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Multistable structure in a Korteweg-de Vries system driven by a solitary wave
27 April 1998
PHYSICS
ELSEVIER
Physics Letters A 241 (1998)
LETTERS
A
159-162
Multistable structure in a Korteweg-de Vries system driven by a solitary wave Luqun Zhou a, Kaifen He b,a, Z.Q. Huang a a Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China b CCAST (Worth Laboratory), PO. Box 8730, Beijing IooO80, China Received 28 July 1997; revised manuscript received 30 October 1997; accepted Communicated by A.R. Bishop
for publication
26 January
1998
Abstract
The dynamical behavior of a Korteweg-cle Vries (KdV) system driven by a solitary wave is studied. A hierarchy of hystereses, i.e. multistability, for the steady wave momenta is found in the system. Winding number bifurcation occurs for the steady wave solution on every hysteresis. @ 1998 Elsevier Science B.V. PACS: 03.40.K
Multistabilities are observed in many physical systems, such as the multiatomic optical systems [ l] and discharge plasmas [ 2,3]. In the present Letter we investigate multistability with a forced KdV (fKdV) system. The equation we consider is 4, + &xxx + fMx
= -r9
- EF(X - vt),
(1)
where a and f are given constants, y is a damping coefficient, E is the driving amplitude, and CpXdenotes &j/ax, 4r is &b/at, &,,, is J3q5/6’x3. F(x - vt) is the driving force. Many interesting phenomena are observed in fKdV equations. For example, Cox and Mortell [4] and Rozmus et al. [ 51 investigated an fKdV equation driven by a sinusoidal wave and found steady and cyclic states. In Ref. [6] for the nonlinear drift wave and KdV equation driven by sinusoidal waves, it is found, respectively, that the “wave energy” corresponding to a steady state of the system traces a hysteretic curve when the driving amplitude is cyclically varied. In Refs. [7,8], interactions of a solitary 0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00088-7
wave of the unperturbed KdV equation (y = 0) with a static soliton type (sech* ( X) ) force and a Gaussian shape (exp( --x2) ) force are discussed, respectively. Phenomena such as passage, repulsion and trapping have been studied. Grimshaw and Tian [ 9,101 and Malkov [ 111 have studied analytically and numerically the KdV-Burgers equation driven by a sinusoidal wave. Periodic and chaotic behavior is observed. In the present Letter, we use a particle-like solitary wave as the external driving force to investigate its effect on a nonlinear system. An example of such solitary waves has the form of sn*( k, x) . We adopt this form with a vanishing constant term, i.e. F(x)
= ~A,cos(n/3x), fl=l
(2)
where L = an-/P(k) is the driving period. When k + 0, it becomes a sinusoidal wave. In general, Eq. (2) is different from a sinusoidal wave. Even if v, E are fixed, the parameter k can still be varied
160
L. Zhou
et d/Physics
Letters
0 -4 0
2
4
6
8
10
12
14
159-162
16
X le+l
1
le+O
3
le-I
3
le-2
;
I
j
@s(5) = 21 c, cos( n/X) n=l
-(aB3n3 0
4
2
6
8
10
II
Fig. 1. The driving Fourier amplitudes.
force F(x).
