- Email: [email protected]
Evolution of radiating solitons described by the fifth-order Korteweg-de Vries type equations
27 July 1998
PHYSICS
ELSEYIER
Physics Letters A 244 (1998)
LETTERS
A
394-396
Evolution of radiating solitons described by the fifth-order Korteweg-de Vries type equations V.I. Karpman ’ Racah Institute of Physics. Hebrew University, Jerusalem Received
10 March 1998; accepted
for publication
91904. Israel
17 March 1998
Communicated by V.M. Agranovich
Abstract The time dependence of velocities and amplitudes of radiating solitons, described type equations is investigated. @ 1998 Elsevier Science B.V.
1. The resonant radiation of solitons due to highorder dispersive effects leads to their damping. The goal of the present paper is an investigation of the evolution of soliton parameters, caused by the radiation, if they are described by the Sth-order Kortewegde Vries (kdV) -type equations
a,u + U”&U + pa$
+ y$u
= 0
(1)
(y is small) with integer p 2 1. At p = 1 and p = 2 we have, respectively, the Sth-order KdV and modified KdV (MKdV) equations. For definiteness, we assume /? > 0. Then the condition of radiation is y > 0 [ 1,2]. At y = 0, the soliton solution of Eq. (1) is uos = a 1’PFo(5),
5 = la/P11’2[x
- x0(t) I?
*qJ>o,
24=
by the fifth-order
a”p[Fo(5)+ f(5,
t) I
Korteweg-de
Vries
(4)
3
where f(&, t) at sufficiently large 5 and t describes the soliton resonant radiation. At p = 1, the decrease of the soliton velocity caused by its radiation is described by the following differential equation [ I], da
_ -I&-*
z-
F
5’2p3~y~)-‘12exp
(-
z).
0 (5)
A similar equation (with a different numerical coefficient) was obtained in Ref. [ 31. (See also the pioneering work [4] where, however, there is another power of a before the exponent). Eq. (5) is valid at the condition
dt
Fo(&
= [i(p
+ l)(p
yu/p2 +2)11ipsech2ip(~~~).
<< 1 ,
(6)
(3)
At small positive y, the soliton solution (2) is replaced by ’ E-mail: [email protected]. 037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00309-O
which is assumed everywhere in this paper. To solve Eq. (5), we introduce a new dependent variable y = 2%-P/( ya)
‘I2
(7)
VI. Karpman/Physics
Letter.? A 244 (1998)
Then Eq. (5) is reduced to
(8) !‘I1
This equation defines y(t); yo denotes y(t) at t = 0. As far as a decreases, y > ye. Eq. (5) is applicable if
394396
395
-7r < argy < 7r. One can also write some relationships between X,(y) and the incomplete r-function. However, they are not used in the present paper and will not be discussed here. From (12), (13) and (9) it follows that y-“e!’ _ y;“eYO z Dt,
(15)
where 1,
y >
?(I >> 1 .
(9) D = B,P5/2y-3/2.
Consider
.
(16)
now p > 1. Instead of (5), we have [ 21 Assuming,
in addition to (9)) that
da dt=
y > II In y,
xexp(-5).
(17)
y0 >> nlny0,
we have from ( 15)
where const is positive and does not depend on p and y. (It is constructed from rr, e, etc.) Generally, const N lo’-103. As far as da/dt < 0, Eq. ( 10) makes sense only at p < 4. (At p > 4, the radiating solitons are unstable [ 51.) Introducing, instead of (7), the new variable
y(t)
= y0 fln(l
-t t/&h),
where t,h is a characteristic tch = yo-“e!OD-’
(18) time, defined by
N p-5/2y3/2y;neY0
(19)
In terms of a,
(11) we reduce ( 10) to
s !
-‘llUe7, &,
17
=
B,p5/2y-3/2r,
(21)
(12)
?‘o
(9) and
where y and ya, as before, satisfy conditions B,, N I-10. 2. Integrating by parts the left-hand and (12), we can write
The characteristic time t& is large and, therefore, comparing the above results with computer and laboratory experiments one should, first of all, consider the case t << t,-h. Then from (21) it follows that
sides of (8) )
/
(t/&h
T”e” drl = X,(Y) - L(Yo)
<
(13)
3
1) .
(22)
In the opposite limit case
?‘o
where II = 4/p 3 0 and X,(y) asymptotic expansion at large y
(
x,,(y) = ypnev1 +
F+ n(n
is defined by the azaa
+ 1) y2
1+~ln(;i;;)]P2 c
[ f...
.
(14)
>
This relation uniquely determines y”X,( y) in the whole complex plane of y and, respectively, X,(y) at
(t/&h
>
1).
(23)
tch, the wavetrain becomes inhomogeneous (because its local amplitude depends on the soliton amplitude a at the moment of radiation; therefore, the
At t >
396
VI. Karpman/Physics
characteristic length of the homogeneity of radiated wavetrain is &, N &hug, where U, is the wave group velocity). This work was partly supported by the Israel Ministry of Science.
Letters A 244 (1998) 394-396
References
[I] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. 121 V.I. Karpman, 131 ES. Benilov, ( 1993) 270. [4] Y Pomeau, A. 127. [ 51 V.I. Karpman.
