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Stability of moving bubbles in a system of interacting bosons
Volume 135, number 2
PHYSICS LETTERS A
13 February 1989
STABILITY OF MOVING BUBBLES IN A SYSTEM OF INTERACTING BOSONS
I.V. BARASHENKOV, T.L. BOYADJIEV, I.V. PUZYNIN and T. ZHANLAV Joint Inst itutefor Nuclear Research, LCTA, Head Post Office P.O. Box 79, 101000 Moscow, USSR Received 7 September 1988; accepted for publication 30 November 1988 Communicated by A.R. Bishop
3 — v.’5 nonlinear Schrodinger equation describing a boson gas with two- and three-body The static bubble-like soliton of the v.’ interactions, is known to be unstable. Here we study the stability of moving “bubbles”. Our conclusion is that there exists a certain critical velocity, v~such that the “bubble” is stable for v> v, and unstable otherwise.
I. Introduction A gas of bosons interacting via the two-body attractive and three-body repulsive ô-function potential is described quasiclassically by the so-called w~nonlinear Schrodinger equation [1]:
sions. (In the case of lumps this property is not surprising but it becomes nontrivial for solitons with non-vanishing boundary conditions. To compare, note that the kinks and vortices of the repulsive w~ NLS do not have localized stationary analogues in D = 3.) However, regardless of the dimension, the
iw 2—a 4=O, (1) 1+E~w—a1w+a3wki’I 5wIwI where a 2/0x~+...+92/ôx~.An 3, cx5>O, ô we prefer to work with equivalent scaled and form which reads [2]
quiescent “bubbles” proved unstable [2,9]. In the present study we address ourselves to the question of whether the travelling “bubble” can be stable. Here our treatment will be restricted to the one-dimensional situation since in this case the cor-
~
responding solution of eq. (2) is known explicitly [2]:
—
(2)
Apart from the physical systems that can be modelled by the Bose gas with this type of interaction (such as superfluid helium [3]) the NLS arises in a number of independent applications in~
cluding quantum crystals [4], one-dimensional ferromagnetic [51and molecular chains [6], nonlinear optics [7], nuclear hydrodynamics [8] and many others. The physically interesting dimension D varies from 1 to 3 while the appropriate boundary conditions can be both zero and of the “condensate” type, i.e. ~(x, t)—~Oor 9’(x,tfl—’l
as
X2-400.
(3)
In ref. [21 eq. (2) was found to possess, under the condition (3), a new type of soliton solutions that were called “bubbles”. These nontopological solitons exist at 0
t)=q~(5~) cosh(~/2—i,u)
~‘~‘~‘
~
[(2—A)
=
(A2 + v2) — 1/2 + cosh ~]1/2’
(4)
where X-=x—vt, ~=(c2—v2)112.~, c=2(1—A)”2 stands for the velocity of sound, and v is the velocity of the soliton, —c< v< c. The “twisting angle” ~z is also defined through v: sin2
2—v2)/(A2+v2)]”2, 1u=~v[(c
—
7t/4 < /~< ~t/4.
In subsequent publications we plan to analyse the two- and three-dimensional situations which we expect to have much in. common with the D= 1 case. 125
Volume 135, number 2
PHYSICS LETTERS A
13 February 1989
2. Linearized stability
0.5
Linearizing eq. (2) in the vicinity of the “bubble” (4) and taking the infinitesimal perturbation in the form
0.4
J, g, )~real, we arrive at the eigenvalue problem
0.2 0.1
with
H~y=)Jy(x),
(5)
y(±oo)=0,
(6)
where liv
—
A 05
03
~.2~O8
0.0 0.0
0.1
0.2 0.3 tI/C
0.4
Fig. 1. The instability growth rate A, versus v/c, the velocity of the soliton in units ofthe sound velocity (c=2~/i~~).
d ~d2 I+v—~J— U~(x),
(7)
fact that for v~v~the eigenvalue 2 1(v) immerses into 1H~which occupies the continuous spectrum of J is, as v—+v~ imaginary axis of That —0 and A 1(v)—~0the functionsf gbecome non-localized. Indeed, asymptotically as x—~±c~ we have r(±oc)=cosj~,j(±oo)=Rsin4uso that
I is the 2 x 2 identity matrix,
~..
~
(0 =
_i) ‘
and
2
2F~rj fF+2F~r
2F~rj F+2F~J2).