(a) The spatial forms; (b) the
to simulate the different shapes of a solitary driving wave. In the limit k --f 1, it becomes a soliton. As is well known, a soliton can be regarded as a particle. Since the solitary wave has a projection in every cosine component, it can drive all modes of the system simultaneously, while a sinusoidal wave drives only one mode directly. The energy transferring among the modes will be more complicated. In this case the system’s solutions can still be well organized into a steady state. In the following, k = 0.999 and L = 7.8 is used. The spatial form of the driving wave and its Fourier amplitudes, A,,, are shown in Fig. 1. Ifweput[=x-vt,r=t,Eq. (1) becomes cb, + a&&
- r+s + .fMJf + r+ + EF(5)
The wave momentum
= 0,
(3)
of the system in one period L,
L
4*(x,
= ;
J’
+ sn sin( nP5)
I,
(5)
which has the same period and velocity as the driving force. Setting &#J/& = 0 in Eq. (3) and substituting Eqs. (2) and (5) into it, one derives a set of equations for {c,, sn} (n= 1,2, . . . , co), as follows,
1e-3
E(r)
(1998)
pending on the values of E. A constant Es corresponds to a steady-state solution c#Q({) of the system, where C& satisfies a&/& = 0. Further investigation shows that the momentum Es as a function of E has a multistable structure. With this simulating method, however, one can only find stable solutions. In order to search for the unstable ones as well, we set
-2
4
A 241
t) dx,
(4)
0
is a constant of motion in the absence of an external force and of damping. We arbitrarily select a = - 1.O, f = -6.0, y = 0.1, and u = 0.1. The pseudospectrum method [ 121 is used in simulating Eq. ( 1) . It is found that E(t) asymptotically tends to a constant, E,,de-
+ uPn)s,+yc, +0.5f@1 C
qsp lfl'=ll
- c/q,) + EA, = 0, + 0.5fPnC (sIcl' I-l'=n
+ O.Sf@z C (-crcp- slsp) = 0. I-l'=ll
(6)
For given E, one can get the solutions {c,, s,}(n= 1,2,... , N) numerically. Here N = 50 modes are used. Then the steady-state solution 4, (5) of Eq. (3) (&#JJc% = 0) and the corresponding momentum Es can be obtained. The momenta Es versus driving amplitude E are plotted in Fig. 2 for o = 0.1. One can see that the variation of Es with E is characteristic of a hierarchy of hystereses. In other words, the system displays multistabilities. In the plot four stairs of hystereses are presented. It is interesting that every stair of hystereses is characteristic for a winding number which is defined as the number of the trajectory winding in phase space (cf. Fig. 3, right column) [ 131. For example, in the 1st stair the winding number is one,while in the 2nd stair the winding number is two, and so on. In Figs. 3, we give three typical examples of the states in the 2nd, 3rd and 4th stairs, respectively. The positions of the states are denoted in Fig. 2 by triangles. The left column in Figs. 3 shows the spatial forms, and the right one is the corresponding phase-space plots. One can
L. Zhou et al./Physics
Letters A 241 (19981 159-162
(a)
(a)
1
2nd (b)
-4 18
0.01
1.oo
0.10
10.00
E Fig. 2. Steady momentum Es versus the driving logarithmic coordinates for u = 0.1.
l!!b
6
amplitude
l
in
see that, when stepping up to a higher stair, an additional winding is developed in the phase-space plot. This phenomenon can be called “winding number bifurcation” or “adding”. Now, let us focus on one stair, say the 3rd stair of the hystereses, for instance. In this case, with varying E along the curve through the stair, a “new” hump will appear and quickly grows while its width becomes narrower and narrower, but the amplitudes of the two “old” peaks change only slowly. When approaching the 4th stair, the amplitude of this hump becomes comparable to the two old peaks and all of them are compressed to the left side; in the mean time another new hump appear on the right side (see the left column in Fig. 3). Winding number bifurcations are observed in different physical systems [ 141. In the present Letter, we show for the first time that a hierarchy of hystereses or multistability appears in a nonlinear system driven by a solitary wave and winding number bifurcation can be closely related to it. Different values of k, i.e. different shapes of the solitary wave, are also tested, and multistabilities can be found as well. It is found that the larger k, the smaller the widths between two turning points of the stairs in the hystereses are. When k approaches 1, this tendency is more obvious. The winding number adding here is different from the so-called period adding phenomenon. Period adding means a sequence of periodic solutions (L + 2L +
0
(1
?
4
h
II
5 Fig. 3. Left column: the steady periodic solution with the same period as the driving force; right column: the phase-space trajectory corresponding to the left column, with parameter (1 = 0.1 and (a) l = 0.7, (b) l = 6.0, (c) E = 21.0.
