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The effect of nonlinear gradient terms on localized states near a weakly inverted bifurcation
Volume 146, number 5
PHYSICS LETTERS A
28 May 1990
THE EFFECT OF NONLINEAR GRADIENT TERMS ON LOCALIZED STATES NEAR A WEAKLY INVERTED BIFURCATION Robert J. DEISSLER a and Helmut R. BRAND b,a
b
Centerfor Nonlinear Studies, MS-B 258, Los A/amos National Laboratory, University of California, LosA/amos, NM 87545, USA FB 7, Physik, Universität Essen, D-4300 Essen 1, FRG
Received 12 February 1990; accepted for publication 2 April 1990 Communicated by A.R. Bishop
We study the effect of nonlinear gradient terms on localized states in a complex Ginzburg—Landau equation near a weakly inverted bifurcation. We find that these terms can greatly affect the speed ofthe state as well as cause an asymmetry. We also study the interaction ofcounterpropagating localized states in the coupled equations for counterpropagating waves. A comparison with recent experiments in binary fluid convection is made.
Recently the existence of confined states in binary fluid convection in an annulus [1,21 has been reported. By confined state one means in this case that only part ofthe annulus is filled with convective rolls, whereas the rest of the cell is apparently free of convection. As first pointed out in ref. [3], the existence of such a state may be partially understood in terms of a slug [4,5], a localized structurebetween two stable regimes. In other words, the system is subcritical (with two basins of attraction) allowing for the coexistence of two different states namely convective rolls and a stable conduction state. It is the stable conduction state on either side of the rolls which confines then the azimuthal direction. One of the least understood features of the confined state in binary convection is the fact that the velocity of the envelope of the confined state is orders of magnitude smaller than the velocity of the rolls within the confined state or zero, even though the group velocity of a linear wavepacket [61 is of the same order of magnitude as the phase velocity of the rolls. In this note we investigate this problem by studying the effect of nonlinear gradient terms in the cornplex Ginzburg—Landau equation and find that the velocity of the envelope can indeed by greatly reduced by incorporating these terms. We further study —
252
the interaction of counterpropagating localized states in the framework of coupled complex Ginzburg— Landau equations. Counterpropagating states were previously studied in ref. [3] with different pararneter values. Here we use the parameter values of ref. [7], in which a single Ginzburg—Landau equation (eq. (1) below without the nonlinear gradient terms) has been studied. The new feature observed in ref. [71is the existence of localized states with a unique size and shape for a wide range of initial conditions. We note that for other parameter values chaotic localized states can exist [4,51.As in ref. [3] we find annihilation of colliding localized states for stabilizing crosscoupling and a transition from subcritical to absolutely unstable supercritical behavior for destabilizing crosscoupling. The new feature which is studied in detail in ref. [8] is the soliton-like behavior of the collision of two one-particle states which return to the same size and shape after the interaction. We stress that these one-particle states are not solitons, since the process is occurring in a strongly dissipative system and we note that this is the first example of this type of behavior in such a strongly dissipative system. The equation for traveling waves in one direction which we study first reads
Volume 146, numberS
PHYSICS LETTERS A
3~A+ vô~A XA + (Yr + iy, )A~~ +
(fir
—
~r
+ if?
1) A I — (or + 1O~) I + iA 2A~— CUr +i/L
1) IA I
I
4A
2A*
1 )A
(1)
opposite to the group velocity). opposite signs of the nonlinear gradient terms theFor one-particle state moves at a speed larger than the group velocity. For sufficiently large values the nonlinear gradient terms will cause the localized state to damp to zero.
x.
