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Interaction of perturbed solitons and the structure of oscillatory shocks
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
INTERACTION OF PERTURBED SOLITONS AND THE STRUCTURE OF OSCILLATORY SHOCKS V.!. KARPMAN Institute for Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), Moscow Region, 142092, USSR Received 15 February 1979
A detailed description ofoscillatory shocks satisfying the perturbed Korteweg—de Vries (KdV) equation is presented on the basis of a simple treatment of the system of interacting perturbed solitons.
t -U Shock waves with oscillatory front structures can exist in different dispersive media, in particular, in plasmas [1,2], nonlinear electric transmission lines [3] dispersive media with viscosity and heat conductivity [4] etc. In many cases such shocks may be described by the perturbed KdV equation
I
,
,
ut_6uu~+u~~~=eR[u]
(1)
,
where R is an operator acting on u(x, t). In this note we study the shock-like solutions of eq. (1) with sufficiently small e. Substituting in eq. (1) u = u(x Vt),
~
—
u
-+
u(x
-~
_oo),
u
O(x
-+
00),
(2)
Fig. 1. The profile of the oscillatory shock wave.
(3)
First we note that the action of the perturbation eR [u] on a single soliton is described by the equations [5—8]
we have u,~ 3u2 —
—
Vu = —e
j’ R [u(x’, t)]
~‘.
x
By assuming here x
-+
_oo,
we obtain
u(x, t) = u 5(z, k(t)) + 6u(x, t), 2 sech2z, z = k(x (t)) u5(z, k) = —2k dk/dt = —eA(k), d~/dt= 4k2 eB(k), —~
u=_~v_y,eT
TJR[u(x,t)]dx.
i~,
(4)
—
If = 0, a solution of eq. (3), satisfying the condition U -+0 at x oo has a soliton form. For finite -~
(but small) and the same boundary condition at x -*oo, the solution (if it exists) is ofthe form shown in fig. 1, i.e., it is a shock wave with an oscillatory structure. The front part of it may be considered as a sequence of solitons with the same velocity V. In what follows we describe the shock structure in terms of eR [u].
A(k)=~
f
B(k)~~j
7
R[uj sech2zdz,
(5) (6)
(7)
R[u 5]
2z + tanhz ÷tanh2z)dz.
—
x (z sech
,
(8) 163
Volume 71A, number 2,3
PHYSICS LETTERS
As for the variation of the soliton shape 6u(x, t), we discuss here only its “tail part” described by the expressions [6]
d~m/dt= 4k~,+ l12k~ —
(6u)
lim
?Iu
=
k2eq
30 April 1979 1
l6k~exp(—2kmrrn)
exp(—2krn ~~1rm
—
i)
rn—i
—
E
6
(14)
k?q(k~) CB(krn) —
i=1
,
x-* —
q
=
4k5
7
Here m = 2,3 .R [u
2z dz.
(9)
5] tanh
—=
Actually, ~u reduces to the tail at several soliton lengthes behind the soliton [6—8] (see also ref. [9] 2! where it was demonstrated numerically for R = a ax2). The length of the tail, approximately, grows proportionally to the time after “switching on” of the perturbation.
eqs. (7)—(9). The considered soliton system is stationary if dk~/ dt = 0 and d~ 0/dt= V. In that case one obtains from eqs. (11 )—(14) i iog[_~-_ A (k~)] (15)
E
(10)
the interaction between the solitons may be treated by perturbation theory, using for the evolution of the nth soliton eqs. (6)—(8) with eR [u5] replaced by eR [u 5~] + 6(a/ax)(u5~u,(0+1)) i- 6(a/ax) (uS(fl —1 )U5~)
2q(k,)
+
6
~i1~ kj
=
—
=
~
,
64k~ ~ 112(l
k 1
Suppose now that there is a sequence of solitons with centers ~n (~n> ~fl+i). If
and A (k), B(k), and q(k) are given by
eA(k
+
V
2
+
eB(k
1)/2V
1)/2V),
k~—k~~1 =~ eA(k~)/k,~ —~eq(k0)k0 (n e
=
(16) (17)
1,2,...). Suppose now that
fR
2z dz >0, [U5]
eq ~ 0.
(18)
sech
Then k~> k~÷1,and the difference between two adjacent maxima of the shock profile (i.e. between the apices of the neighbouring solitons) is
(a/ax) (u
—(u0 ~
—2ek~q(k~).
