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The influence of noise on critical collapse in the nonlinear Schrödinger equation
a __
14 August 1995
__ i!B
PHYSICS
Physics Letters A 204 (1995)
ELSEWER
LETTERS
A
121-127
The influence of noise on critical collapse in the nonlinear Schriidinger equation K.@. Rasmussen ‘, Yu.B. Gaididei b, 0. Bang’, P.L. Christiansen” ’ Institute of Mathematical Modelling, The Technicul University of Denmark, DK-2800 Lyngby, Denmark h Institute for Theoretical Physics, Metrologicheskaya Street 14 B, 252 143 Kiev 143. Ukraine ’ Lubomtoire de Physique. Ecole Normule Supirieure de Lyon, 46 All& d’halie, 69364 Lyon Cedex 07. France
for publication 15June Bishop
Received 25 August 1994; accepted
1995
Communicated by A.R.
Abstract Multiplicative Gaussian white noise is included in the nonlinear Schrodinger equation of one and two dimensions in the critical case, in which collapse occurs. The collective coordinate method is used to derive analytical results concerning the effects of the noise on the collapse process. This approach makes it possible to distinguish between the one- and two-dimensional case. The analytical results are in good agreement with numerical results obtained directly from the partial differential equation.
1. Introduction The inlluence of noise on the soliton solution to the nonlinear Schriidinger equation (NLS) [ l] has attracted some attention in the literature. Examples are applications in the field of optical fiber communication [2] and in biomolecular dynamics of Davydov solitons under the influence of thermal fluctuations (see, e.g., Ref. [ 31). Recently, the two-dimensional NLS equation, including thermal fluctuations, has been suggested as a model for energy transport and localization in organic thin films [4,5]. In this Letter we study the influence of noise on collapse [ 61. We study the critical case ad = 2, in one and two dimensions (d = 1,2), (T being the degree of nonlinearity. The two-dimensional collapse is important in the study of nonlinear optics, where the two-dimensional NLS describes propagation of electromagnetic beams in media whose refraction index 037%9601195/.W9.50
increases with the field amplitude [ 7,8]. Letting v = 2, in the one-dimensional case we obtain critical collapse also in this system. The point of this Letter is to investigate the differences between the one- and two-dimensional critical cases. In Section 2 we present an analytical approach based on collective coordinates [ 5,9]. In Section 3 we compare our analytical results to results obtained directly by numerical simulation of the partial differential equation.
2. Collective coordinate approach Introducing multiplicative tion we obtain
ircIl+ V2* + IIcI12”q +7(x, t)$ = 0,
@ 1995 El sevier Science B.V. All rights reserved
ssD10375-9601(95)00490-4
noise in the NLS equa-
(1)
122
K.0. Rasmussen et al./Physics
where 7(.x, t) is assumed to be Gaussian with zero mean
white noise
Letters A 204 (1995) 121-127
and the coefficient SmJI =
(rl(.& r)> = 0, (~(X,t)7j+‘,t’))
=DS(x-x’)C?(r-t’).
(2)
Here and in the following ( ) denotes the ensemble average, and D is the variance of the noise. For higher dimensions x = (XI, x2,. . . , xd). In the noise free case of D = 0 the number N and the Hamiltonian H, A’{$} = /
H{@) =
]+I2 ddx,
(3)
2a+2 - v(r, t) [#I2 rd-’ dr, l -: (TI@l >
where r = dm.
(5)
Using the trial function,
= A(t)sech[r/B(t)]
exp[ia(t)r2
+ i/?(t)], (6)
where A, B, a, and B are real functions, we obtain for the critical case, cd = 2, the following equation for the width B(t), [5] 4
B3B=A+-
J[
1 - (r/B)
sd+l,2 x
sech2(r/B)v(r,
t)rd-’
where the parameter d&Csd+1.2 -
(9)
As a consequence of the self-similar trial function, Eq. (6), A is a constant in time. (As pointed out by several authors [ 10-121, Eq. (7) can be made an exact description of the collapse by introduction of a subtle time dependence in A.) For A < 0 FCq. (7) clearly describes a collapse process. As argued in Ref. [S], Eq. (7), and the simpler equation t
(4)
Im(+*9t) +Mr12 J( --
+(r,r>
x”’ sech”( x) dx. s
ddx, lW12 -$---[+12n+2) SC
are conserved. Under the influence of the multiplicative noise N is still conserved, while H is not. We shall treat the problem of collapse influenced by noise analytically, applying the method of collective coordinates [ 91. The equation of motion, Eq. ( I), can under the assumption of isotropy, be derived from the following Lagrangian, L=
s,,,,, is given by
sd-I,2 sd-1,2u+2
1 + a)sd”_t , 2
NC
, 1
= 0,
(h(t)h(t’))
= 2y8(t
- t’),
(II)
with the variance 2y given by Y = gDl&,
(12)
,2.