(8)
Next, in (5)— (8) r and j stand for the real and imaginary part of eq. (4): ~(~=r(2)+ij(~), and p= ~I =(r2+j2)”2. Finally, F=F(p)=(p—l)(2A+1—3p), F~= 2 (A + 2— 3p), y (f g)T, and the tilde over x has been omitted in (5)—(8). The eigenvalue problem (5), (6) was analysed numerically at 20 equally spaced values of A in the range (0, 1). Having taken into account the symmetry HV(x)=H(_V)(—x), we confined ourselves to positive velocities v. For each A a single positive eigenvalue )~= was found, depending continuously on v. The function 2 1(v) which looks similar for each A, is depicted in fig. 1. It is seen that there is a certain critical velocity v~such that for v>~v~only the zero eigenvalue A. 0(v) 0 exists corresponding to translational The soliton is therefore stableonly for v>~v,~.Tosymmetry. describethe stability domain it remains to find the precise value of v,~. The difficulty in determining v~is related to the 126
Uv(±~)=c2(\Rsinucosll cos2~i
~sin~cos~z\ sin2ji
)
and the characteristic equation for eq. (5) is readily calculated to be k4—c2k2+(t
2=0. (9) 1+vk) It is not difficult to show that (at least in a finite vicinity of )~= 0) eq. (9) has four real roots, two of them being positive and two negative. Let k±be the two roots with the minimal moduli, k+>.0 and k <0. Then y (f g)T obeys y(x)—~y~exp(—k÷x)asx—~±oo.
(10)
Now it is straightforward to obtain from (9) that when )~—p0 we have k+ —0 implying that the solulion {A 1, y} of the problem (5), (6) ceases to exist at v~. ne which cannot1 1(v) therefore determine v~simply as v= such vO for vanishes; some sort of extrapolation procedure has to be applied instead. Our extrapolation was based on a Taylor series expansion 2+,c 2+... (11) 1i(v)=,ci(v—v~) 2(v— v~) in the vicinity of v~(for v~va). The validity of (11) was verified numerically.
Volume 135, number 2
PHYSICS LETTERS A
3. Numerical computation
13 February 1989 0.5
Suppose v approaches v~—0 and k+ —~‘0.Then if we wanted the boundary conditions (6) to be satisfied to some reasonable accuracy, we would have to extend the integration interval to infinity. This cornputational difficulty can be circumvented by invoking the asymptotics (10) and passing to the conditions of the form (dy/~+k+y)I~=±00=0.
0.4 0.3 ~ 02 0 1 0.0 0.0
Next, for 21-40 eq. (9) yields k±=
±(c~v)
0.2
0.4
0.6
0.8
1.0
A
‘A~+O(2~)
Fig. 2. The critical velocity v, versusA.
so that in the vicinity of the critical velocity the latter conditions simplify to
{dy/dx±(c~v)’2y}I~ ±00=0.
(12)
Our computational policy was to solve the inverse spectral problem, i.e., for several sufficiently small values of 2 we solved the problem (5), (12) to determine the(11) corresponding v. We have found that the expansion is indeed correct, and that ic~~ 0. The critical velocity was obtained then by the linear extrapolation of the resulting curve 2 1(v) to 11=0. The values of v~pertaining to different A are collected in table 1; the dependence v~ (A) is illustrated by fig. 2.
4. Discussion So we have shown that the one-dimensional bubble-like solitons stabilize when moving sufficiently fast. It would be interesting to find out whether analogous critical velocities exist for two- and three-dimensional “bubbles” which are also known to be unstable at rest. In this connection it is worthwhile to recall [2] that propagation of transonic “bubbles” is Table 1 The critical velocities (velocities of stabilization), v,, for different values of the parameter A. v, is given in units of the sound velocity, c=2~/iT.
_____________________________________________ A
va/c
A
v,/c
A
va/c
0.1 0.2 0.3
0.1301 0.1943 0.2461
0.4 0.5 0.6
0.2911 0.3317 0.3692
0.7 0.8 0.9
0.4044 0.4377 0.4695
governed approximately by the Korteweg—de Vries equation at D= 1, and by the two- and three-dimensional Kadomtsev—Petviashvili equations at D = 2 and 3. On the other hand, the KdV soliton as well as the two-dimensional KP lump is known to be stable while the three-dimensional lump is unstable [10]. This suggests the critical velocity such that stathe 5 NLS that “bubble” is unstable forv~ v< v~and ble atw v>~v~,exists only for D= 1 and 2. The threedimensional “bubble”, conversely, is expected to be —
unstable for any velocity. Finally, let us remark that when we analyse NLS equations, the dependence of the soliton’s stability on its velocity is inherent only for solitons with nonvanishing boundary conditions, i.e., asx—+±oo. M (13) ~,(x,t)-+e~’ Really, consider the case of the vanishingconditions ~(x,
t) —p ~ 0 and suppose we have a lump ~ (x, moving with velocity v. The Galilean transformation t)
~
t)—~(x,t) =exp[—3iv(x+fvt)]ço(x+vt,t)
(14)
takes then the soliton q~to the rest frame. Furthermore, this transformation can be applied to any nearby solution, the stability problem being reduced to the one for v=0. On the other hand, in the case of the boundary conditions (13) the transformation (14) does not take the travelling soliton to the static one; the dependence on the velocity is more complicated here, see e.g. eq. (4). 127
Volume 135, number 2
PHYSICS LETTERS A
Acknowledgement It is a pleasure to thank Professor V.G. Makhankov for his interest to this work and for continual encouragement.