(19)
5,,)
1=1
Here the second and third terms describe the interaction of the nth soliton with its nearest neighbours and the last term appears due to the influence of the tails of the n 1 first solitons on the nth soliton. If, in addition, the amplitudes of the adjacent solitons are close to each other (1k n kn—i irn ~ 1), the equations describing the soliton evolution take the form dk 1/dt = 64k~exp(-~-2k1r1) eA(k~), (11) dkrn/dt=—64k~iexp(_2kmirm_i) —
—
Eqs. (15)—(l7) and (19) give a complete description of the “soliton part” of the shock front. They are valid under the conditions 2k~r0 1, eq (k~)I 1 (20) which are necessary for applicability of perturbation theory. As a simple example, consider KdV—Burgers 2/ax2). Then, the computing A (k) equation (i.e. R = a ~
~
,
(12)
and B(k) by means of eqs. 3/l5,q (6.14) = —8/15k, and (6.22) from ref. [7] ,we haveA(k) = 8k k~~ 1 k1 —en/2 (21a) 1 log[ r0 (2k0) 14~(k~/k~+1 1)] (21b)
(13)
u~÷1U1
—
,
÷64k~exp(_2krnrm)
d~1/dt= 4k?
164
—
—
eA(km),
16k? exp(—2k1r1)
—
eB(k1),
—
—
22
+ j~ en(k1
—
~ en).
,
(22)
Volume hA, number 2,3
PHYSICS LETTERS
Here n = 1,2 In particular, at n ~ N = k1 /2, 1~1 0.75, rN 2.36knT —1.36k 2 —1 ,UN ~. ~ “~Nf 1 ~23, -~
-~
—.. ,~
/
-
At n > N, the first of conditions (20) is violated. Therefore, one may regard the number of solitons in the shock front approximately equal to N. From eqs. (4) and (16) we obtain v —1.33k? which is very close to the UN from eq. (23). Thus, profile oscillations following sidered as solitons linear. with numbers n k1 /2 may be conTo investigate these oscillations in the general case, we substitute into eq. (3) u u + ii (x Vt), where ii is considered as small. After linearization, and taking into account eq. (4), we have —
+
f
Vu =e
R[v+u(x’)]dx’.
(24)
0)] cx, a = V
+
eu1 (x), .
,
.
we obtain in lowest order in
/
f
dxR[v + D(y) cosax]
=
0,
—Inn Il/a
dxR Eu + D(y) cos ax] sin ax = 0,
(25)
—
2ira
dD
+
inla
j
—~
—~
References
x
—
1/2
where y
Eq. (26) should be solved under the boundary condition: D 0 at y _oo• The existence of such a solution, as well as the fulfillment of relations (25) are necessary conditions for the possibility of a shock under the action of the perturbation eR [u]. The other conditions assumed above are eqs. (18) and the convergence of the integral defining 1’ (eq. (4)). The last one was already discussed in ref. [10].e >0 all these con2/ax2, In particular, forand R =eq. a (26) gives ditions are fulfilled D(y) = D ex (y/2) 0 P which coincides with the known asymptotics for that case (e.g., ref. [4], § 23).
[1] R.Z. Sagdeev, Soy. Phys. Tech. Phys. 6(1962)867. [2] R.Z. Sagdeev and A.A. Galeev. Nonlinear plasma theory (Benjamin, New York, 1969).
—~
Making the substitution ü (x) = D(y) cos [a(x
30 April 1979
dxR [u+ D(y) cos ax] cos ax = 0. (26)
[3] A.V. Gaponov, L.A. Ostrovski and G.I. Freidman, Radiophys. Quantum Electron 10 (1967) 1376. [4] V.1. Karpman, Nonlinear waves in dispersive media (Pergamon, Oxford, 1975). [5] V.!. Karpman and E.M. Maslov, Zh. Eksp. Teor. Fiz. 73 (1977) 537 [Soy. Phys. JETP 46 (1977) 281]. [6] V.1. Karpman and E.M. Maslov, Zh. Eksp. Teor. Fiz. 75 (1978) 504. [7] V.1: Karpman, Proc. Chalmers Symp. on Solitons, Phys. Scripta 18, to be published. [8] D.J. Kaup and A.C. Newell, Proc. Roy. Soc. A361 (1978) [9]
413.
J.C. Fernandez, G. Reinisch, A.
Bondeson and J. Weiland,
Phys. Lett. 66A (1978) 175.
[10] D. Ffirsch and R.N. Sudan, Phys. Fluids 14 (1971) 1033.