Thus we replace Eq. (7) by Eq. ( 10) in the following. The power p in Eq. (10) equals 2 and 2 in one and two dimensions, respectively. In order to remove the singularities of Eq. ( IO), it is convenient to introduce the transformation B = Y1iz yielding 2YY - Y2 = 4( A + hY’3-P’/2).
(13)
Examining this equation analytically, we first note that the exact solution of Eq. (13) for h G 0, becomes Y(t) = +t:
- ?),
c
tc=fi,
t 14)
(8)
I’(t)
sd-1,4
C
(27r)d-‘(
(h(t))
(7)
A is defined as -
have the same Fokker-Planck equation for the appropriate probability density, provided the noise h(t) is delta correlated,
initial conditions being Y( t = 0) = Yo, and I’( t = 0) = 0. A has been assumed negative, since only such values are relevant in the collapse case. Assuming that the form of the solution will be approximately the same for nonzero noise we see that close to collapse, t N t,, the first term of Eq. ( 13) will be close to zero since Y is finite and the second term will be finite. Therefore it is permitted as a first step in analyzing Eq. (13) to neglect the first term of Eq. ( 13). This leads to
tanh(r/B)]
dr,
10)
1 _
&J-P+, 214
K.0. Rasmussen et al. /Physics
123
Letters A 204 (1995) 121-127
Here it is assumed that Y(3-p)/2h << IAl, which isreasonable at times close to the collapse time. Introducing
layed by the noise, such that the new collapse tc (D) , becomes
Y = y + 6.
fc(D)=
r = (Y)
(16)
t,(D 1 _
=O)
time,
(24)
(j~/~g,,(A(‘/~’
we get to leading orders in 6 ?=
-2m+
1
-JT;?i2
1 -(3-p’/2
s=my
3 - P @..)F’l -d/f
(17)
(18)
h(t).
Eq. ( 18) has the solution
dtl y(3-P”2(tl)h(t,).
(19)
Using Eq. ( 11) we are now able to calculate the correlation function (h( t)s( t)),
This expression also allows us to give the variance, which should be sufficient to stop the collapse process (tc = cQ), (25)
D = ;1A13’2s;.2.
As can be seen from Fig. 2 the analytical results derived above are not in too good agreement with the numerically obtained results. Therefore we will give a better approach in the following. The problem with this approach is that it is only usable in the twodimensional case. Introducing B=B+S,
Br
(B)
(26)
into Eq. ( 10) will give in leading order of S (h(t)S(t))
= -$+?)“(t),
(20)
which inserted into Eq. (18) gives ?=-2m+-
3 -
PF2’-”
2141
.
(21)
From Eq. (2 1) it is evident that collapse, y = 0, can only occur when 4 < 2~. The result of this approach is thus that total collapse cannot occur in one dimension (p = i ) when white noise is present. However, in one dimension
the stationary
width, B, (? = 0), is given
by
(22) Thus a certain degree of localization may still occur for large initial amplitudes ( IAJ > 0)) and/or weak noise, where B, becomes small. The process can therefore resemble a collapse process initially. If we turn to the two-dimensional case, p = f, collapse may still occur in the usual form. The appropriate solution of Eq. (21) is in this case t;(t)
= 4(A([t, - (1 - 3y/4\A(3’2)tJ,
z3g
_-_ + 3B B(S2) + 3B2(&)
..-3 6B
+
3i?*%
= -\A\ + (ha),
(27) (28)
= Bh.
From Eq. (28) we get = (ha) - 3(6’)% 3,
82(68)
(29)
and Eq. (27) takes the form B38 - 6B $8’)
= -[Al - 2(hc?).
From Eq. (14) and Fig. 2 it is reasonable that B(t)
= Bo( 1 - r2/tz)1/2,
(30) to assume
(31)
where t, is the unknown collapse time we are looking for. After some algebra, which is given in full length in Appendix A, we can derive the following (32)
(ha) = -&> 0
(S2)N
$$i2( t).
(33)
0
(23) Introducing
which satisfies the same initial conditions as approximation ( 15). We see that the collapse process is de-
Eqs. (32),
K*( 1 - &13)
(33) into Eq. (30) given us
= IAl - ~0,
(34)
K.O. Rasmussen et al. /Physics Letters A 204 (I 99s) 121-127
124
1.25
0.25
0.00
0.50 n
Fig. I. Width B versus time in two dimensions. The solid line indicates the analytical solution of EQ. (10) for h z 0. Numerical solutions are for D = 0.05 (dashed line), D = 0.2 (dotted line) and D = 0.4 (chain-dashed line).
where 0 = r,/B$ For small y an approximate is obviously given as
solution
Fig. 2. Solid line: collapse time r, versus noise variance D according to Eq. (24); chain-dashed line: predictions of E!q. (35). Squares indicate values obtained from numerical simulation of Fq, (13).
least the form of the results is correctly predicted by both methods and therefore it can be expected that the form of the one-dimensional results also is correct.