The new feature is the presence of the nonlinear gradient terms, IA 2A~and A 2A which occur to the same order as the quintic term IAI4A, as first discussed in refs. [9,10]. For our numerical computations we use fourth order spatial differencing and second order Runge—Kutta in time as described in detail in ref. [11]. Fig. 1 a shows the one-particle state first observed by Thual and Fauve [7] with a group velocity v~1. This state is moving to the right with speed 1. Fig. lb shows the resulting state under the influence of the nonlinear gradient terms. The yelocity of the particle has been reduced by a factor of about 7 for this choice of parameter values (cornpare fig. lb). In addition the state is skewed to the left as one would expect, since the nonlinear gradient terms have a greater influence on the portions with larger amplitude. For certain combinations of parameter values we can get the one-particle state to stand still. For larger values ofthe nonlinear gradient terms the one-particle state moves backwards (i.e. ~,
~
a
We now come to the comparison with recent experiments in binary fluid convection by Niemela, Ahlers and Cannell [2]. In these experiments a localized state forms, which initially has a velocity which is orders of magnitude smaller than the linear group velocity. Since a signature of the effect of nonlinear gradient terms is an asymmetry of the envelope of the localized state, more precise measurements could determine, whether there is indeed a sufficiently large asymmetry present in the experiments. In the experiment the state slows down exponentially with time [2] and eventually stops a!together. This slowing down is a feature which probably cannot be captured in the context of a cornplex Ginzburg—Landau equation without mean flow effects. We now study the collision of counterpropagating one-particle states in the coupled complex Ginzburg—Landau equations describing counterpropagating waves. For completeness we consider all the terms which arise to the order of the quintic term. In the numerical simulations some of the terms will be set t9iA+VôxA=XA+(Yr+~Yi)Axx equal to zero (compare below),
‘t)
d
~
________________________________
o
io
20
30
40
50
60
(‘er
x
~L-
b
~
0
10I
(Ar
+iA~)IA I 2A~
_(fi~+jfij)B*ABx_(~~+j)AB~,
40
50
(2)
a
___________
20 30 I~I~I~I~ X
+ii~) BI2A—
_(lLr+j~Li)A2A~_(Xr+jXi)IBl2Ax
_______________________
_____
28 May 1990
60
Fig. 1. Plot of the one-particle solution (v= I), (a) without nonlinear gradient terms, (b) including nonlinear gradient terms —1). In all the plots we have chosen the parametervaluesusedinref.[7]:~=—0.l,flr=—3,fl,=—l,y,=l, = 0, ô,= 2.75, ô~ = — 1, and we have set all the coefficients ofthe nonlinear gradient terms to zero unless specified otherwise. In additionwehavechosen eters varying from one plot ~,=l. to the Thisleaves~,andvastheparamnext.
1B—~o~B=xA+ (Yr+~yi)B~x 2B_(Or+~Oi)IBI4B +(fir+jfii)IBI ~ + +
Cur (fir
+ ~ILi )B2B~+ (Ir + ~ IA I 2B~ +i/2 1)A*BA~+ (‘~r+j1~)BAAx~.
(3)
(ArA?ZrPi
In writing down these equations, we have discarded quintic, spatially 2IAI2A, homogeneous IBI4A)crosscoupling in eqs. (2), terms (3). (like Fig.those 2 showsIBIthe collision of two counterpropa—‘
253
Volume 146, number 5
PHYSICS LETTERS A
________________
C
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o
28 May 1990 “1
a
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~
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x Fig. 2. Complete annihilation of two one-particle solutions (‘~r=3, I ), (a) initial approach, (b) interaction, (c) one-particle states decay to zero,
gating one-particle states at successive times for stabilizing crosscoupling. During the interaction the amplitudes of the particles are sufficiently reduced to bring the amplitude below the separation between growth and decay and the particles decay giving complete annihilation. If the states are sufficiently different in size just before the interaction, ‘only one of the states is annihilated as is seen in fig. 3. If the stabilizing crosscoupling is decreased, the amplitude of the state will not be decreased as much during the interaction and the localized solutions will reemerge after the collision and return to their initial size and shape. Fig. 4 shows such an interaction with nonlinear gradient terms included. We note that for suffi ciently small group velocity with or without nonlinear gradient terms the two states can stop moving during the early stages of the interaction giving rise to a composite stationary state, and do not interpenetrate. If one were to observe experimentally such a composite stationary state, one would have in different parts of the state waves traveling to the right and to the left, respectively. Fig. 5 shows the interaction of two colliding one-
254
I
80
_________________
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‘I
60
x
-.
0
120
x
60
80
100
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x Fig. 3. Annihilation of a one-particle solution (~r=3,v=2), (a) initial approach, (b) interaction, (c) the surviving one-particle solution propagates to the right.
I
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a
H
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Fig. 4. Interpenetration of two nonsymmetric one-particle states (~~r2,5,v=2, 2r iI~r/2,~), (a) initial configuration, (b) interaction, (c) final state.
Volume 146, numberS
PHYSICS LETTERS A
linear gradient terms on localized states arising in a complex Ginzburg—Landau equation for a weakly inverted bifurcation. We find that these terms can greatly affect the velocity of the state and also cause
_________________________________
a / .‘.