References [1] AS. Kovalev and A.M. Kosevich, Fiz. Nizk. Temp. 2 (1976) 913. [2] IV. Barashenkov and V.G. Makhankov, Phys. Lett. A 128 (1988) 52. [3] V.L. Ginsburg and L.P. Pitayevski, Soy. Phys. JETP 7 (1968) 858; L.P. Pitayevski, Soy. Phys. JETP 13 (1961) 451; V.L. Ginsburg and A.A. Sobianin, Usp. Fiz. Nauk 120 (1976) 153.
128
13 February 1989
[4] D. Pushkarov and ZI. Kojnov, Zh. Eksp. Teor. Fiz. 74 (1978) 1845. [5] Kh.I. Pushkarov and M.T. Pnmatarova, ICTP, Trieste, preprintlC/84/131 (1984). [6] Kh.I. Pushkarov and M.T. Primatarova, Phys. Stat. Sol. (b) 123 (1984) 573. [7] V.E. Zakharov, V.V. Sobolev and V.S. Synakh, Soy. Phys. JETP33 (1971) 77; yE. Zakharov and V.S. Synakh, Zh. Eksp. Teor. Fiz. 68 (1975) 940; Kh.I. Pushkarov, D.I. Pushkarov and IV. Tomov, Opt. Quantum Electron. 11(1979) 471. [8] V.G. Kartavenko, Soy. J. Nucl. Phys. 40 (1984) 240. [9] IV. Barashenkov, A.D. Gocheva, V.G. Makhankov and IV. Puzynin, JINR, Dubna, preprint P17-88-411 (1988); Physica D, to be published. [10] E.A. Kuznetsov,AM. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103.
PHYSICS LETTERS A
13 February 1989
STABILITY OF MOVING BUBBLES IN A SYSTEM OF INTERACTING BOSONS
I.V. BARASHENKOV, T.L. BOYADJIEV, I.V. PUZYNIN and T. ZHANLAV Joint Inst itutefor Nuclear Research, LCTA, Head Post Office P.O. Box 79, 101000 Moscow, USSR Received 7 September 1988; accepted for publication 30 November 1988 Communicated by A.R. Bishop
3 — v.’5 nonlinear Schrodinger equation describing a boson gas with two- and three-body The static bubble-like soliton of the v.’ interactions, is known to be unstable. Here we study the stability of moving “bubbles”. Our conclusion is that there exists a certain critical velocity, v~such that the “bubble” is stable for v> v, and unstable otherwise.
I. Introduction A gas of bosons interacting via the two-body attractive and three-body repulsive ô-function potential is described quasiclassically by the so-called w~nonlinear Schrodinger equation [1]:
sions. (In the case of lumps this property is not surprising but it becomes nontrivial for solitons with non-vanishing boundary conditions. To compare, note that the kinks and vortices of the repulsive w~ NLS do not have localized stationary analogues in D = 3.) However, regardless of the dimension, the
iw 2—a 4=O, (1) 1+E~w—a1w+a3wki’I 5wIwI where a 2/0x~+...+92/ôx~.An 3, cx5>O, ô we prefer to work with equivalent scaled and form which reads [2]
quiescent “bubbles” proved unstable [2,9]. In the present study we address ourselves to the question of whether the travelling “bubble” can be stable. Here our treatment will be restricted to the one-dimensional situation since in this case the cor-
~
responding solution of eq. (2) is known explicitly [2]:
—
(2)
Apart from the physical systems that can be modelled by the Bose gas with this type of interaction (such as superfluid helium [3]) the NLS arises in a number of independent applications in~
cluding quantum crystals [4], one-dimensional ferromagnetic [51and molecular chains [6], nonlinear optics [7], nuclear hydrodynamics [8] and many others. The physically interesting dimension D varies from 1 to 3 while the appropriate boundary conditions can be both zero and of the “condensate” type, i.e. ~(x, t)—~Oor 9’(x,tfl—’l
as
X2-400.