(35) It is also easy to calculate the D where the relevant solution to Eq. (34) becomes complex such that the collapse ceases to exist, D
=
24,214/3'2
15&
.
(36)
We have solved the two-dimensional version of Eq. ( 13) numerically using an ordinary fifth order RungeKutta scheme. The results are shown in Fig. 1, where the solid curve is the noise free solution of Eq. (10) (see Eq. ( 14) ), and the numerical results are for D = 0.05 (dashed line), D = 0.2, (dotted line), and D = 0.4 (chain-dashed line), respectively. It is seen that the effect of the noise is, as predicted, to slow down the collapse process, and it shows that an assumption like Eq. (3 1) is reasonable. In addition the solid line in Fig. 2 shows the collapse time according to Eq. (24)) and the dashed line shows the result of Eq. (35)) while the squares indicate numerically obtained values. We see that none of the analytical approaches agrees perfectly with the numerical results. It seems that the last approach agrees quite well for small noise variances, which is in agreement with the approximations of the approach. Furthermore it should be noted that the analytical approaches only differ by a numerical prefactor and neither of the prefactors seem to be completely right. Thus we may expect that at
3. Numerical
results
We now investigate numerically the influence of noise on critical collapse (ad = 2) in both one and two dimensions. The results were obtained using a second order split-step Fourier scheme [ 141. In one dimension the noise was generated in Fourier space using a method given in Ref. [ 151. The large number of grid points in two dimensions demands that the noise must be generated in each step of the calculation. This is accomplished using the method given in Ref. [ 161. The one-dimensional results were obtained using the following perturbed ground state solution [ 61 as initial condition, $(x,t
= 0) =&,(x)
= Ao x 3”4sech1i2(2n),
(37)
where A0 = 1 is the ground state solution to the NLS equation without noise (H{@a} = 0). The same idea is used to obtain an initial condition in two dimensions, but in this case the exact expression for the ground state solution is not known. However, it is possible to obtain an approximate expression for the ground state solution [ 171, given as Jl(r,t r=
=0) II
X:+X;,
=@a(r)
=AaAsech(r/B), (38)
K.0.
Rasmussen et al./Physics
125
Letters A 204 (1995) 121-127
where
450 - (a) 0
,’ ,’
300 ,’
For both initial conditions it is true that collapse will occur when A0 > 1. Investigating the interplay between the coherent collapse process and the incoherent noise process numerically, we need a way to determine whether the excitation is losing or gaining coherence (collapse). To determine this we use a quantity used for a similar purpose by Kadantsev et al. [ 181
l@12rr+2 ddX.
R= .i
,’ /’ ,’ ,’ ,’
150-
i
”
(40)
Collapse will reflect itself as an increase in R, while the opposite will be true if the excitation is scattered. The purpose of the numerical simulations is then to investigate whether it is true, as the analytical work predicts, that noise can prevent a collapse process, and if this is the case, to determine how strong the noise must be. In Fig. 3a we show the result of the one-dimensional simulations with a solid line. Above the line the noise is strong enough to destroy the collapse. This result has been obtained using the quantity R (Eq. (40)) such that above (below) the solid line R is a decreasing (increasing) function of time. The predictions of Eq. (24) are given by the dashed line, which has been obtained as a result of the following interpretation: When B, > Bo we interpret this as a scattering of the initial excitation, while B, < Bo is interpreted as a localization (collapse) process. This means that the curve B,( Ao) = Bo is the curve separating the domain of collapse from that of scattering, and thus the analytical equivalent to the numerically obtained results. Excellent agreement is observed for low initial amplitudes. The increasing discrepancy as the initial amplitude increases may be ascribed to the self-similar constraint of the analytical approach. Since the noise strength needed is becoming large when the initial amplitude is increased, the excitation will be more distorted when the initial amplitude increases. This is clearly a violation of the self-similarity assumption of the analytical approach. We further note that the separating curve is of the form D cx A: according to Eq. (22) and Eq. (8), which is a very strong dependence on the initial amplitude Ao,
,’
0
/;
1 04
/’ ,’ ,’
11-. ,’
,’
10
,’
,’
,’
,’
,’
5
,,
‘//
/,_,>’
__-’
Fig. 3. (a) Critical variance versus initial amplitude A,) in the one-dimensional case. Solid line: results obtained by solving Eq. ( 1) numerically; dashed line: predictions of Eq. (24). (b) Critical variance versus initial amplitude A() in the two-dimensional case. Solid line: results obtained by solving Eq. ( 1) numerically; dashed line: predictions of Eq. (25); chain-dashed line: results of FQ. (36).