[9,15],andwhenN-+m,chaos . . . +NL-+...) will emerge. Winding number adding, however, shows that a new hump develops and its width is different from that of the old peaks. It is interesting to investigate whether chaos would occur when the number of the peaks is large enough. This project is supported by the National Nature Science Foundation (No. 19675006), the National Basic Research Project “Nonlinear Science” and the Education Committee of the State Council through the Foundation of Doctoral Training.
References [ 11 S.Ya. Kilin, T.V. Krinitskaga, J. Exp. Theor. Phys. 76 ( 1993) 573. [2] Y. Jiang, H. Wang, C. Yu, Chin. Phys. Lett. 5 ( 1988) 201. 13) H. Sun, L. Ma, L. Wang, Phys. Rev. E 51 ( 199.5) 3475. [4] E.A. Cox, M.P. Nortell, J. Fluid Mech. 162 (1986) 99. [ 51 W. Rozmus, M. Casanova, D. Pesme, A. Heron, J.-C. Adam, Phys. Fluids B 4 ( 1992) 576.
162
L. Zhou et ai. /Physics
161 K. He, A. Salat, Phys. Lett. A 132 (1988) 175. 171 R. Grimshaw, E. Pelinovsky, X. Tiau, Physica D 77 405. 181 L.C. Redekopp. Z. You, Phys. Rev. Lett. 74 (1995) 191 R. Grimshaw, X. Tian, Proc. R. Sot. London A 445 1. 1101 X. Tian, R. Grimshaw, Int. J. Bifurcation Chaos 5 1221.
( 1994) 5158. (1994) (1995)
Letters A 241 (1998) 159-162
[ 111M.A. Malkov, Physica D 95 (1996) 62. 1121 S.A. Orszag, Studies Appl. Mathematics L (1971) 293. [ 131 U. Parlitz, W. Lauterbom, Phys. Lett. A 107 (1985) 35 1, [ 141 K. He, A. Salat, Acta. Phys. Sin. 39 (1990) 204 (in Chinese). [ 15 I J.H. Peng, W.L. Zhou, X. Zhang, lnt. Workshop on Statistical Physics (BNN Press, 1992) p. 328.
PHYSICS
ELSEVIER
Physics Letters A 241 (1998)
LETTERS
A
159-162
Multistable structure in a Korteweg-de Vries system driven by a solitary wave Luqun Zhou a, Kaifen He b,a, Z.Q. Huang a a Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China b CCAST (Worth Laboratory), PO. Box 8730, Beijing IooO80, China Received 28 July 1997; revised manuscript received 30 October 1997; accepted Communicated by A.R. Bishop
for publication
26 January
1998
Abstract
The dynamical behavior of a Korteweg-cle Vries (KdV) system driven by a solitary wave is studied. A hierarchy of hystereses, i.e. multistability, for the steady wave momenta is found in the system. Winding number bifurcation occurs for the steady wave solution on every hysteresis. @ 1998 Elsevier Science B.V. PACS: 03.40.K
Multistabilities are observed in many physical systems, such as the multiatomic optical systems [ l] and discharge plasmas [ 2,3]. In the present Letter we investigate multistability with a forced KdV (fKdV) system. The equation we consider is 4, + &xxx + fMx
= -r9
- EF(X - vt),
(1)
where a and f are given constants, y is a damping coefficient, E is the driving amplitude, and CpXdenotes &j/ax, 4r is &b/at, &,,, is J3q5/6’x3. F(x - vt) is the driving force. Many interesting phenomena are observed in fKdV equations. For example, Cox and Mortell [4] and Rozmus et al. [ 51 investigated an fKdV equation driven by a sinusoidal wave and found steady and cyclic states. In Ref. [6] for the nonlinear drift wave and KdV equation driven by sinusoidal waves, it is found, respectively, that the “wave energy” corresponding to a steady state of the system traces a hysteretic curve when the driving amplitude is cyclically varied. In Refs. [7,8], interactions of a solitary 0375-9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00088-7
wave of the unperturbed KdV equation (y = 0) with a static soliton type (sech* ( X) ) force and a Gaussian shape (exp( --x2) ) force are discussed, respectively. Phenomena such as passage, repulsion and trapping have been studied. Grimshaw and Tian [ 9,101 and Malkov [ 111 have studied analytically and numerically the KdV-Burgers equation driven by a sinusoidal wave. Periodic and chaotic behavior is observed. In the present Letter, we use a particle-like solitary wave as the external driving force to investigate its effect on a nonlinear system. An example of such solitary waves has the form of sn*( k, x) . We adopt this form with a vanishing constant term, i.e. F(x)
= ~A,cos(n/3x), fl=l
(2)
where L = an-/P(k) is the driving period. When k + 0, it becomes a sinusoidal wave. In general, Eq. (2) is different from a sinusoidal wave. Even if v, E are fixed, the parameter k can still be varied
160
L. Zhou
et d/Physics
Letters
0 -4 0
2
4
6
8
10
12
14
159-162
16
X le+l
1
le+O
3
le-I
3
le-2
;
I
j
@s(5) = 21 c, cos( n/X) n=l
-(aB3n3 0
4
2
6
8
10
II
Fig. 1. The driving Fourier amplitudes.
