- Email: [email protected]
Stationary solitons and stabilization of the collapse described by KdV-type equations with high nonlinearities and dispersion
8 May 1995
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 200 (1995) 423-428
Stationary solitons and stabilization of the collapse described by KdV-type equations with high nonlinearities and dispersion V.I. Karpman a71,J.-M. Vanden-Broeck
b
a Center for Plasma Physics and Nonlinear Dynamics, Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel b Department of Mathematics
and Center for the Mathematical Sciences, University of Wisconsin-Madison,
Received 16 December 1994; accepted for publication Communicated by A.R. Bishop
9 February
WI 53706, USA
1995
Abstract Solitons of fifth order KdV-type equations with high nonlinearities are investigated numerically by finite difference schemes. It is shown that the soliton asymptotics may be both monotonically and oscillatory decaying, in agreement with analytical predictions. In the absence of higher order dispersion (i.e. without the fifth order derivative in the equation), solitons with sufficiently high nonlinearities in the equations are shown to be unstable with respect to collapse-type instabilities, which agrees with the general theory of collapse. On the other hand, the instabilities have not been detected in the presence of fifth order dispersion, which shows that the latter plays a stabilizing role.
1. Introduction.
Basic equations
We study steady solitary waves described fifth-order KdV-type equations du du d3U ~+“Up~+P~+Y~=O’ u=u(x-@cut),
u+O(x--,
by the
a5u (la) km),
(lb) with integer p > 0 and crp > 0. Then, without loss of generality, one can assume that CY= /3 = 1. (The case CYP< 0 will be considered in a separate paper by Karpman.) The last term in (la) describes higher order dispersive effects and may have an important influence, in particular, on the properties of solitons. For p = 1, Eq. (la) is the fifth order KdV equation
which was considered, for instance in Refs. [l-6]. It has applications in fluid mechanics (e.g., shallow water waves with surface tension [1,2]), plasma physics [ll, etc. The soliton solutions (lb) which it is convenient to write in the form u=
]Cz]““I;(&
F-0
(6+
5‘= ]a]“2(X-at), *to),
y
a>0
(3)
or y>O,
a
l>1/2,
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00173-5
(4)
where E = I ya I l’*. As it is well known,
1 E-mail: [email protected].
(2)
with p = 1 can exist only if [4,5]
y>O,
a>O,
(5) a > 0 when y = 0. For (6)
424
V.I. Karpman, J.-M. VandL,n-Broeck/Physics
the solitons are radiating, if p < 4 [4,6]; in this case the soliton velocity a depends on time, and a = a(t) is a decreasing function found in Refs. [4,6] for E -=c 1, when the soliton lifetime is large. For p = 2, Eq. (la) is the fifth order MKdV equation which may be of interest also in fluid mechanics, plasma physics and other fields where the “classical” MKdV equation has applications. The radiation and respective attenuation of solitons in the case p = 2 have been studied in Ref. [6]. Stationary solitons in this case, discussed analytically in Ref. [5], will be found numerically below, along with those for p & 4. Eq. (la> with p > 4 has a considerable theoretical interest in connection with the general problem of collapse of nonlinear waves. Indeed, it is known that the soliton solutions of (la) for y = 0 that can be written as u,(6)=
[~IaI(P+l)(P+2)]1'Zp
Xsech2/J’(ip,$), (7) are unstable with respect to collapse-type instabilities if p 2 4 [7,8]. As for y # 0, the instability remains in case (6), i.e. for radiating solitons [6]. In this paper, we demonstrate numerically that there are in fact stationary solutions of (la), satisfying (lb) and (3) with p equal to 2, 3, 4, 5 and 6. In case (4), the solitons are found for p = 3, 5 (in addition to p = 1, found in Refs. [1,2]). As for even p, one can show analytically that solitons satisfying (1) and (4) do not exist if sgn(a) = sgn( p) [9]. Along with the existence of solitons, satisfying (l), we also investigate (numerically) their stability. The results, presented below, do not show any signs of instability for cases (3) and (4). From this it follows that the fifth order dispersion probably stabilizes the collapse instability in these cases. Now let us consider basic equations in scaled form. Substituting (2) into (la), we have after one integration the ordinary differential equation defining the stationary solitons sgn(y)e2F”(
e) +F”(
5) + - l
P+l
Fp+‘(
Zj)
- sgn( a)F( <) = 0. (8) The advantage of scaling (2) is that now (8) contains only one parameter, namely E. The signs of y and a have to be taken according to (3) or (4).