Similarly, we have in Fig. 3b depicted the simulations in two dimensions as a solid line together with the analytical predictions of Eq. (25) as a dashed line and of Eq. (36) by a chain-dashed line. It is seen that the analytical predictions of Eq. (2.5) agree reasonably well with the numerically obtained results, while the predictions of Eq. (36) are not too good. It should be kept in mind that the method leading to Eq. (36) is the more precise of the two with respect to Eq. ( IO). It clearly gives the best results. This indicates that the discrepancy between the analytical and the numerical results stems mainly from the method applied to reduce the problem given by the partial differential equation ( 1) to the ordinary differential equation ( IO). Some further approximations have also been imposed in the two-dimensional case since we had to assume the noise to be radially symmetrical in the analytical approach, which has not been necessary in the numerical simulations. In spite of the discrepancies we
K.0. Rasmussen et al./Physics
126
may still trust the analytical approach to have given the correct form of the separating curve, which in this two-dimensional case is D DCAi. Thus a much weaker dependence than was seen in one dimension.
Letters A 204 (1995) 121-127
with Wronskian (44) Thus from Eq. (42) we get T
4. Conclusion It has been shown both analytically and numerically that the collapse process may be prevented by introducing a multiplicative noise term in the equation of motion. Some differences between the one-and twodimensional cases were found. It seems that the onedimensional collapse process is significantly stronger than the equivalent one in two dimensions since the strength of the noise must be much larger in one dimension than in two dimensions to have any effect.
- w(T)w(T~)l.
x [wI(TI)w(T)
(45)
It is seen from Eq. (45) that w can diverge at T --f 1. To avoid this we have to choose the constant c2 as follows, I
t2
c2=--c
%
J
h(Tltc)
dT,
(46)
J1’--TTiWI(G),
0
and obtain from Eqs (46) Acknowledgement
T
We wish to thank A.C. Scott. F. If and J.J. Rasmussen for many helpful discussions and valuable suggestions. Yu.B. Gaididei acknowledges The Technical University of Denmark for a Guest Professorship, and 0. Bang the CEC for financial support under contract No. SCl-CT91-0705. Finally we acknowledge financial support from The Danish Research Council through contract No. 52 1055 I - 1.
t2
w=--c
4
(s ‘Jiq
h(Tltc)
dT
I
+
s
h(Tltc)
dT
T
where we put ct = 0. Using Eq. (47) we are now able to calculate (ha) and (S2),
(
0
Using the assumption
(3 1) in Eq. (30) we get (41)
(S2) = &,
1
IT,, 3
I
( ll+;)3
[,
0
+;(I
-T*)log (
where T = t/tc is the normalized time. If we put S = w&?? we obtain from Eq. (41)
(~-T2)?!&2+?=
-T l+T
Near the collapse (T 4 and (33).
1 + T2) + 2T2
>I
(49)
1) these simplify to Eqs. (32)
ht,2 1 - T2
Bn’JC-F’ (42)
The homogeneous part of Eq. (42) is the Legendre equation. It has the two linearly independent solutions 11-T WI = 21+T’
(47)
I $=iy
(hi3= -2~;
Appendix A
w~(TI)wI(T)
0
2T w2 = , _T23
(43)
Ill V.E. Zakharov
and A.B. Shabat, Zh. Eksp. Tear. Fiz. 61 (1971) 118 [Sov. Phys. JETP 34 (1972) 621. 121J.N. Elgin, Phys. Lett. A 110 (1985) 441; 181 (1993) 54; Opt. Len. 18 (1993) 10. 131 P.L. Christiansen and A.C. Scott, eds., in: NATO ASI Series B: Physics, Vol. 243. Davydov’s soliton revisited (Plenum, New York. 1990).
K.O. Rasmussen et al/Physics 14 1 0.
Bang, PL Christiansen,
Gaididei,
151 0.
E If, K.O. Rasmussen and Yu.B.
Phys. Rev. E 49 (1994)
Bang, PL. Christiansen.
Gnididei,
submitted
IO Appl.
E. Carmire
Anal.
AI!
8 (1991) I
I 1 L.
Fokker-Planck
Phys. Rev. Len.
and R.V. Khokhlov,
1.
Physica D 52 (1991)
BergC and D. Pesme. Phys. Rev. E 48 (1993)
R684.
2.
(Springer,
Berlin,
(1979)
Ph.D.
Thesis,
Lyngby
( 1993).
18. The 381.
Phys. Rev. B 32
M. Bonnedal
The
Technical
and M. Lisak.