force F(x).
(a) The spatial forms; (b) the
to simulate the different shapes of a solitary driving wave. In the limit k --f 1, it becomes a soliton. As is well known, a soliton can be regarded as a particle. Since the solitary wave has a projection in every cosine component, it can drive all modes of the system simultaneously, while a sinusoidal wave drives only one mode directly. The energy transferring among the modes will be more complicated. In this case the system’s solutions can still be well organized into a steady state. In the following, k = 0.999 and L = 7.8 is used. The spatial form of the driving wave and its Fourier amplitudes, A,,, are shown in Fig. 1. Ifweput[=x-vt,r=t,Eq. (1) becomes cb, + a&&
- r+s + .fMJf + r+ + EF(5)
The wave momentum
= 0,
(3)
of the system in one period L,
L
4*(x,
= ;
J’
+ sn sin( nP5)
I,
(5)
which has the same period and velocity as the driving force. Setting &#J/& = 0 in Eq. (3) and substituting Eqs. (2) and (5) into it, one derives a set of equations for {c,, sn} (n= 1,2, . . . , co), as follows,
1e-3
E(r)
(1998)
pending on the values of E. A constant Es corresponds to a steady-state solution c#Q({) of the system, where C& satisfies a&/& = 0. Further investigation shows that the momentum Es as a function of E has a multistable structure. With this simulating method, however, one can only find stable solutions. In order to search for the unstable ones as well, we set
-2
4
A 241
t) dx,
(4)
0
is a constant of motion in the absence of an external force and of damping. We arbitrarily select a = - 1.O, f = -6.0, y = 0.1, and u = 0.1. The pseudospectrum method [ 121 is used in simulating Eq. ( 1) . It is found that E(t) asymptotically tends to a constant, E,,de-
+ uPn)s,+yc, +0.5f@1 C
qsp lfl'=ll
- c/q,) + EA, = 0, + 0.5fPnC (sIcl' I-l'=n
+ O.Sf@z C (-crcp- slsp) = 0. I-l'=ll
(6)
For given E, one can get the solutions {c,, s,}(n= 1,2,... , N) numerically. Here N = 50 modes are used. Then the steady-state solution 4, (5) of Eq. (3) (&#JJc% = 0) and the corresponding momentum Es can be obtained. The momenta Es versus driving amplitude E are plotted in Fig. 2 for o = 0.1. One can see that the variation of Es with E is characteristic of a hierarchy of hystereses. In other words, the system displays multistabilities. In the plot four stairs of hystereses are presented. It is interesting that every stair of hystereses is characteristic for a winding number which is defined as the number of the trajectory winding in phase space (cf. Fig. 3, right column) [ 131. For example, in the 1st stair the winding number is one,while in the 2nd stair the winding number is two, and so on. In Figs. 3, we give three typical examples of the states in the 2nd, 3rd and 4th stairs, respectively. The positions of the states are denoted in Fig. 2 by triangles. The left column in Figs. 3 shows the spatial forms, and the right one is the corresponding phase-space plots. One can
L. Zhou et al./Physics
Letters A 241 (19981 159-162
(a)
(a)
1
2nd (b)
-4 18
0.01
1.oo
0.10
10.00
E Fig. 2. Steady momentum Es versus the driving logarithmic coordinates for u = 0.1.
