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Modified Korteweg-De Vries equation for spatially inhomogeneous plasmas
Volume 70A, number 1
5 February 1979
PHYSICS LETTERS
MODIFIED KORTEWEG-DE VRIES EQUATION FOR SPATIALLY INHOMOGENEOUS PLASMAS N. Nagesha RAO and Ram K. VARMA Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India Received 27 June 1977
The problem of propagation of ion-acoustic KdV solitons in weakly, spatially inhomogeneous plasmas is considered taking into account self-consistently the zeroth-order velocity and the potential existing in the system due to the presence of the inhomogeneity. An explicit soliton solution of the modified KdV equation is obtained.
The propagation of ion-acoustic KdV solitons in homogeneous plasmas has been studied extensively over the last several years [1—4].The propagation in a spatially inhomogeneous plasma was first considered by Nishikawa and Kaw [5], and reconsidered recentiy by Gell and Gomberoff [6]. There are certain inadequacies and inconsistencies in both these treatments. Firstly, as is physically clear and as we shall show here, aa spatial inhomogeneity the zeroth der implies zeroth-order ion fluid in velocity. Noneorof the above authors has taken this into account consistently. Secondly, the coordinate stretching employed in these analyses is not the one appropriate for a spatially inhomogeneous medium as, for instance, discussed [7]. [6] have compared the results Cell by andAsano Gomberoff of their calculation with the experimental results of John and Saxena [8] on the propagation of solitons in a spatially mhomogeneous medium. As has been discussed by Rao and Varma [9] however, the choice of coordinate stretching appropriate for an experimentally observed soliton is determined, even in the homogeneous case, by the method of launching the soliton in the experiment. In this paper, we carry out a consistent analysis of the problem taking the zeroth-order velocities and potentials into account and using the correct coordinate stretchings for the problem. The basic equations in the dimensionless form are [2] •
.
.
.
,
au/at + u au/ax + açb/ax
=
a2p/ax2 g(x9)e~+ n
=
(2)
0.
(3)
—
Following Asano [7], the right set of stretched coordinates to be used for a spatially inhomogeneous plasma is 3/2x (4 1/2 =e X 0(x’) j
[L~ ~
_____
—
where X0 is the velocity of the moving frame to be determined later self-consistently. It can be noted that the stretched coordinates used by Gell 312t.and Gomberoff areSince of theA form ~ = 112 (x—t), n = e 0 and the equilibrium ion density n0 are functions of x only, we have ~
,~
~
—
t
(JflQ
UA0/UtV,
But from eqs. (4) a/at give
ax ia
=
o
a
a
a/ak. Hence eqs. (5)
_~h/2 =
no! ~
0
0
6 .
( )
We now carry out the reductive perturbation analysis of eqs. (l)—(3). For this, the quantities n, u and ~ are expanded as -~
2v uu0+eu1+e
an/at+(a/ax)[nvj=o,
0,
2+...,
(1) 9
Volume 70A, number 1
5 February 1979
PHYSICS LETTERS
where u0 and ~ are the zeroth-order velocity and potential, respectively, existing in the system due to the presence of the spatial inhomogeneity. Substituting these expansions into eqs. (l)—(3) and combining the zeroth-order equations thus obtained with eqs. (5), weget av0!a~=o, a~0/a~=o. (7)
—
The second-order equations are 8n i a + [n0v~ + n1 v1 + n2 u1~] ~—
6
a +~—[n0v1+n1v0]=0,
av2 The next higher order equations in e are an1 1 a a + ~ [n0u1 + n1v0] + [n0u0] —
~—
av1
au1
~0 +—
—
+v0
a~ x0 a~
0,
(8)
~ a~1 a~0 a~ x0 a~+—0, a~
av0
—+—-
(9)
—
— —
n041
+
n1
—
0.
~
+
au2
÷ [vo -~-j+ v1
— -~-~-
=
~
~
(14)
au1
~—
~
au~
Lv0
-
1 ~
av~ + V1
~
~
+
-~
a~j x0 ~ !
22~0
2~ 1 ~2
—
0
ai~
—
—0
15 ‘
~
16
(10)
Integrating these equations and using the boundary condition that the plasma is homogeneous at the boundary, that is, as ~ -÷ 00, n1 = u1 = ~ n0 A0 = 1 and v0 = 0, we have u1 =Th1—~T, ~1Th0u1—~R,
io~i~ (11)
Using the relation A0 = 1 -I- u0, we are able to elimmate all the second-order quantities exactly. Substituting for u1 and n1 in terms of~1from eqs. (11), we get the following modified KdV equation:
a~1 ~ ~
~
a~1 ~
~
2n0X~j ~
where
~o a T(i~)”——j~(flOVo)~
_____
R(~)=
x0
I
1 ~
a~ a~~
(~v0
-~-~ +
-~—~) .