University
of
Phys. Fluids
22
1838.
V.N. Kadantsev, Sol. (b)
Vol.
1984).
and J.F. Ladik, J. Math. Phys. 17 (1976)
1171 D. Anderson.
[ 181
2082.
285.
1512.
Bang.
Denmark,
and M. Lisak, J. Opt. Sot. Am. B Fraiman,
equation
A IS1 (1990)
series in synergetics,
If, M.P Surensen and PI_. Christiansen,
(1985)
Usp.
I9 I Sov. Phys. JETP 35 ( 1968) 609
and G.M.
Phys. Lett. in: Springer
1141 M.J. Ablowitz
and C.H. Townes,
Sukhorrukov
D. Anderson
IO] A.1 Smirnov
H. Risken,
[ I61 0.
Fiz. Nauk 93 ( 1967) 191 M. Desaix,
V.M. Malkin,
[ 151F.
Phys. Ser. 33 ( 1986) 481.
13 ( 1964) 479. 18 1 S.A. Akhmanov.
[ 12) [ 131
F. If, K.O. Rasmussen and Yu.B.
161 1.1. Rasmussen and K. Rypdal, 17 1 R.Y. Chiao,
4627.
127
Letters A 204 (1995) 121-127
I43
L.N. Lupichov
(1987)
and A.V.
569; 147 (1988)
Savin. Phys. Slat. I.55
14 August 1995
__ i!B
PHYSICS
Physics Letters A 204 (1995)
ELSEWER
LETTERS
A
121-127
The influence of noise on critical collapse in the nonlinear Schriidinger equation K.@. Rasmussen ‘, Yu.B. Gaididei b, 0. Bang’, P.L. Christiansen” ’ Institute of Mathematical Modelling, The Technicul University of Denmark, DK-2800 Lyngby, Denmark h Institute for Theoretical Physics, Metrologicheskaya Street 14 B, 252 143 Kiev 143. Ukraine ’ Lubomtoire de Physique. Ecole Normule Supirieure de Lyon, 46 All& d’halie, 69364 Lyon Cedex 07. France
for publication 15June Bishop
Received 25 August 1994; accepted
1995
Communicated by A.R.
Abstract Multiplicative Gaussian white noise is included in the nonlinear Schrodinger equation of one and two dimensions in the critical case, in which collapse occurs. The collective coordinate method is used to derive analytical results concerning the effects of the noise on the collapse process. This approach makes it possible to distinguish between the one- and two-dimensional case. The analytical results are in good agreement with numerical results obtained directly from the partial differential equation.
1. Introduction The inlluence of noise on the soliton solution to the nonlinear Schriidinger equation (NLS) [ l] has attracted some attention in the literature. Examples are applications in the field of optical fiber communication [2] and in biomolecular dynamics of Davydov solitons under the influence of thermal fluctuations (see, e.g., Ref. [ 31). Recently, the two-dimensional NLS equation, including thermal fluctuations, has been suggested as a model for energy transport and localization in organic thin films [4,5]. In this Letter we study the influence of noise on collapse [ 61. We study the critical case ad = 2, in one and two dimensions (d = 1,2), (T being the degree of nonlinearity. The two-dimensional collapse is important in the study of nonlinear optics, where the two-dimensional NLS describes propagation of electromagnetic beams in media whose refraction index 037%9601195/.W9.50
increases with the field amplitude [ 7,8]. Letting v = 2, in the one-dimensional case we obtain critical collapse also in this system. The point of this Letter is to investigate the differences between the one- and two-dimensional critical cases. In Section 2 we present an analytical approach based on collective coordinates [ 5,9]. In Section 3 we compare our analytical results to results obtained directly by numerical simulation of the partial differential equation.
2. Collective coordinate approach Introducing multiplicative tion we obtain
ircIl+ V2* + IIcI12”q +7(x, t)$ = 0,
@ 1995 El sevier Science B.V. All rights reserved
ssD10375-9601(95)00490-4
noise in the NLS equa-
(1)
122
K.0. Rasmussen et al./Physics
where 7(.x, t) is assumed to be Gaussian with zero mean
white noise
Letters A 204 (1995) 121-127
and the coefficient SmJI =
(rl(.& r)> = 0, (~(X,t)7j+‘,t’))
=DS(x-x’)C?(r-t’).
(2)
Here and in the following ( ) denotes the ensemble average, and D is the variance of the noise. For higher dimensions x = (XI, x2,. . . , xd). In the noise free case of D = 0 the number N and the Hamiltonian H, A’{$} = /
H{@) =
]+I2 ddx,
(3)
2a+2 - v(r, t) [#I2 rd-’ dr, l -: (TI@l >
where r = dm.