Letters A 200 (1995) 423-428
We first find the asymptotic behavior of the solitons as t+ fa. Neglecting the nonlinear term in (S), we have u(S)
=C,
(5+
cxp(r&5)
+C,
exp(T&t)
*m>,
(9)
where, for the case (3), k,,, = &[(I
+ 2~)~‘~ T (1 - 2~)“~]
(10)
and for the case (41, k,,,=
A[(Zt--
1)“2ki(2~+
I)“~].
(11)
Thus, in case (3), the asymptotic behavior as 5 -+ + ~0 must be monotonic when E < 3 and oscillatory when E > 3. In case (41, the asymptotics is always oscillatory. These conclusions, naturally, do not depend on p. Solutions with this behavior when p = 1 were obtained numerically by Kawahara [l]. We shall show that our solutions for integer p > 1 also behave in this way. (In particular, this also happens for p > 4.1 To investigate the soliton stability, we shall use the same scaling as in (2), where now a is the initial soliton velocity (for t > 0, the soliton parameters may change due to a possible instability.) Thus we write CL= lal”PF(l, I al%,
l=
Substituting G
+Fpg
r),
(12)
7= I a13’2t.
(13)
(12) and (13) into (la), we obtain d3 + ;)iF+sgn(y)t2FF=0,
d5
(14) where E is defined in (5). For stationary
solitons
F(i,~)=F(!:-sgn(a)7)=F(5).
(15) For the investigations of the soliton stability, one can also use another nonstationary equation, namely, one for the function F, expressed through the variables 5 and r, i.e. F( 5, 7). Evidently, this function satisfies the equation dF
3
5
~--sgn(u)~+FP~~+sgn(y)s2~ = 0
(16)
V.I. Karpman, J.-M. Vanden-Broeck/Physics
and describes the evolution in the frame of the initial soliton (i.e. in the frame moving with velocity a). The efficiency of this equation was demonstrated in the study of soliton radiation [6]. For the study of soliton stability it is sometimes convenient to write (16) in the form
z +f
4
sgn( y) e2$ i
1 + pFP+’ p+l
2
+$-sgn(n)r
(17)
=O. 1
Here, the expression in the large parentheses cides with the left-hand side of (8).
2. Numerical 2.1. Stationary
coin-
procedures solutions
Stationary solutions can be obtained by solving either (8) or Eq. (16) without the time derivative term, i.e., sgn( y)
l2F”“’ + F”’ + FPF’ - sgn( a)F’ = 0,
-+(N-
l)e+
(I-
l)e,
I=l,...,N (19)
and the corresponding
unknowns (20)
FI = F( 51).
Here e is the interval of discretization. We approximate the first, third and fifth derivatives in (18) by seven-point centered difference formulas. For the end points we assume that F, = 0 for I < 0 and I > N. By satisfying (18) at the mesh points (191, we obtain N nonlinear algebraic equations for the N unknowns F1 for I = 1,. . . , N. This system is solved by Newton’s method. 2.2. Time dependent
first introduce ll;=
the time step A and the mesh points
-+(M-l)e+(Z-l)e,
I=1
,...,
M. (21)
Here e is the interval of discretization. The unknowns are F( &;, t). As in the previous section we approximate the derivatives with respect to t by seven-point centered difference formulas. We then start with initial data, i.e. values of F( &, 0) and compute the values of F( &, t) for t > 0 by a fourth order Runge-Kutta method. For a given value of e, the time step A has to be small enough to ensure numerical stability. Also e needs to be small enough for the spatial derivatives to be accurately approximated by the finite difference formulas. Most of the computations in this paper were done with e = 0.13 and A = 0.000016. We checked that all the results presented are accurate at least to graphical accuracy.