l!!b
6
amplitude
l
in
see that, when stepping up to a higher stair, an additional winding is developed in the phase-space plot. This phenomenon can be called “winding number bifurcation” or “adding”. Now, let us focus on one stair, say the 3rd stair of the hystereses, for instance. In this case, with varying E along the curve through the stair, a “new” hump will appear and quickly grows while its width becomes narrower and narrower, but the amplitudes of the two “old” peaks change only slowly. When approaching the 4th stair, the amplitude of this hump becomes comparable to the two old peaks and all of them are compressed to the left side; in the mean time another new hump appear on the right side (see the left column in Fig. 3). Winding number bifurcations are observed in different physical systems [ 141. In the present Letter, we show for the first time that a hierarchy of hystereses or multistability appears in a nonlinear system driven by a solitary wave and winding number bifurcation can be closely related to it. Different values of k, i.e. different shapes of the solitary wave, are also tested, and multistabilities can be found as well. It is found that the larger k, the smaller the widths between two turning points of the stairs in the hystereses are. When k approaches 1, this tendency is more obvious. The winding number adding here is different from the so-called period adding phenomenon. Period adding means a sequence of periodic solutions (L + 2L +
0
(1
?
4
h
II
5 Fig. 3. Left column: the steady periodic solution with the same period as the driving force; right column: the phase-space trajectory corresponding to the left column, with parameter (1 = 0.1 and (a) l = 0.7, (b) l = 6.0, (c) E = 21.0.
[9,15],andwhenN-+m,chaos . . . +NL-+...) will emerge. Winding number adding, however, shows that a new hump develops and its width is different from that of the old peaks. It is interesting to investigate whether chaos would occur when the number of the peaks is large enough. This project is supported by the National Nature Science Foundation (No. 19675006), the National Basic Research Project “Nonlinear Science” and the Education Committee of the State Council through the Foundation of Doctoral Training.
References [ 11 S.Ya. Kilin, T.V. Krinitskaga, J. Exp. Theor. Phys. 76 ( 1993) 573. [2] Y. Jiang, H. Wang, C. Yu, Chin. Phys. Lett. 5 ( 1988) 201. 13) H. Sun, L. Ma, L. Wang, Phys. Rev. E 51 ( 199.5) 3475. [4] E.A. Cox, M.P. Nortell, J. Fluid Mech. 162 (1986) 99. [ 51 W. Rozmus, M. Casanova, D. Pesme, A. Heron, J.-C. Adam, Phys. Fluids B 4 ( 1992) 576.
162
L. Zhou et ai. /Physics
161 K. He, A. Salat, Phys. Lett. A 132 (1988) 175. 171 R. Grimshaw, E. Pelinovsky, X. Tiau, Physica D 77 405. 181 L.C. Redekopp. Z. You, Phys. Rev. Lett. 74 (1995) 191 R. Grimshaw, X. Tian, Proc. R. Sot. London A 445 1. 1101 X. Tian, R. Grimshaw, Int. J. Bifurcation Chaos 5 1221.
( 1994) 5158. (1994) (1995)
Letters A 241 (1998) 159-162
[ 111M.A. Malkov, Physica D 95 (1996) 62. 1121 S.A. Orszag, Studies Appl. Mathematics L (1971) 293. [ 131 U. Parlitz, W. Lauterbom, Phys. Lett. A 107 (1985) 35 1, [ 141 K. He, A. Salat, Acta. Phys. Sin. 39 (1990) 204 (in Chinese). [ 15 I J.H. Peng, W.L. Zhou, X. Zhang, lnt. Workshop on Statistical Physics (BNN Press, 1992) p. 328.