________________________________ 0
20
40
60
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100
120
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an asymmetry of the one-particle state. In addition we have studied the interaction of counterpropagating localized states in coupled Ginzburg—Landau equations describing a weakly subcritical bifurcation and we have found (1) an annihilation of localized states, (2) transitions from subcritical to absolutely unstable and (3) soliton-like interactions for which the states after the collision have the same size and shape as before the collision. It is a pleasure to thank Guenter Ahlers for stimulating discussions. H.R.B. thanks the Deutsche Forschungsgemeinschaft for support. The work done at
-~
d
0
28 May 1990
‘
80
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120
Fig. 5. Transition subcritical to absolutely unstable (~,= 2, v= 1), (a) initial state, (b) interaction starts, (c) broadening of the state filling up the cell gradually.
the Center for Nonlinear Studies, Los Alamos National Laboratory has been performed under the auspices of the United States Department of Energy. References [1] P. Kolodner, D. Bensimon and C.M. Surko, Phys. Rev. Lett.
particle states for a destabilizing crosscoupling. In this case the interaction induces a transition from subcritical to absolutely unstable and the state eventually fills the entire sy stem We now discuss the comparison with recent experiments in binary convection [2]. As observed in these experiments it is possible to have two counterpropagating localized states. Upon interaction one of the localized states is annihilated, with the larger state surviving. The mechanism for this behavior is a stabilizing interaction bringing the amplitude of one state below the amplitude separating growth from decay, similar to that occurring in fig. 3. In this note we have investigated the effect of non-
60 (1988) 1723. G. Ahlers and D.S. Cannell, Bull. Am. Soc. 33 (1988) 2261; and to be published. [31R.J. Deissler, [4] Deisslerand J. Stat. H.R.Phys. Brand, 54Phys. (1989)1459. Lett. A 130 (1988) 293. [51R.J. Deissler, Phys. Lett. A 120 (1987) 334.
[21J. Niemela,
[6] C.M. Surko and P. Kolodner, Phys. Rev. Lett. 58 (1987) 2055.
[71 0. Thual and S. Fauve, J. Phys. (Pars) 49 (1988,) 1829. [8] H.R. Brand and R.J. Deissler, submitted for publication.
[9]H.R. Brand, P.S. Lomdahl and A.C. Newell, Phys. Lett. A 118 (1986) 67. [10] H.R. Brand, P.S. Lomdahl and A.C. Newell, Physica D 23 (1986) 345. [11] R.J. Deissler, J. Stat. Phys. 40 (1985) 371.
255
PHYSICS LETTERS A
28 May 1990
THE EFFECT OF NONLINEAR GRADIENT TERMS ON LOCALIZED STATES NEAR A WEAKLY INVERTED BIFURCATION Robert J. DEISSLER a and Helmut R. BRAND b,a
b
Centerfor Nonlinear Studies, MS-B 258, Los A/amos National Laboratory, University of California, LosA/amos, NM 87545, USA FB 7, Physik, Universität Essen, D-4300 Essen 1, FRG
Received 12 February 1990; accepted for publication 2 April 1990 Communicated by A.R. Bishop
We study the effect of nonlinear gradient terms on localized states in a complex Ginzburg—Landau equation near a weakly inverted bifurcation. We find that these terms can greatly affect the speed ofthe state as well as cause an asymmetry. We also study the interaction ofcounterpropagating localized states in the coupled equations for counterpropagating waves. A comparison with recent experiments in binary fluid convection is made.
Recently the existence of confined states in binary fluid convection in an annulus [1,21 has been reported. By confined state one means in this case that only part ofthe annulus is filled with convective rolls, whereas the rest of the cell is apparently free of convection. As first pointed out in ref. [3], the existence of such a state may be partially understood in terms of a slug [4,5], a localized structurebetween two stable regimes. In other words, the system is subcritical (with two basins of attraction) allowing for the coexistence of two different states namely convective rolls and a stable conduction state. It is the stable conduction state on either side of the rolls which confines then the azimuthal direction. One of the least understood features of the confined state in binary convection is the fact that the velocity of the envelope of the confined state is orders of magnitude smaller than the velocity of the rolls within the confined state or zero, even though the group velocity of a linear wavepacket [61 is of the same order of magnitude as the phase velocity of the rolls. In this note we investigate this problem by studying the effect of nonlinear gradient terms in the cornplex Ginzburg—Landau equation and find that the velocity of the envelope can indeed by greatly reduced by incorporating these terms. We further study —
252
the interaction of counterpropagating localized states in the framework of coupled complex Ginzburg— Landau equations. Counterpropagating states were previously studied in ref. [3] with different pararneter values. Here we use the parameter values of ref. [7], in which a single Ginzburg—Landau equation (eq. (1) below without the nonlinear gradient terms) has been studied. The new feature observed in ref. [71is the existence of localized states with a unique size and shape for a wide range of initial conditions. We note that for other parameter values chaotic localized states can exist [4,51.As in ref. [3] we find annihilation of colliding localized states for stabilizing crosscoupling and a transition from subcritical to absolutely unstable supercritical behavior for destabilizing crosscoupling. The new feature which is studied in detail in ref. [8] is the soliton-like behavior of the collision of two one-particle states which return to the same size and shape after the interaction. We stress that these one-particle states are not solitons, since the process is occurring in a strongly dissipative system and we note that this is the first example of this type of behavior in such a strongly dissipative system. The equation for traveling waves in one direction which we study first reads
Volume 146, numberS
PHYSICS LETTERS A
3~A+ vô~A XA + (Yr + iy, )A~~ +
(fir
—
~r
+ if?