165
PHYSICS LETTERS
30 April 1979
INTERACTION OF PERTURBED SOLITONS AND THE STRUCTURE OF OSCILLATORY SHOCKS V.!. KARPMAN Institute for Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), Moscow Region, 142092, USSR Received 15 February 1979
A detailed description ofoscillatory shocks satisfying the perturbed Korteweg—de Vries (KdV) equation is presented on the basis of a simple treatment of the system of interacting perturbed solitons.
t -U Shock waves with oscillatory front structures can exist in different dispersive media, in particular, in plasmas [1,2], nonlinear electric transmission lines [3] dispersive media with viscosity and heat conductivity [4] etc. In many cases such shocks may be described by the perturbed KdV equation
I
,
,
ut_6uu~+u~~~=eR[u]
(1)
,
where R is an operator acting on u(x, t). In this note we study the shock-like solutions of eq. (1) with sufficiently small e. Substituting in eq. (1) u = u(x Vt),
~
—
u
-+
u(x
-~
_oo),
u
O(x
-+
00),
(2)
Fig. 1. The profile of the oscillatory shock wave.
(3)
First we note that the action of the perturbation eR [u] on a single soliton is described by the equations [5—8]
we have u,~ 3u2 —
—
Vu = —e
j’ R [u(x’, t)]
~‘.
x
By assuming here x
-+
_oo,
we obtain
u(x, t) = u 5(z, k(t)) + 6u(x, t), 2 sech2z, z = k(x (t)) u5(z, k) = —2k dk/dt = —eA(k), d~/dt= 4k2 eB(k), —~
u=_~v_y,eT
TJR[u(x,t)]dx.
i~,
(4)
—
If = 0, a solution of eq. (3), satisfying the condition U -+0 at x oo has a soliton form. For finite -~
(but small) and the same boundary condition at x -*oo, the solution (if it exists) is ofthe form shown in fig. 1, i.e., it is a shock wave with an oscillatory structure. The front part of it may be considered as a sequence of solitons with the same velocity V. In what follows we describe the shock structure in terms of eR [u].
A(k)=~
f
B(k)~~j
7
R[uj sech2zdz,
(5) (6)
(7)
R[u 5]
2z + tanhz ÷tanh2z)dz.
—
x (z sech
,
(8) 163
Volume 71A, number 2,3
PHYSICS LETTERS
As for the variation of the soliton shape 6u(x, t), we discuss here only its “tail part” described by the expressions [6]
d~m/dt= 4k~,+ l12k~ —
(6u)
lim
?Iu
=
k2eq
30 April 1979 1
l6k~exp(—2kmrrn)
exp(—2krn ~~1rm
—
i)
rn—i
—
E
6
(14)
k?q(k~) CB(krn) —
i=1
,
x-* —
q
=
4k5
7
Here m = 2,3 .R [u
2z dz.
(9)
5] tanh
—=
Actually, ~u reduces to the tail at several soliton lengthes behind the soliton [6—8] (see also ref. [9] 2! where it was demonstrated numerically for R = a ax2). The length of the tail, approximately, grows proportionally to the time after “switching on” of the perturbation.
eqs. (7)—(9). The considered soliton system is stationary if dk~/ dt = 0 and d~ 0/dt= V. In that case one obtains from eqs. (11 )—(14) i iog[_~-_ A (k~)] (15)
E
(10)
the interaction between the solitons may be treated by perturbation theory, using for the evolution of the nth soliton eqs. (6)—(8) with eR [u5] replaced by eR [u 5~] + 6(a/ax)(u5~u,(0+1)) i- 6(a/ax) (uS(fl —1 )U5~)
2q(k,)
+
6
~i1~ kj
=
—
=
~
,
64k~ ~ 112(l
k 1
Suppose now that there is a sequence of solitons with centers ~n (~n> ~fl+i). If
and A (k), B(k), and q(k) are given by
eA(k
+
V
2
+
eB(k
1)/2V
1)/2V),
k~—k~~1 =~ eA(k~)/k,~ —~eq(k0)k0 (n e
=
(16) (17)
1,2,...). Suppose now that
fR
2z dz >0, [U5]
eq ~ 0.
(18)
sech
Then k~> k~÷1,and the difference between two adjacent maxima of the shock profile (i.e. between the apices of the neighbouring solitons) is
(a/ax) (u
—(u0 ~
—2ek~q(k~).