3. Numerical
results
(18)
where the primes denote full derivatives with respect to 5. After one integration (18) reduces to (8). Here, we use Eq. (18). First we introduce the mesh points E,=
425
Letters A 200 (1995) 423-428
solutions
In the present paper we solve Eq. (14). To compute time dependent solutions of this equation, we
We used the scheme of Section 2.1 to compute stationary solutions. We first look for stationary solutions satisfying (18). Such solutions were computed before by Kawahara [l] and Hunter and Vanden-Broeck [2] when p = 1. We used them as the initial guess to compute solutions with p > 1. Solutions for p = 3, E = 0.72 and p = 5, E = 2.82 are shown in Figs. la and lb. In Fig. la, the oscillations in the far field appear clearly. The distance between succesive maxima and minima is seen to be in good agreement with the value 2E?r (26 + I)‘/2
= 2.9
predicted by (11). In Fig. lb, the oscillations in the far field are smaller but are still visible on the scale of the graph. Next we look for solutions satisfying (3). We used the exact solution (7) for y = 0 as an initial guess to compute solutions with y # 0. Our numerical results showed that solitons could only be obtained for y < 0. For y > 0, the scheme does not converge as N increases. This agrees with conditions
V.I. Karpman, J.-M. Vanden-Broeck/Physics
I -15
-10
-5
0
5
10
15
Fig. 1. Solutions satisfying
bl
20
25
Letters A 200 (1995) 423-42X
-18
’ -,DD
1 -80
-60
-40
-20
0
20
40
60
80
100
(4) for p = 3 and E = 0.72 (a) and for p = 5 and E = 2.82 (b).
1
Fig. 2. Solutions satisfying (3) for p = 2 and E = 0.66 (a), p = 4 and E = 0.72 (b), p = 5 and E = 0.24 (c) and p = 6 and E = 0.73 Cd). The solid curves are the stationary solutions. The broken curves arc the solutions obtained from (14) by marching in time. Before starting the marching in time, the initial soliton profile was multiplied by a factor 6 = 1.0001 in (a), cc>, Cd) and by a factor 6 = 1.001 in (b).
V.I. Karpman, J.-M. Vanden-Broeck/Physics
Letters A 200 (1995) 423-428
427
1
:_I_: -15
-14
-13
-12
-11
-10
-9
-8
-7
6
-2e-06’-15
Fig. 3. Portions of the tail of the solid curve of Fig. 2d on an expanded
-14.5
-14
s
-13 5
_;3
~,2.5
vertical scale (a) and on a further expanded scale (b).
25,
(a)
4,
Fig. 4. Solutions for y = 0 and p = 2 (a), p = 4 (b), p = 5 (c) and p = 6 Cd). The solid curves correspond to the solution (7). The broken curves are the solutions obtained by marching in time. Before starting the marching in time, the initial profile was multiplied by a factor 6 = 1.0001 in (a), (c), (d) and by a factor 6 = 1.001 in (b>.
428
V.I. Karpman, J.-M. Vanden-Broeck/Phvsics
(3). Typical solutions for p = 2, 4, 5 and 6 are shown by the solid curves in Figs. 2a-2d. All the solutions of Fig. 2 appear to be flat in the far field. However, when the vertical scale is expanded, oscillations appear in the far field when E > 0.5. This is illustrated in Fig. 3. The solid curve of Fig. 2d appears to be flat between - 15 and - 6. However, when the vertical scale is expanded a minimum appears at about x = - 7 (see Fig. 3a). The solution of Fig. 3a appears to be flat between - 15 and - 12.5 but when the vertical scale is further expanded a maximum appears at about x = - 13.7 (see Fig. 3b). This illustrates that the asymptotic behavior is oscillatory when E > 0.5. The distance between successive maxima and minima is in good agreement with the value 2nlZ (26 _ 1)1/Z = 6.7. On the other hand, the tail of the solid curve of Fig. 2c is monotonic even on an expanded scale, in accordance with the asymptotic behavior discussed at the end of Section 1. Finally we look at the stability of the solitons satisfying (14) with y < 0, in connection with the collapse instability. We use the stationary solutions computed by the scheme of Section 2.1 as an initial condition in the scheme of Section 2.2 and we choose M = N. In order to trigger possible instabilities, we multiply all the values of F by a factor 6 > 1, but very close to 1, before starting the marching in time. In most of the calculations we chose S = 1.0001. Typical solutions for y = 0 are shown in Figs. 4a-4d. The solution with p = 2 is stable (see Fig. 4a), and those for p = 4, 5, 6 are unstable (see Figs. 4b-4d). This is in accordance with the results of Refs. [7,8]. On the other hand, no instabilities appear in our calculations with y < 0 and p = 2, 3, 4, 5, 6. In these calculations we also multiplied the initial soliton profile by a factor S > 1 but very close to 1, before starting the marching in time. Within our accuracy, the corresponding solutions of (14) are the solitons shown in Fig. 2. Therefore the inclusion of fifth order dispersion plays a stabilizing role. Our results do not prove that the solutions are stable at all times. On the other hand, they show that the instabilities can be avoided over a finite interval of time by choosing E sufficiently large.