Eliminating n1 and v1 in eqs. (11), g~can be written as ~ 111 ~ ~ ~12, In the above equation, the right-hand side depends only on the zeroth-order quantitieswhereas the lefthand side is a first-order quantity. In order that the first-order quantities are not determined by the zerothorder quantities, make theput expression Ø~an indeterminate one. we That is, we both the for numerator and the denominator equal to zero separately. Setting the denominator equal to zero, we get A 0 = (1 + u0). Using this in the numerator and setting it equal to zero, the following equation is obtained: ax a~ / i an ~ 8n2-i-——2÷(X a~ \fl A0 — 0--i)(— _-_9) =0. (13) — —
,.
~
—
0
.
an /
This gives a self-consistent relation between n0, u0 and 10
~
~
a~1 ~
an0~
(17)
Comparing this equation with the one derived earher [51,we note the presence of an extra’term with the coefficient (X~~X0!~0).Further, the coefficients of the nonlinear term and the third-order term of the KdV equation get modified by the presence of A0. These changes modify the width and the velocity of the soliton as it propagates in the medium. Eq. (17) can be easily solved by making some transformations similar to those given in refs. [5,7]. The explicit solution is obtained as 3a ri a\ 1/2 2L~~~) ~ V?20 SeCh (18) d 1”4 (1 lnNd”4)(1 + In A X N0 0) ~ =—~
—
—
N0
where N0 = n0?$ and hence a function of x only. It is obvious from the above solution that the amplitude dependence of the soliton with respeetto the equilibrium ion density n0 is the same as obtained in the earlier calculation [5]. But the width and the velo-
Volume 70A,
number 1
PHYSICS LETTERS
city of the soliton are different. This is so because the argument of the function sech, which gives the propagation characteristics of the soliton, is modified now by the presence of A0. Finally, we would like to point out that contrary to the claims made by the authors in ref. [6], the results of their theoretical calculations do not agree with the experimental results as quoted in ref. [8]. References
5 February 1979
[21 H. Washimi and T. Taniuti, Phys. Rev. Lett. 17 966. [31 C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095.
(1966) —
[41 R.C.
Davidson, Methods in nonlinear plasma theory (Academic Press, New York, 1972).
[5] K. Nishikawa and P.K. Kaw, Phys. Lett. SOA (1975) ~ [61 Y. Gell and L. Gomberoff, Phys. Lett. 60A (1977) 125. 171 N. Asano, Prog. Theor. Phys. Suppl. 55 (1974) 52. [81 P.1. John and Y.C. Saxena, Phys. Lett. 56A (1976) 385. [9] N.N. Rao and R.K. Varma, Pramana 8 (1977) 427.
[1] N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240.
11
5 February 1979
PHYSICS LETTERS
MODIFIED KORTEWEG-DE VRIES EQUATION FOR SPATIALLY INHOMOGENEOUS PLASMAS N. Nagesha RAO and Ram K. VARMA Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India Received 27 June 1977
The problem of propagation of ion-acoustic KdV solitons in weakly, spatially inhomogeneous plasmas is considered taking into account self-consistently the zeroth-order velocity and the potential existing in the system due to the presence of the inhomogeneity. An explicit soliton solution of the modified KdV equation is obtained.
The propagation of ion-acoustic KdV solitons in homogeneous plasmas has been studied extensively over the last several years [1—4].The propagation in a spatially inhomogeneous plasma was first considered by Nishikawa and Kaw [5], and reconsidered recentiy by Gell and Gomberoff [6]. There are certain inadequacies and inconsistencies in both these treatments. Firstly, as is physically clear and as we shall show here, aa spatial inhomogeneity the zeroth der implies zeroth-order ion fluid in velocity. Noneorof the above authors has taken this into account consistently. Secondly, the coordinate stretching employed in these analyses is not the one appropriate for a spatially inhomogeneous medium as, for instance, discussed [7]. [6] have compared the results Cell by andAsano Gomberoff of their calculation with the experimental results of John and Saxena [8] on the propagation of solitons in a spatially mhomogeneous medium. As has been discussed by Rao and Varma [9] however, the choice of coordinate stretching appropriate for an experimentally observed soliton is determined, even in the homogeneous case, by the method of launching the soliton in the experiment. In this paper, we carry out a consistent analysis of the problem taking the zeroth-order velocities and potentials into account and using the correct coordinate stretchings for the problem. The basic equations in the dimensionless form are [2] •
.
.
.
,
au/at + u au/ax + açb/ax
=
a2p/ax2 g(x9)e~+ n
=
(2)
0.
(3)
—
Following Asano [7], the right set of stretched coordinates to be used for a spatially inhomogeneous plasma is 3/2x (4 1/2 =e X 0(x’) j
[L~ ~
_____
—
where X0 is the velocity of the moving frame to be determined later self-consistently. It can be noted that the stretched coordinates used by Gell 312t.and Gomberoff areSince of theA form ~ = 112 (x—t), n = e 0 and the equilibrium ion density n0 are functions of x only, we have ~
,~
~
—
t
(JflQ
UA0/UtV,
But from eqs. (4) a/at give
ax ia
=
o
a
a
a/ak. Hence eqs. (5)
_~h/2 =
no! ~
0
0
6 .