(5)
Using the trial function,
= A(t)sech[r/B(t)]
exp[ia(t)r2
+ i/?(t)], (6)
where A, B, a, and B are real functions, we obtain for the critical case, cd = 2, the following equation for the width B(t), [5] 4
B3B=A+-
J[
1 - (r/B)
sd+l,2 x
sech2(r/B)v(r,
t)rd-’
where the parameter d&Csd+1.2 -
(9)
As a consequence of the self-similar trial function, Eq. (6), A is a constant in time. (As pointed out by several authors [ 10-121, Eq. (7) can be made an exact description of the collapse by introduction of a subtle time dependence in A.) For A < 0 FCq. (7) clearly describes a collapse process. As argued in Ref. [S], Eq. (7), and the simpler equation t
(4)
Im(+*9t) +Mr12 J( --
+(r,r>
x”’ sech”( x) dx. s
ddx, lW12 -$---[+12n+2) SC
are conserved. Under the influence of the multiplicative noise N is still conserved, while H is not. We shall treat the problem of collapse influenced by noise analytically, applying the method of collective coordinates [ 91. The equation of motion, Eq. ( I), can under the assumption of isotropy, be derived from the following Lagrangian, L=
s,,,,, is given by
sd-I,2 sd-1,2u+2
1 + a)sd”_t , 2
NC
, 1
= 0,
(h(t)h(t’))
= 2y8(t
- t’),
(II)
with the variance 2y given by Y = gDl&,
(12)
,2.
Thus we replace Eq. (7) by Eq. ( 10) in the following. The power p in Eq. (10) equals 2 and 2 in one and two dimensions, respectively. In order to remove the singularities of Eq. ( IO), it is convenient to introduce the transformation B = Y1iz yielding 2YY - Y2 = 4( A + hY’3-P’/2).
(13)
Examining this equation analytically, we first note that the exact solution of Eq. (13) for h G 0, becomes Y(t) = +t:
- ?),
c
tc=fi,
t 14)
(8)
I’(t)
sd-1,4
C
(27r)d-‘(
(h(t))
(7)
A is defined as -
have the same Fokker-Planck equation for the appropriate probability density, provided the noise h(t) is delta correlated,
initial conditions being Y( t = 0) = Yo, and I’( t = 0) = 0. A has been assumed negative, since only such values are relevant in the collapse case. Assuming that the form of the solution will be approximately the same for nonzero noise we see that close to collapse, t N t,, the first term of Eq. ( 13) will be close to zero since Y is finite and the second term will be finite. Therefore it is permitted as a first step in analyzing Eq. (13) to neglect the first term of Eq. ( 13). This leads to
tanh(r/B)]
dr,
10)
1 _
&J-P+, 214
K.0. Rasmussen et al. /Physics
123
Letters A 204 (1995) 121-127
Here it is assumed that Y(3-p)/2h << IAl, which isreasonable at times close to the collapse time. Introducing
layed by the noise, such that the new collapse tc (D) , becomes
Y = y + 6.
fc(D)=
r = (Y)
(16)
t,(D 1 _
=O)
time,
(24)
(j~/~g,,(A(‘/~’
we get to leading orders in 6 ?=
-2m+
1
-JT;?i2
1 -(3-p’/2
s=my
3 - P @..)F’l -d/f
(17)
(18)
h(t).
Eq. ( 18) has the solution
dtl y(3-P”2(tl)h(t,).
(19)
Using Eq. ( 11) we are now able to calculate the correlation function (h( t)s( t)),
This expression also allows us to give the variance, which should be sufficient to stop the collapse process (tc = cQ), (25)
D = ;1A13’2s;.2.
As can be seen from Fig. 2 the analytical results derived above are not in too good agreement with the numerically obtained results. Therefore we will give a better approach in the following. The problem with this approach is that it is only usable in the twodimensional case. Introducing B=B+S,
Br
(B)
(26)
into Eq. ( 10) will give in leading order of S (h(t)S(t))
= -$+?)“(t),
(20)
which inserted into Eq. (18) gives ?=-2m+-
3 -
PF2’-”
2141
.
(21)
From Eq. (2 1) it is evident that collapse, y = 0, can only occur when 4 < 2~. The result of this approach is thus that total collapse cannot occur in one dimension (p = i ) when white noise is present. However, in one dimension
the stationary
width, B, (? = 0), is given
by
(22) Thus a certain degree of localization may still occur for large initial amplitudes ( IAJ > 0)) and/or weak noise, where B, becomes small. The process can therefore resemble a collapse process initially. If we turn to the two-dimensional case, p = f, collapse may still occur in the usual form. The appropriate solution of Eq. (21) is in this case t;(t)
= 4(A([t, - (1 - 3y/4\A(3’2)tJ,
z3g
_-_ + 3B B(S2) + 3B2(&)
..-3 6B
+
3i?*%
= -\A\ + (ha),
(27) (28)
= Bh.