Letters A 200 (199.5) 423-428
4. Conclusions We have demonstrated numerically that Eq. (la) at p > 1 has indeed soliton solutions satisfying conditions (3) and (4) in agreement with the predictions of Ref. [5]. Similar to the case p = 1, the soliton asymptotic behavior at large 15 1may be both monotonic and oscillatory, in accordance with (10) and (11). We have shown numerically that the fifth order derivative term in (la) is of critical importance for the soliton stability at sufficient high p. At y = 0 and p 2 4 the soliton solutions are unstable, in agreement with the theory of collapse instability [7,8]. On the other hand, no instability has been detected in our calculations at y < 0. This agrees with an analytical investigation of the stability of solitons of Eq. (la) [lo]. Finally, it should be mentioned that the results obtained can be, in principle, checked experimentally by modelling Eq. (la) in electronic transmission line
[ill. Acknowledgement The work of the second author was supported in part by the National Science Foundation, as well as the Lady Davis Foundation. He gratefully acknowledges the hospitality of the Hebrew University of Jerusalem, Tel Aviv University and Technion.
References [I] T. Kawahara, J. Phys. Sot. Japan 33 (1972) 260. [2] J.K. Hunter and J.-M. Vanden-Broeck, J. Fluid Mech. 134 (1983) 205. [3] Y. Pomeau, A. Ramani and B. Grammaticos, Physica D 31 (1988) 127. [4] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. [5] V.I. Karpman, Phys. Lett. A 186 (1994) 300. [6] V.I. Karpman, Phys. Lett. A 186 (1994) 303. [7] E.A. Kuznetsov, A.M. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103. [8] J.J. Rasmussen and K. Rypdal, Phys. Ser. 33 (1986) 481. [9] V.I. Karpman, to be published. [lOI V.I. Karpman, to be published. [ll] R.K. Dodd, J.C. Eilbeck, J.D. Gibbons and H.C. Morris, Solitons and nonlinear wave equations (Academic Press, New York, 1984) ch. 5.5.
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 200 (1995) 423-428
Stationary solitons and stabilization of the collapse described by KdV-type equations with high nonlinearities and dispersion V.I. Karpman a71,J.-M. Vanden-Broeck
b
a Center for Plasma Physics and Nonlinear Dynamics, Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel b Department of Mathematics
and Center for the Mathematical Sciences, University of Wisconsin-Madison,
Received 16 December 1994; accepted for publication Communicated by A.R. Bishop
9 February
WI 53706, USA
1995
Abstract Solitons of fifth order KdV-type equations with high nonlinearities are investigated numerically by finite difference schemes. It is shown that the soliton asymptotics may be both monotonically and oscillatory decaying, in agreement with analytical predictions. In the absence of higher order dispersion (i.e. without the fifth order derivative in the equation), solitons with sufficiently high nonlinearities in the equations are shown to be unstable with respect to collapse-type instabilities, which agrees with the general theory of collapse. On the other hand, the instabilities have not been detected in the presence of fifth order dispersion, which shows that the latter plays a stabilizing role.
1. Introduction.
Basic equations
We study steady solitary waves described fifth-order KdV-type equations du du d3U ~+“Up~+P~+Y~=O’ u=u(x-@cut),
u+O(x--,
by the
a5u (la) km),
(lb) with integer p > 0 and crp > 0. Then, without loss of generality, one can assume that CY= /3 = 1. (The case CYP< 0 will be considered in a separate paper by Karpman.) The last term in (la) describes higher order dispersive effects and may have an important influence, in particular, on the properties of solitons. For p = 1, Eq. (la) is the fifth order KdV equation
which was considered, for instance in Refs. [l-6]. It has applications in fluid mechanics (e.g., shallow water waves with surface tension [1,2]), plasma physics [ll, etc. The soliton solutions (lb) which it is convenient to write in the form u=
]Cz]““I;(&
F-0
(6+
5‘= ]a]“2(X-at), *to),
y
a>0
(3)
or y>O,
a
l>1/2,
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00173-5
(4)