1) A I — (or + 1O~) I + iA 2A~— CUr +i/L
1) IA I
I
4A
2A*
1 )A
(1)
opposite to the group velocity). opposite signs of the nonlinear gradient terms theFor one-particle state moves at a speed larger than the group velocity. For sufficiently large values the nonlinear gradient terms will cause the localized state to damp to zero.
x.
The new feature is the presence of the nonlinear gradient terms, IA 2A~and A 2A which occur to the same order as the quintic term IAI4A, as first discussed in refs. [9,10]. For our numerical computations we use fourth order spatial differencing and second order Runge—Kutta in time as described in detail in ref. [11]. Fig. 1 a shows the one-particle state first observed by Thual and Fauve [7] with a group velocity v~1. This state is moving to the right with speed 1. Fig. lb shows the resulting state under the influence of the nonlinear gradient terms. The yelocity of the particle has been reduced by a factor of about 7 for this choice of parameter values (cornpare fig. lb). In addition the state is skewed to the left as one would expect, since the nonlinear gradient terms have a greater influence on the portions with larger amplitude. For certain combinations of parameter values we can get the one-particle state to stand still. For larger values ofthe nonlinear gradient terms the one-particle state moves backwards (i.e. ~,
~
a
We now come to the comparison with recent experiments in binary fluid convection by Niemela, Ahlers and Cannell [2]. In these experiments a localized state forms, which initially has a velocity which is orders of magnitude smaller than the linear group velocity. Since a signature of the effect of nonlinear gradient terms is an asymmetry of the envelope of the localized state, more precise measurements could determine, whether there is indeed a sufficiently large asymmetry present in the experiments. In the experiment the state slows down exponentially with time [2] and eventually stops a!together. This slowing down is a feature which probably cannot be captured in the context of a cornplex Ginzburg—Landau equation without mean flow effects. We now study the collision of counterpropagating one-particle states in the coupled complex Ginzburg—Landau equations describing counterpropagating waves. For completeness we consider all the terms which arise to the order of the quintic term. In the numerical simulations some of the terms will be set t9iA+VôxA=XA+(Yr+~Yi)Axx equal to zero (compare below),
‘t)
d
~
________________________________
o
io
20
30
40
50
60
(‘er
x
~L-
b
~
0
10I
(Ar
+iA~)IA I 2A~
_(fi~+jfij)B*ABx_(~~+j)AB~,
40
50
(2)
a
___________
20 30 I~I~I~I~ X
+ii~) BI2A—
_(lLr+j~Li)A2A~_(Xr+jXi)IBl2Ax
_______________________
_____
28 May 1990
60
Fig. 1. Plot of the one-particle solution (v= I), (a) without nonlinear gradient terms, (b) including nonlinear gradient terms —1). In all the plots we have chosen the parametervaluesusedinref.[7]:~=—0.l,flr=—3,fl,=—l,y,=l, = 0, ô,= 2.75, ô~ = — 1, and we have set all the coefficients ofthe nonlinear gradient terms to zero unless specified otherwise. In additionwehavechosen eters varying from one plot ~,=l. to the Thisleaves~,andvastheparamnext.
1B—~o~B=xA+ (Yr+~yi)B~x 2B_(Or+~Oi)IBI4B +(fir+jfii)IBI ~ + +
Cur (fir
+ ~ILi )B2B~+ (Ir + ~ IA I 2B~ +i/2 1)A*BA~+ (‘~r+j1~)BAAx~.
