy to the discussion of the linear case
in Section II B.
0 2
When (cid:0)0 (cid:28) (cid:0)(0)+ER(0)(cid:11)qjzj it is z_(t) ' 0 and z(t) is close to the instantaneous (cid:12)xed
point zc(t) de(cid:12)ned by V(t;zc(t)) = 0. If zc(t) is such a (cid:12)xed point we have
(cid:0)(0)2 ! (cid:12)(cid:12)Z c (cid:12)(cid:12)2
+ER0 (0)2(cid:11)2q2jzcj4 jzcj2 = V02(cid:12)(cid:12) (cid:22)(x;t)e(0;x) dx(cid:12)(cid:12) (41)
4 b
and conversely if jzcj solves Eq. (41) then we get the unique (cid:12)xed point of modulus jzcj
R
c
iV0 b (cid:22)(x;t)e(0;x) dx
zc = 0 2 : (42)
(cid:0)(0)=2 +iER(0)(cid:11)qjzcj
9
Since the left hand side of Eq. (41) is a strictly increasing function of jzcj it is clear that V
has a unique (cid:12)xed point given by Eq. (41) and Eq. (42). In this situation we do not have
oscillations in jzc(t)j and therefore in Q(t) as shown in Fig. 1.
Independently on the validityof the previous inequality,we distinguish two zones for the
0 2
(cid:12)xed point behavior. a) the linear zone : ER(0)(cid:11)qjzcj (cid:28) (cid:0)(0)=2. In this zone we have
R
c
2iV0 b (cid:22)(x;t)e(0;x) dx
zc(t) ' : (43)
(cid:0)(0)
Using this expression the linear zone is de(cid:12)ned by the condition
8ER0 (0)(cid:11)qjFj2e(cid:0)((cid:0)20)2(t(cid:0)t0)2 (cid:28) (cid:0)(0)3 : (44)
Due to the time dependence of the driving term we are sure to be in the linear zone at the
0 2
beginning and at the end of the process. b) the nonlinear zone: ER(0)(cid:11)qjzcj (cid:29) (cid:0)(0). In
this zone from Eq. (41) we have
(cid:12)(cid:12) Rc (cid:12)(cid:12)1=3
(cid:12)V0 b (cid:22)(x;t)e(0;x) dx(cid:12)
jzc(t)j ' (cid:12)(cid:12) ER0 (0)(cid:11)q (cid:12)(cid:12) (45)
and from Eq. (42)
Z
c
argzc(t) ' arg (cid:22)(x;t)e(0;x) dx : (46)
b
Therefore the nonlinear zone is de(cid:12)ned by
(ER0 (0)(cid:11)q)1=3jFj2=3e(cid:0)31((cid:0)20)2(t(cid:0)t0)2 (cid:29) (cid:0)(0) : (47)
Notice that the argument of zc changes by (cid:25)=2 when we go from the linear to the nonlinear
zone. When the driving term reaches its maximum, zc(t) becomes approximatively station-
ary and then returns to the origin following a curve close to the outgoing trajectory. This
kind of behavior is well illustrated in Fig. 4 and 5.
0 2
When (cid:0)0 (cid:29) (cid:0)(0)+ER(0)(cid:11)qjzj we have the other limitingbehavior characterized by the
approximate equation
z(t) ' iV0e(cid:0)(cid:0)(20)t(cid:0)iER0 (0)(cid:11)qR0tjz(t0)j2 dt0 Z tZ c(cid:22)(x;t0)e(0;x) dx dt0 (48)
0 b
and therefore
jz(t)j2 = jFj2e(cid:0)(cid:0)(0)t(cid:18)Z te(cid:0)12((cid:0)20)2(t0(cid:0)t0)2 dt0(cid:19)2 (49)
0
which is as in the linear case without oscillations in Q(t). The corresponding condition can
be written more explicitely as
(cid:0)0 (cid:29) (cid:0)(0)+ER0 (0)(cid:11)qjFj2e(cid:0)(cid:0)(0)t(cid:18)Z te(cid:0)21((cid:0)20)2(t0(cid:0)t0)2 dt0(cid:19)2 : (50)
0
10