where E = I ya I l’*. As it is well known,
1 E-mail: [email protected].
(2)
with p = 1 can exist only if [4,5]
y>O,
a>O,
(5) a > 0 when y = 0. For (6)
424
V.I. Karpman, J.-M. VandL,n-Broeck/Physics
the solitons are radiating, if p < 4 [4,6]; in this case the soliton velocity a depends on time, and a = a(t) is a decreasing function found in Refs. [4,6] for E -=c 1, when the soliton lifetime is large. For p = 2, Eq. (la) is the fifth order MKdV equation which may be of interest also in fluid mechanics, plasma physics and other fields where the “classical” MKdV equation has applications. The radiation and respective attenuation of solitons in the case p = 2 have been studied in Ref. [6]. Stationary solitons in this case, discussed analytically in Ref. [5], will be found numerically below, along with those for p & 4. Eq. (la> with p > 4 has a considerable theoretical interest in connection with the general problem of collapse of nonlinear waves. Indeed, it is known that the soliton solutions of (la) for y = 0 that can be written as u,(6)=
[~IaI(P+l)(P+2)]1'Zp
Xsech2/J’(ip,$), (7) are unstable with respect to collapse-type instabilities if p 2 4 [7,8]. As for y # 0, the instability remains in case (6), i.e. for radiating solitons [6]. In this paper, we demonstrate numerically that there are in fact stationary solutions of (la), satisfying (lb) and (3) with p equal to 2, 3, 4, 5 and 6. In case (4), the solitons are found for p = 3, 5 (in addition to p = 1, found in Refs. [1,2]). As for even p, one can show analytically that solitons satisfying (1) and (4) do not exist if sgn(a) = sgn( p) [9]. Along with the existence of solitons, satisfying (l), we also investigate (numerically) their stability. The results, presented below, do not show any signs of instability for cases (3) and (4). From this it follows that the fifth order dispersion probably stabilizes the collapse instability in these cases. Now let us consider basic equations in scaled form. Substituting (2) into (la), we have after one integration the ordinary differential equation defining the stationary solitons sgn(y)e2F”(
e) +F”(
5) + - l
P+l
Fp+‘(
Zj)
- sgn( a)F( <) = 0. (8) The advantage of scaling (2) is that now (8) contains only one parameter, namely E. The signs of y and a have to be taken according to (3) or (4).
Letters A 200 (1995) 423-428
We first find the asymptotic behavior of the solitons as t+ fa. Neglecting the nonlinear term in (S), we have u(S)
=C,
(5+
cxp(r&5)
+C,
exp(T&t)
*m>,
(9)
where, for the case (3), k,,, = &[(I
+ 2~)~‘~ T (1 - 2~)“~]
(10)
and for the case (41, k,,,=
A[(Zt--
1)“2ki(2~+
I)“~].
(11)
Thus, in case (3), the asymptotic behavior as 5 -+ + ~0 must be monotonic when E < 3 and oscillatory when E > 3. In case (41, the asymptotics is always oscillatory. These conclusions, naturally, do not depend on p. Solutions with this behavior when p = 1 were obtained numerically by Kawahara [l]. We shall show that our solutions for integer p > 1 also behave in this way. (In particular, this also happens for p > 4.1 To investigate the soliton stability, we shall use the same scaling as in (2), where now a is the initial soliton velocity (for t > 0, the soliton parameters may change due to a possible instability.) Thus we write CL= lal”PF(l, I al%,
l=
Substituting G
+Fpg
r),
(12)
7= I a13’2t.
(13)
(12) and (13) into (la), we obtain d3 + ;)iF+sgn(y)t2FF=0,
d5
(14) where E is defined in (5). For stationary
solitons
F(i,~)=F(!:-sgn(a)7)=F(5).
(15) For the investigations of the soliton stability, one can also use another nonstationary equation, namely, one for the function F, expressed through the variables 5 and r, i.e. F( 5, 7). Evidently, this function satisfies the equation dF
3
5
~--sgn(u)~+FP~~+sgn(y)s2~ = 0
(16)
V.I. Karpman, J.-M. Vanden-Broeck/Physics
and describes the evolution in the frame of the initial soliton (i.e. in the frame moving with velocity a). The efficiency of this equation was demonstrated in the study of soliton radiation [6]. For the study of soliton stability it is sometimes convenient to write (16) in the form
z +f
4
sgn( y) e2$ i
1 + pFP+’ p+l
2
+$-sgn(n)r
(17)
=O. 1
Here, the expression in the large parentheses cides with the left-hand side of (8).