(3)
(ArA?ZrPi
In writing down these equations, we have discarded quintic, spatially 2IAI2A, homogeneous IBI4A)crosscoupling in eqs. (2), terms (3). (like Fig.those 2 showsIBIthe collision of two counterpropa—‘
253
Volume 146, number 5
PHYSICS LETTERS A
________________
C
~I
o
28 May 1990 “1
a
I
~
20
_________________
(
P
40
60
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100
120
0
20
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a /
-~
60
80
100
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_________________
b
b
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o
20
40
60
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x Fig. 2. Complete annihilation of two one-particle solutions (‘~r=3, I ), (a) initial approach, (b) interaction, (c) one-particle states decay to zero,
gating one-particle states at successive times for stabilizing crosscoupling. During the interaction the amplitudes of the particles are sufficiently reduced to bring the amplitude below the separation between growth and decay and the particles decay giving complete annihilation. If the states are sufficiently different in size just before the interaction, ‘only one of the states is annihilated as is seen in fig. 3. If the stabilizing crosscoupling is decreased, the amplitude of the state will not be decreased as much during the interaction and the localized solutions will reemerge after the collision and return to their initial size and shape. Fig. 4 shows such an interaction with nonlinear gradient terms included. We note that for suffi ciently small group velocity with or without nonlinear gradient terms the two states can stop moving during the early stages of the interaction giving rise to a composite stationary state, and do not interpenetrate. If one were to observe experimentally such a composite stationary state, one would have in different parts of the state waves traveling to the right and to the left, respectively. Fig. 5 shows the interaction of two colliding one-
254
I
80
_________________
C
‘I
60
x
-.
0
120
x
60
80
100
120
x Fig. 3. Annihilation of a one-particle solution (~r=3,v=2), (a) initial approach, (b) interaction, (c) the surviving one-particle solution propagates to the right.
I
~--
a
H
d
~ I
0
i~i
I
20
40
60
80
100
120
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40
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80
100
120
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X
Fig. 4. Interpenetration of two nonsymmetric one-particle states (~~r2,5,v=2, 2r iI~r/2,~), (a) initial configuration, (b) interaction, (c) final state.
Volume 146, numberS
PHYSICS LETTERS A
linear gradient terms on localized states arising in a complex Ginzburg—Landau equation for a weakly inverted bifurcation. We find that these terms can greatly affect the velocity of the state and also cause
_________________________________
a / .‘.
________________________________ 0
20
40
60
80
100
120
X
b -~
o
20
40
-
60
--
80
100
120
Ui 20
‘
40
60
an asymmetry of the one-particle state. In addition we have studied the interaction of counterpropagating localized states in coupled Ginzburg—Landau equations describing a weakly subcritical bifurcation and we have found (1) an annihilation of localized states, (2) transitions from subcritical to absolutely unstable and (3) soliton-like interactions for which the states after the collision have the same size and shape as before the collision. It is a pleasure to thank Guenter Ahlers for stimulating discussions. H.R.B. thanks the Deutsche Forschungsgemeinschaft for support. The work done at
-~
d
0
28 May 1990
‘
80
‘
120
Fig. 5. Transition subcritical to absolutely unstable (~,= 2, v= 1), (a) initial state, (b) interaction starts, (c) broadening of the state filling up the cell gradually.
the Center for Nonlinear Studies, Los Alamos National Laboratory has been performed under the auspices of the United States Department of Energy. References [1] P. Kolodner, D. Bensimon and C.M. Surko, Phys. Rev. Lett.
particle states for a destabilizing crosscoupling. In this case the interaction induces a transition from subcritical to absolutely unstable and the state eventually fills the entire sy stem We now discuss the comparison with recent experiments in binary convection [2]. As observed in these experiments it is possible to have two counterpropagating localized states. Upon interaction one of the localized states is annihilated, with the larger state surviving. The mechanism for this behavior is a stabilizing interaction bringing the amplitude of one state below the amplitude separating growth from decay, similar to that occurring in fig. 3. In this note we have investigated the effect of non-
60 (1988) 1723. G. Ahlers and D.S. Cannell, Bull. Am. Soc. 33 (1988) 2261; and to be published. [31R.J. Deissler, [4] Deisslerand J. Stat. H.R.Phys. Brand, 54Phys. (1989)1459. Lett. A 130 (1988) 293. [51R.J. Deissler, Phys. Lett. A 120 (1987) 334.
[21J. Niemela,
[6] C.M. Surko and P. Kolodner, Phys. Rev. Lett. 58 (1987) 2055.
[71 0. Thual and S. Fauve, J. Phys. (Pars) 49 (1988,) 1829. [8] H.R. Brand and R.J. Deissler, submitted for publication.
[9]H.R. Brand, P.S. Lomdahl and A.C. Newell, Phys. Lett. A 118 (1986) 67. [10] H.R. Brand, P.S. Lomdahl and A.C. Newell, Physica D 23 (1986) 345. [11] R.J. Deissler, J. Stat. Phys. 40 (1985) 371.
255