(3)
In ref. [21 eq. (2) was found to possess, under the condition (3), a new type of soliton solutions that were called “bubbles”. These nontopological solitons exist at 0
t)=q~(5~) cosh(~/2—i,u)
~‘~‘~‘
~
[(2—A)
=
(A2 + v2) — 1/2 + cosh ~]1/2’
(4)
where X-=x—vt, ~=(c2—v2)112.~, c=2(1—A)”2 stands for the velocity of sound, and v is the velocity of the soliton, —c< v< c. The “twisting angle” ~z is also defined through v: sin2
2—v2)/(A2+v2)]”2, 1u=~v[(c
—
7t/4 < /~< ~t/4.
In subsequent publications we plan to analyse the two- and three-dimensional situations which we expect to have much in. common with the D= 1 case. 125
Volume 135, number 2
PHYSICS LETTERS A
13 February 1989
2. Linearized stability
0.5
Linearizing eq. (2) in the vicinity of the “bubble” (4) and taking the infinitesimal perturbation in the form
0.4
J, g, )~real, we arrive at the eigenvalue problem
0.2 0.1
with
H~y=)Jy(x),
(5)
y(±oo)=0,
(6)
where liv
—
A 05
03
~.2~O8
0.0 0.0
0.1
0.2 0.3 tI/C
0.4
Fig. 1. The instability growth rate A, versus v/c, the velocity of the soliton in units ofthe sound velocity (c=2~/i~~).
d ~d2 I+v—~J— U~(x),
(7)
fact that for v~v~the eigenvalue 2 1(v) immerses into 1H~which occupies the continuous spectrum of J is, as v—+v~ imaginary axis of That —0 and A 1(v)—~0the functionsf gbecome non-localized. Indeed, asymptotically as x—~±c~ we have r(±oc)=cosj~,j(±oo)=Rsin4uso that
I is the 2 x 2 identity matrix,
~..
~
(0 =
_i) ‘
and
2
2F~rj fF+2F~r
2F~rj F+2F~J2).
(8)
Next, in (5)— (8) r and j stand for the real and imaginary part of eq. (4): ~(~=r(2)+ij(~), and p= ~I =(r2+j2)”2. Finally, F=F(p)=(p—l)(2A+1—3p), F~= 2 (A + 2— 3p), y (f g)T, and the tilde over x has been omitted in (5)—(8). The eigenvalue problem (5), (6) was analysed numerically at 20 equally spaced values of A in the range (0, 1). Having taken into account the symmetry HV(x)=H(_V)(—x), we confined ourselves to positive velocities v. For each A a single positive eigenvalue )~= was found, depending continuously on v. The function 2 1(v) which looks similar for each A, is depicted in fig. 1. It is seen that there is a certain critical velocity v~such that for v>~v~only the zero eigenvalue A. 0(v) 0 exists corresponding to translational The soliton is therefore stableonly for v>~v,~.Tosymmetry. describethe stability domain it remains to find the precise value of v,~. The difficulty in determining v~is related to the 126
Uv(±~)=c2(\Rsinucosll cos2~i
~sin~cos~z\ sin2ji
)
and the characteristic equation for eq. (5) is readily calculated to be k4—c2k2+(t
2=0. (9) 1+vk) It is not difficult to show that (at least in a finite vicinity of )~= 0) eq. (9) has four real roots, two of them being positive and two negative. Let k±be the two roots with the minimal moduli, k+>.0 and k <0. Then y (f g)T obeys y(x)—~y~exp(—k÷x)asx—~±oo.
(10)
Now it is straightforward to obtain from (9) that when )~—p0 we have k+ —0 implying that the solulion {A 1, y} of the problem (5), (6) ceases to exist at v~. ne which cannot1 1(v) therefore determine v~simply as v= such vO for vanishes; some sort of extrapolation procedure has to be applied instead. Our extrapolation was based on a Taylor series expansion 2+,c 2+... (11) 1i(v)=,ci(v—v~) 2(v— v~) in the vicinity of v~(for v~va). The validity of (11) was verified numerically.
Volume 135, number 2
PHYSICS LETTERS A
3. Numerical computation
13 February 1989 0.5
Suppose v approaches v~—0 and k+ —~‘0.Then if we wanted the boundary conditions (6) to be satisfied to some reasonable accuracy, we would have to extend the integration interval to infinity. This cornputational difficulty can be circumvented by invoking the asymptotics (10) and passing to the conditions of the form (dy/~+k+y)I~=±00=0.