From Eq. (28) we get = (ha) - 3(6’)% 3,
82(68)
(29)
and Eq. (27) takes the form B38 - 6B $8’)
= -[Al - 2(hc?).
From Eq. (14) and Fig. 2 it is reasonable that B(t)
= Bo( 1 - r2/tz)1/2,
(30) to assume
(31)
where t, is the unknown collapse time we are looking for. After some algebra, which is given in full length in Appendix A, we can derive the following (32)
(ha) = -&> 0
(S2)N
$$i2( t).
(33)
0
(23) Introducing
which satisfies the same initial conditions as approximation ( 15). We see that the collapse process is de-
Eqs. (32),
K*( 1 - &13)
(33) into Eq. (30) given us
= IAl - ~0,
(34)
K.O. Rasmussen et al. /Physics Letters A 204 (I 99s) 121-127
124
1.25
0.25
0.00
0.50 n
Fig. I. Width B versus time in two dimensions. The solid line indicates the analytical solution of EQ. (10) for h z 0. Numerical solutions are for D = 0.05 (dashed line), D = 0.2 (dotted line) and D = 0.4 (chain-dashed line).
where 0 = r,/B$ For small y an approximate is obviously given as
solution
Fig. 2. Solid line: collapse time r, versus noise variance D according to Eq. (24); chain-dashed line: predictions of E!q. (35). Squares indicate values obtained from numerical simulation of Fq, (13).
least the form of the results is correctly predicted by both methods and therefore it can be expected that the form of the one-dimensional results also is correct.
(35) It is also easy to calculate the D where the relevant solution to Eq. (34) becomes complex such that the collapse ceases to exist, D
=
24,214/3'2
15&
.
(36)
We have solved the two-dimensional version of Eq. ( 13) numerically using an ordinary fifth order RungeKutta scheme. The results are shown in Fig. 1, where the solid curve is the noise free solution of Eq. (10) (see Eq. ( 14) ), and the numerical results are for D = 0.05 (dashed line), D = 0.2, (dotted line), and D = 0.4 (chain-dashed line), respectively. It is seen that the effect of the noise is, as predicted, to slow down the collapse process, and it shows that an assumption like Eq. (3 1) is reasonable. In addition the solid line in Fig. 2 shows the collapse time according to Eq. (24)) and the dashed line shows the result of Eq. (35)) while the squares indicate numerically obtained values. We see that none of the analytical approaches agrees perfectly with the numerical results. It seems that the last approach agrees quite well for small noise variances, which is in agreement with the approximations of the approach. Furthermore it should be noted that the analytical approaches only differ by a numerical prefactor and neither of the prefactors seem to be completely right. Thus we may expect that at
3. Numerical
results
We now investigate numerically the influence of noise on critical collapse (ad = 2) in both one and two dimensions. The results were obtained using a second order split-step Fourier scheme [ 141. In one dimension the noise was generated in Fourier space using a method given in Ref. [ 151. The large number of grid points in two dimensions demands that the noise must be generated in each step of the calculation. This is accomplished using the method given in Ref. [ 161. The one-dimensional results were obtained using the following perturbed ground state solution [ 61 as initial condition, $(x,t
= 0) =&,(x)
= Ao x 3”4sech1i2(2n),
(37)
where A0 = 1 is the ground state solution to the NLS equation without noise (H{@a} = 0). The same idea is used to obtain an initial condition in two dimensions, but in this case the exact expression for the ground state solution is not known. However, it is possible to obtain an approximate expression for the ground state solution [ 171, given as Jl(r,t r=
=0) II
X:+X;,
=@a(r)
=AaAsech(r/B), (38)
K.0.
Rasmussen et al./Physics
125
Letters A 204 (1995) 121-127
where
450 - (a) 0
,’ ,’
300 ,’
For both initial conditions it is true that collapse will occur when A0 > 1. Investigating the interplay between the coherent collapse process and the incoherent noise process numerically, we need a way to determine whether the excitation is losing or gaining coherence (collapse). To determine this we use a quantity used for a similar purpose by Kadantsev et al. [ 181
l@12rr+2 ddX.