2. Numerical 2.1. Stationary
coin-
procedures solutions
Stationary solutions can be obtained by solving either (8) or Eq. (16) without the time derivative term, i.e., sgn( y)
l2F”“’ + F”’ + FPF’ - sgn( a)F’ = 0,
-+(N-
l)e+
(I-
l)e,
I=l,...,N (19)
and the corresponding
unknowns (20)
FI = F( 51).
Here e is the interval of discretization. We approximate the first, third and fifth derivatives in (18) by seven-point centered difference formulas. For the end points we assume that F, = 0 for I < 0 and I > N. By satisfying (18) at the mesh points (191, we obtain N nonlinear algebraic equations for the N unknowns F1 for I = 1,. . . , N. This system is solved by Newton’s method. 2.2. Time dependent
first introduce ll;=
the time step A and the mesh points
-+(M-l)e+(Z-l)e,
I=1
,...,
M. (21)
Here e is the interval of discretization. The unknowns are F( &;, t). As in the previous section we approximate the derivatives with respect to t by seven-point centered difference formulas. We then start with initial data, i.e. values of F( &, 0) and compute the values of F( &, t) for t > 0 by a fourth order Runge-Kutta method. For a given value of e, the time step A has to be small enough to ensure numerical stability. Also e needs to be small enough for the spatial derivatives to be accurately approximated by the finite difference formulas. Most of the computations in this paper were done with e = 0.13 and A = 0.000016. We checked that all the results presented are accurate at least to graphical accuracy.
3. Numerical
results
(18)
where the primes denote full derivatives with respect to 5. After one integration (18) reduces to (8). Here, we use Eq. (18). First we introduce the mesh points E,=
425
Letters A 200 (1995) 423-428
solutions
In the present paper we solve Eq. (14). To compute time dependent solutions of this equation, we
We used the scheme of Section 2.1 to compute stationary solutions. We first look for stationary solutions satisfying (18). Such solutions were computed before by Kawahara [l] and Hunter and Vanden-Broeck [2] when p = 1. We used them as the initial guess to compute solutions with p > 1. Solutions for p = 3, E = 0.72 and p = 5, E = 2.82 are shown in Figs. la and lb. In Fig. la, the oscillations in the far field appear clearly. The distance between succesive maxima and minima is seen to be in good agreement with the value 2E?r (26 + I)‘/2
= 2.9
predicted by (11). In Fig. lb, the oscillations in the far field are smaller but are still visible on the scale of the graph. Next we look for solutions satisfying (3). We used the exact solution (7) for y = 0 as an initial guess to compute solutions with y # 0. Our numerical results showed that solitons could only be obtained for y < 0. For y > 0, the scheme does not converge as N increases. This agrees with conditions
V.I. Karpman, J.-M. Vanden-Broeck/Physics
I -15
-10
-5
0
5
10
15
Fig. 1. Solutions satisfying
bl
20
25
Letters A 200 (1995) 423-42X
-18
’ -,DD
1 -80
-60
-40
-20
0
20
40
60
80
100
(4) for p = 3 and E = 0.72 (a) and for p = 5 and E = 2.82 (b).
1
Fig. 2. Solutions satisfying (3) for p = 2 and E = 0.66 (a), p = 4 and E = 0.72 (b), p = 5 and E = 0.24 (c) and p = 6 and E = 0.73 Cd). The solid curves are the stationary solutions. The broken curves arc the solutions obtained from (14) by marching in time. Before starting the marching in time, the initial soliton profile was multiplied by a factor 6 = 1.0001 in (a), cc>, Cd) and by a factor 6 = 1.001 in (b).
V.I. Karpman, J.-M. Vanden-Broeck/Physics
Letters A 200 (1995) 423-428
427
1
:_I_: -15
-14
-13
-12
-11
-10
-9
-8
-7
6
-2e-06’-15
Fig. 3. Portions of the tail of the solid curve of Fig. 2d on an expanded
-14.5
-14
s
-13 5
_;3
~,2.5
vertical scale (a) and on a further expanded scale (b).
25,
(a)
4,
Fig. 4. Solutions for y = 0 and p = 2 (a), p = 4 (b), p = 5 (c) and p = 6 Cd). The solid curves correspond to the solution (7). The broken curves are the solutions obtained by marching in time. Before starting the marching in time, the initial profile was multiplied by a factor 6 = 1.0001 in (a), (c), (d) and by a factor 6 = 1.001 in (b>.