0.4 0.3 ~ 02 0 1 0.0 0.0
Next, for 21-40 eq. (9) yields k±=
±(c~v)
0.2
0.4
0.6
0.8
1.0
A
‘A~+O(2~)
Fig. 2. The critical velocity v, versusA.
so that in the vicinity of the critical velocity the latter conditions simplify to
{dy/dx±(c~v)’2y}I~ ±00=0.
(12)
Our computational policy was to solve the inverse spectral problem, i.e., for several sufficiently small values of 2 we solved the problem (5), (12) to determine the(11) corresponding v. We have found that the expansion is indeed correct, and that ic~~ 0. The critical velocity was obtained then by the linear extrapolation of the resulting curve 2 1(v) to 11=0. The values of v~pertaining to different A are collected in table 1; the dependence v~ (A) is illustrated by fig. 2.
4. Discussion So we have shown that the one-dimensional bubble-like solitons stabilize when moving sufficiently fast. It would be interesting to find out whether analogous critical velocities exist for two- and three-dimensional “bubbles” which are also known to be unstable at rest. In this connection it is worthwhile to recall [2] that propagation of transonic “bubbles” is Table 1 The critical velocities (velocities of stabilization), v,, for different values of the parameter A. v, is given in units of the sound velocity, c=2~/iT.
_____________________________________________ A
va/c
A
v,/c
A
va/c
0.1 0.2 0.3
0.1301 0.1943 0.2461
0.4 0.5 0.6
0.2911 0.3317 0.3692
0.7 0.8 0.9
0.4044 0.4377 0.4695
governed approximately by the Korteweg—de Vries equation at D= 1, and by the two- and three-dimensional Kadomtsev—Petviashvili equations at D = 2 and 3. On the other hand, the KdV soliton as well as the two-dimensional KP lump is known to be stable while the three-dimensional lump is unstable [10]. This suggests the critical velocity such that stathe 5 NLS that “bubble” is unstable forv~ v< v~and ble atw v>~v~,exists only for D= 1 and 2. The threedimensional “bubble”, conversely, is expected to be —
unstable for any velocity. Finally, let us remark that when we analyse NLS equations, the dependence of the soliton’s stability on its velocity is inherent only for solitons with nonvanishing boundary conditions, i.e., asx—+±oo. M (13) ~,(x,t)-+e~’ Really, consider the case of the vanishingconditions ~(x,
t) —p ~ 0 and suppose we have a lump ~ (x, moving with velocity v. The Galilean transformation t)
~
t)—~(x,t) =exp[—3iv(x+fvt)]ço(x+vt,t)
(14)
takes then the soliton q~to the rest frame. Furthermore, this transformation can be applied to any nearby solution, the stability problem being reduced to the one for v=0. On the other hand, in the case of the boundary conditions (13) the transformation (14) does not take the travelling soliton to the static one; the dependence on the velocity is more complicated here, see e.g. eq. (4). 127
Volume 135, number 2
PHYSICS LETTERS A
Acknowledgement It is a pleasure to thank Professor V.G. Makhankov for his interest to this work and for continual encouragement.
References [1] AS. Kovalev and A.M. Kosevich, Fiz. Nizk. Temp. 2 (1976) 913. [2] IV. Barashenkov and V.G. Makhankov, Phys. Lett. A 128 (1988) 52. [3] V.L. Ginsburg and L.P. Pitayevski, Soy. Phys. JETP 7 (1968) 858; L.P. Pitayevski, Soy. Phys. JETP 13 (1961) 451; V.L. Ginsburg and A.A. Sobianin, Usp. Fiz. Nauk 120 (1976) 153.
128
13 February 1989
[4] D. Pushkarov and ZI. Kojnov, Zh. Eksp. Teor. Fiz. 74 (1978) 1845. [5] Kh.I. Pushkarov and M.T. Pnmatarova, ICTP, Trieste, preprintlC/84/131 (1984). [6] Kh.I. Pushkarov and M.T. Primatarova, Phys. Stat. Sol. (b) 123 (1984) 573. [7] V.E. Zakharov, V.V. Sobolev and V.S. Synakh, Soy. Phys. JETP33 (1971) 77; yE. Zakharov and V.S. Synakh, Zh. Eksp. Teor. Fiz. 68 (1975) 940; Kh.I. Pushkarov, D.I. Pushkarov and IV. Tomov, Opt. Quantum Electron. 11(1979) 471. [8] V.G. Kartavenko, Soy. J. Nucl. Phys. 40 (1984) 240. [9] IV. Barashenkov, A.D. Gocheva, V.G. Makhankov and IV. Puzynin, JINR, Dubna, preprint P17-88-411 (1988); Physica D, to be published. [10] E.A. Kuznetsov,AM. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103.