( )
We now carry out the reductive perturbation analysis of eqs. (l)—(3). For this, the quantities n, u and ~ are expanded as -~
2v uu0+eu1+e
an/at+(a/ax)[nvj=o,
0,
2+...,
(1) 9
Volume 70A, number 1
5 February 1979
PHYSICS LETTERS
where u0 and ~ are the zeroth-order velocity and potential, respectively, existing in the system due to the presence of the spatial inhomogeneity. Substituting these expansions into eqs. (l)—(3) and combining the zeroth-order equations thus obtained with eqs. (5), weget av0!a~=o, a~0/a~=o. (7)
—
The second-order equations are 8n i a + [n0v~ + n1 v1 + n2 u1~] ~—
6
a +~—[n0v1+n1v0]=0,
av2 The next higher order equations in e are an1 1 a a + ~ [n0u1 + n1v0] + [n0u0] —
~—
av1
au1
~0 +—
—
+v0
a~ x0 a~
0,
(8)
~ a~1 a~0 a~ x0 a~+—0, a~
av0
—+—-
(9)
—
— —
n041
+
n1
—
0.
~
+
au2
÷ [vo -~-j+ v1
— -~-~-
=
~
~
(14)
au1
~—
~
au~
Lv0
-
1 ~
av~ + V1
~
~
+
-~
a~j x0 ~ !
22~0
2~ 1 ~2
—
0
ai~
—
—0
15 ‘
~
16
(10)
Integrating these equations and using the boundary condition that the plasma is homogeneous at the boundary, that is, as ~ -÷ 00, n1 = u1 = ~ n0 A0 = 1 and v0 = 0, we have u1 =Th1—~T, ~1Th0u1—~R,
io~i~ (11)
Using the relation A0 = 1 -I- u0, we are able to elimmate all the second-order quantities exactly. Substituting for u1 and n1 in terms of~1from eqs. (11), we get the following modified KdV equation:
a~1 ~ ~
~
a~1 ~
~
2n0X~j ~
where
~o a T(i~)”——j~(flOVo)~
_____
R(~)=
x0
I
1 ~
a~ a~~
(~v0
-~-~ +
-~—~) .
Eliminating n1 and v1 in eqs. (11), g~can be written as ~ 111 ~ ~ ~12, In the above equation, the right-hand side depends only on the zeroth-order quantitieswhereas the lefthand side is a first-order quantity. In order that the first-order quantities are not determined by the zerothorder quantities, make theput expression Ø~an indeterminate one. we That is, we both the for numerator and the denominator equal to zero separately. Setting the denominator equal to zero, we get A 0 = (1 + u0). Using this in the numerator and setting it equal to zero, the following equation is obtained: ax a~ / i an ~ 8n2-i-——2÷(X a~ \fl A0 — 0--i)(— _-_9) =0. (13) — —
,.
~
—
0
.
an /
This gives a self-consistent relation between n0, u0 and 10
~
~
a~1 ~
an0~
(17)
Comparing this equation with the one derived earher [51,we note the presence of an extra’term with the coefficient (X~~X0!~0).Further, the coefficients of the nonlinear term and the third-order term of the KdV equation get modified by the presence of A0. These changes modify the width and the velocity of the soliton as it propagates in the medium. Eq. (17) can be easily solved by making some transformations similar to those given in refs. [5,7]. The explicit solution is obtained as 3a ri a\ 1/2 2L~~~) ~ V?20 SeCh (18) d 1”4 (1 lnNd”4)(1 + In A X N0 0) ~ =—~
—
—
N0
where N0 = n0?$ and hence a function of x only. It is obvious from the above solution that the amplitude dependence of the soliton with respeetto the equilibrium ion density n0 is the same as obtained in the earlier calculation [5]. But the width and the velo-
Volume 70A,
number 1
PHYSICS LETTERS
city of the soliton are different. This is so because the argument of the function sech, which gives the propagation characteristics of the soliton, is modified now by the presence of A0. Finally, we would like to point out that contrary to the claims made by the authors in ref. [6], the results of their theoretical calculations do not agree with the experimental results as quoted in ref. [8]. References
5 February 1979
[21 H. Washimi and T. Taniuti, Phys. Rev. Lett. 17 966. [31 C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095.
(1966) —
[41 R.C.
Davidson, Methods in nonlinear plasma theory (Academic Press, New York, 1972).
[5] K. Nishikawa and P.K. Kaw, Phys. Lett. SOA (1975) ~ [61 Y. Gell and L. Gomberoff, Phys. Lett. 60A (1977) 125. 171 N. Asano, Prog. Theor. Phys. Suppl. 55 (1974) 52. [81 P.1. John and Y.C. Saxena, Phys. Lett. 56A (1976) 385. [9] N.N. Rao and R.K. Varma, Pramana 8 (1977) 427.
[1] N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240.
11