(19)
5,,)
1=1
Here the second and third terms describe the interaction of the nth soliton with its nearest neighbours and the last term appears due to the influence of the tails of the n 1 first solitons on the nth soliton. If, in addition, the amplitudes of the adjacent solitons are close to each other (1k n kn—i irn ~ 1), the equations describing the soliton evolution take the form dk 1/dt = 64k~exp(-~-2k1r1) eA(k~), (11) dkrn/dt=—64k~iexp(_2kmirm_i) —
—
Eqs. (15)—(l7) and (19) give a complete description of the “soliton part” of the shock front. They are valid under the conditions 2k~r0 1, eq (k~)I 1 (20) which are necessary for applicability of perturbation theory. As a simple example, consider KdV—Burgers 2/ax2). Then, the computing A (k) equation (i.e. R = a ~
~
,
(12)
and B(k) by means of eqs. 3/l5,q (6.14) = —8/15k, and (6.22) from ref. [7] ,we haveA(k) = 8k k~~ 1 k1 —en/2 (21a) 1 log[ r0 (2k0) 14~(k~/k~+1 1)] (21b)
(13)
u~÷1U1
—
,
÷64k~exp(_2krnrm)
d~1/dt= 4k?
164
—
—
eA(km),
16k? exp(—2k1r1)
—
eB(k1),
—
—
22
+ j~ en(k1
—
~ en).
,
(22)
Volume hA, number 2,3
PHYSICS LETTERS
Here n = 1,2 In particular, at n ~ N = k1 /2, 1~1 0.75, rN 2.36knT —1.36k 2 —1 ,UN ~. ~ “~Nf 1 ~23, -~
-~
—.. ,~
/
-
At n > N, the first of conditions (20) is violated. Therefore, one may regard the number of solitons in the shock front approximately equal to N. From eqs. (4) and (16) we obtain v —1.33k? which is very close to the UN from eq. (23). Thus, profile oscillations following sidered as solitons linear. with numbers n k1 /2 may be conTo investigate these oscillations in the general case, we substitute into eq. (3) u u + ii (x Vt), where ii is considered as small. After linearization, and taking into account eq. (4), we have —
+
f
Vu =e
R[v+u(x’)]dx’.
(24)
0)] cx, a = V
+
eu1 (x), .
,
.
we obtain in lowest order in
/
f
dxR[v + D(y) cosax]
=
0,
—Inn Il/a
dxR Eu + D(y) cos ax] sin ax = 0,
(25)
—
2ira
dD
+
inla
j
—~
—~
References
x
—
1/2
where y
Eq. (26) should be solved under the boundary condition: D 0 at y _oo• The existence of such a solution, as well as the fulfillment of relations (25) are necessary conditions for the possibility of a shock under the action of the perturbation eR [u]. The other conditions assumed above are eqs. (18) and the convergence of the integral defining 1’ (eq. (4)). The last one was already discussed in ref. [10].e >0 all these con2/ax2, In particular, forand R =eq. a (26) gives ditions are fulfilled D(y) = D ex (y/2) 0 P which coincides with the known asymptotics for that case (e.g., ref. [4], § 23).
[1] R.Z. Sagdeev, Soy. Phys. Tech. Phys. 6(1962)867. [2] R.Z. Sagdeev and A.A. Galeev. Nonlinear plasma theory (Benjamin, New York, 1969).
—~
Making the substitution ü (x) = D(y) cos [a(x
30 April 1979
dxR [u+ D(y) cos ax] cos ax = 0. (26)
[3] A.V. Gaponov, L.A. Ostrovski and G.I. Freidman, Radiophys. Quantum Electron 10 (1967) 1376. [4] V.1. Karpman, Nonlinear waves in dispersive media (Pergamon, Oxford, 1975). [5] V.!. Karpman and E.M. Maslov, Zh. Eksp. Teor. Fiz. 73 (1977) 537 [Soy. Phys. JETP 46 (1977) 281]. [6] V.1. Karpman and E.M. Maslov, Zh. Eksp. Teor. Fiz. 75 (1978) 504. [7] V.1: Karpman, Proc. Chalmers Symp. on Solitons, Phys. Scripta 18, to be published. [8] D.J. Kaup and A.C. Newell, Proc. Roy. Soc. A361 (1978) [9]
413.
J.C. Fernandez, G. Reinisch, A.
Bondeson and J. Weiland,
Phys. Lett. 66A (1978) 175.
[10] D. Ffirsch and R.N. Sudan, Phys. Fluids 14 (1971) 1033.
165