428
V.I. Karpman, J.-M. Vanden-Broeck/Phvsics
(3). Typical solutions for p = 2, 4, 5 and 6 are shown by the solid curves in Figs. 2a-2d. All the solutions of Fig. 2 appear to be flat in the far field. However, when the vertical scale is expanded, oscillations appear in the far field when E > 0.5. This is illustrated in Fig. 3. The solid curve of Fig. 2d appears to be flat between - 15 and - 6. However, when the vertical scale is expanded a minimum appears at about x = - 7 (see Fig. 3a). The solution of Fig. 3a appears to be flat between - 15 and - 12.5 but when the vertical scale is further expanded a maximum appears at about x = - 13.7 (see Fig. 3b). This illustrates that the asymptotic behavior is oscillatory when E > 0.5. The distance between successive maxima and minima is in good agreement with the value 2nlZ (26 _ 1)1/Z = 6.7. On the other hand, the tail of the solid curve of Fig. 2c is monotonic even on an expanded scale, in accordance with the asymptotic behavior discussed at the end of Section 1. Finally we look at the stability of the solitons satisfying (14) with y < 0, in connection with the collapse instability. We use the stationary solutions computed by the scheme of Section 2.1 as an initial condition in the scheme of Section 2.2 and we choose M = N. In order to trigger possible instabilities, we multiply all the values of F by a factor 6 > 1, but very close to 1, before starting the marching in time. In most of the calculations we chose S = 1.0001. Typical solutions for y = 0 are shown in Figs. 4a-4d. The solution with p = 2 is stable (see Fig. 4a), and those for p = 4, 5, 6 are unstable (see Figs. 4b-4d). This is in accordance with the results of Refs. [7,8]. On the other hand, no instabilities appear in our calculations with y < 0 and p = 2, 3, 4, 5, 6. In these calculations we also multiplied the initial soliton profile by a factor S > 1 but very close to 1, before starting the marching in time. Within our accuracy, the corresponding solutions of (14) are the solitons shown in Fig. 2. Therefore the inclusion of fifth order dispersion plays a stabilizing role. Our results do not prove that the solutions are stable at all times. On the other hand, they show that the instabilities can be avoided over a finite interval of time by choosing E sufficiently large.
Letters A 200 (199.5) 423-428
4. Conclusions We have demonstrated numerically that Eq. (la) at p > 1 has indeed soliton solutions satisfying conditions (3) and (4) in agreement with the predictions of Ref. [5]. Similar to the case p = 1, the soliton asymptotic behavior at large 15 1may be both monotonic and oscillatory, in accordance with (10) and (11). We have shown numerically that the fifth order derivative term in (la) is of critical importance for the soliton stability at sufficient high p. At y = 0 and p 2 4 the soliton solutions are unstable, in agreement with the theory of collapse instability [7,8]. On the other hand, no instability has been detected in our calculations at y < 0. This agrees with an analytical investigation of the stability of solitons of Eq. (la) [lo]. Finally, it should be mentioned that the results obtained can be, in principle, checked experimentally by modelling Eq. (la) in electronic transmission line
[ill. Acknowledgement The work of the second author was supported in part by the National Science Foundation, as well as the Lady Davis Foundation. He gratefully acknowledges the hospitality of the Hebrew University of Jerusalem, Tel Aviv University and Technion.
References [I] T. Kawahara, J. Phys. Sot. Japan 33 (1972) 260. [2] J.K. Hunter and J.-M. Vanden-Broeck, J. Fluid Mech. 134 (1983) 205. [3] Y. Pomeau, A. Ramani and B. Grammaticos, Physica D 31 (1988) 127. [4] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. [5] V.I. Karpman, Phys. Lett. A 186 (1994) 300. [6] V.I. Karpman, Phys. Lett. A 186 (1994) 303. [7] E.A. Kuznetsov, A.M. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103. [8] J.J. Rasmussen and K. Rypdal, Phys. Ser. 33 (1986) 481. [9] V.I. Karpman, to be published. [lOI V.I. Karpman, to be published. [ll] R.K. Dodd, J.C. Eilbeck, J.D. Gibbons and H.C. Morris, Solitons and nonlinear wave equations (Academic Press, New York, 1984) ch. 5.5.