R= .i
,’ /’ ,’ ,’ ,’
150-
i
”
(40)
Collapse will reflect itself as an increase in R, while the opposite will be true if the excitation is scattered. The purpose of the numerical simulations is then to investigate whether it is true, as the analytical work predicts, that noise can prevent a collapse process, and if this is the case, to determine how strong the noise must be. In Fig. 3a we show the result of the one-dimensional simulations with a solid line. Above the line the noise is strong enough to destroy the collapse. This result has been obtained using the quantity R (Eq. (40)) such that above (below) the solid line R is a decreasing (increasing) function of time. The predictions of Eq. (24) are given by the dashed line, which has been obtained as a result of the following interpretation: When B, > Bo we interpret this as a scattering of the initial excitation, while B, < Bo is interpreted as a localization (collapse) process. This means that the curve B,( Ao) = Bo is the curve separating the domain of collapse from that of scattering, and thus the analytical equivalent to the numerically obtained results. Excellent agreement is observed for low initial amplitudes. The increasing discrepancy as the initial amplitude increases may be ascribed to the self-similar constraint of the analytical approach. Since the noise strength needed is becoming large when the initial amplitude is increased, the excitation will be more distorted when the initial amplitude increases. This is clearly a violation of the self-similarity assumption of the analytical approach. We further note that the separating curve is of the form D cx A: according to Eq. (22) and Eq. (8), which is a very strong dependence on the initial amplitude Ao,
,’
0
/;
1 04
/’ ,’ ,’
11-. ,’
,’
10
,’
,’
,’
,’
,’
5
,,
‘//
/,_,>’
__-’
Fig. 3. (a) Critical variance versus initial amplitude A,) in the one-dimensional case. Solid line: results obtained by solving Eq. ( 1) numerically; dashed line: predictions of Eq. (24). (b) Critical variance versus initial amplitude A() in the two-dimensional case. Solid line: results obtained by solving Eq. ( 1) numerically; dashed line: predictions of Eq. (25); chain-dashed line: results of FQ. (36).
Similarly, we have in Fig. 3b depicted the simulations in two dimensions as a solid line together with the analytical predictions of Eq. (25) as a dashed line and of Eq. (36) by a chain-dashed line. It is seen that the analytical predictions of Eq. (2.5) agree reasonably well with the numerically obtained results, while the predictions of Eq. (36) are not too good. It should be kept in mind that the method leading to Eq. (36) is the more precise of the two with respect to Eq. ( IO). It clearly gives the best results. This indicates that the discrepancy between the analytical and the numerical results stems mainly from the method applied to reduce the problem given by the partial differential equation ( 1) to the ordinary differential equation ( IO). Some further approximations have also been imposed in the two-dimensional case since we had to assume the noise to be radially symmetrical in the analytical approach, which has not been necessary in the numerical simulations. In spite of the discrepancies we
K.0. Rasmussen et al./Physics
126
may still trust the analytical approach to have given the correct form of the separating curve, which in this two-dimensional case is D DCAi. Thus a much weaker dependence than was seen in one dimension.
Letters A 204 (1995) 121-127
with Wronskian (44) Thus from Eq. (42) we get T
4. Conclusion It has been shown both analytically and numerically that the collapse process may be prevented by introducing a multiplicative noise term in the equation of motion. Some differences between the one-and twodimensional cases were found. It seems that the onedimensional collapse process is significantly stronger than the equivalent one in two dimensions since the strength of the noise must be much larger in one dimension than in two dimensions to have any effect.
- w(T)w(T~)l.
x [wI(TI)w(T)
(45)
It is seen from Eq. (45) that w can diverge at T --f 1. To avoid this we have to choose the constant c2 as follows, I
t2
c2=--c
%
J
h(Tltc)
dT,
(46)
J1’--TTiWI(G),
0
and obtain from Eqs (46) Acknowledgement
T
We wish to thank A.C. Scott. F. If and J.J. Rasmussen for many helpful discussions and valuable suggestions. Yu.B. Gaididei acknowledges The Technical University of Denmark for a Guest Professorship, and 0. Bang the CEC for financial support under contract No. SCl-CT91-0705. Finally we acknowledge financial support from The Danish Research Council through contract No. 52 1055 I - 1.
t2
w=--c
4
(s ‘Jiq
h(Tltc)
dT
I
+
s
h(Tltc)
dT
T
where we put ct = 0. Using Eq. (47) we are now able to calculate (ha) and (S2),
(
0
Using the assumption
(3 1) in Eq. (30) we get (41)
(S2) = &,
1
IT,, 3
I
( ll+;)3
[,
0
+;(I
-T*)log (
where T = t/tc is the normalized time. If we put S = w&?? we obtain from Eq. (41)
(~-T2)?!&2+?=
-T l+T
Near the collapse (T 4 and (33).
1 + T2) + 2T2
>I
(49)
1) these simplify to Eqs. (32)
ht,2 1 - T2
Bn’JC-F’ (42)
The homogeneous part of Eq. (42) is the Legendre equation. It has the two linearly independent solutions 11-T WI = 21+T’
(47)
I $=iy
(hi3= -2~;
Appendix A
w~(TI)wI(T)
0
2T w2 = , _T23
(43)
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