Phys. Lett. A 186 (1994) 303. R. Grimshaw. E.P. Kuznetsova. Physica Ramani, B. Grammaticos. Phys. Lett. A 215 (1996)
D 69
Physica D 31 ( 1988) 257.
PHYSICS
ELSEYIER
Physics Letters A 244 (1998)
LETTERS
A
394-396
Evolution of radiating solitons described by the fifth-order Korteweg-de Vries type equations V.I. Karpman ’ Racah Institute of Physics. Hebrew University, Jerusalem Received
10 March 1998; accepted
for publication
91904. Israel
17 March 1998
Communicated by V.M. Agranovich
Abstract The time dependence of velocities and amplitudes of radiating solitons, described type equations is investigated. @ 1998 Elsevier Science B.V.
1. The resonant radiation of solitons due to highorder dispersive effects leads to their damping. The goal of the present paper is an investigation of the evolution of soliton parameters, caused by the radiation, if they are described by the Sth-order Kortewegde Vries (kdV) -type equations
a,u + U”&U + pa$
+ y$u
= 0
(1)
(y is small) with integer p 2 1. At p = 1 and p = 2 we have, respectively, the Sth-order KdV and modified KdV (MKdV) equations. For definiteness, we assume /? > 0. Then the condition of radiation is y > 0 [ 1,2]. At y = 0, the soliton solution of Eq. (1) is uos = a 1’PFo(5),
5 = la/P11’2[x
- x0(t) I?
*qJ>o,
24=
by the fifth-order
a”p[Fo(5)+ f(5,
t) I
Korteweg-de
Vries
(4)
3
where f(&, t) at sufficiently large 5 and t describes the soliton resonant radiation. At p = 1, the decrease of the soliton velocity caused by its radiation is described by the following differential equation [ I], da
_ -I&-*
z-
F
5’2p3~y~)-‘12exp
(-
z).
0 (5)
A similar equation (with a different numerical coefficient) was obtained in Ref. [ 31. (See also the pioneering work [4] where, however, there is another power of a before the exponent). Eq. (5) is valid at the condition
dt
Fo(&
= [i(p
+ l)(p
yu/p2 +2)11ipsech2ip(~~~).
<< 1 ,
(6)
(3)
At small positive y, the soliton solution (2) is replaced by ’ E-mail: [email protected]. 037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00309-O
which is assumed everywhere in this paper. To solve Eq. (5), we introduce a new dependent variable y = 2%-P/( ya)
‘I2
(7)
VI. Karpman/Physics
Letter.? A 244 (1998)
Then Eq. (5) is reduced to
(8) !‘I1
This equation defines y(t); yo denotes y(t) at t = 0. As far as a decreases, y > ye. Eq. (5) is applicable if
394396
395
-7r < argy < 7r. One can also write some relationships between X,(y) and the incomplete r-function. However, they are not used in the present paper and will not be discussed here. From (12), (13) and (9) it follows that y-“e!’ _ y;“eYO z Dt,
(15)
where 1,
y >
?(I >> 1 .
(9) D = B,P5/2y-3/2.
Consider
.
(16)
now p > 1. Instead of (5), we have [ 21 Assuming,
in addition to (9)) that
da dt=
y > II In y,
xexp(-5).
(17)
y0 >> nlny0,
we have from ( 15)
where const is positive and does not depend on p and y. (It is constructed from rr, e, etc.) Generally, const N lo’-103. As far as da/dt < 0, Eq. ( 10) makes sense only at p < 4. (At p > 4, the radiating solitons are unstable [ 51.) Introducing, instead of (7), the new variable
y(t)
= y0 fln(l
-t t/&h),
where t,h is a characteristic tch = yo-“e!OD-’
(18) time, defined by
N p-5/2y3/2y;neY0
(19)
In terms of a,
(11) we reduce ( 10) to
s !
-‘llUe7, &,
17
=
B,p5/2y-3/2r,
(21)
(12)
?‘o
(9) and
where y and ya, as before, satisfy conditions B,, N I-10. 2. Integrating by parts the left-hand and (12), we can write
The characteristic time t& is large and, therefore, comparing the above results with computer and laboratory experiments one should, first of all, consider the case t << t,-h. Then from (21) it follows that
sides of (8) )
/
(t/&h
T”e” drl = X,(Y) - L(Yo)
<
(13)
3
1) .
(22)
In the opposite limit case
?‘o
where II = 4/p 3 0 and X,(y) asymptotic expansion at large y
(
x,,(y) = ypnev1 +
F+ n(n
is defined by the azaa
+ 1) y2
1+~ln(;i;;)]P2 c
[ f...
.
(14)
>
This relation uniquely determines y”X,( y) in the whole complex plane of y and, respectively, X,(y) at
(t/&h
>
1).
(23)
tch, the wavetrain becomes inhomogeneous (because its local amplitude depends on the soliton amplitude a at the moment of radiation; therefore, the
At t >
396
VI. Karpman/Physics
characteristic length of the homogeneity of radiated wavetrain is &, N &hug, where U, is the wave group velocity). This work was partly supported by the Israel Ministry of Science.
Letters A 244 (1998) 394-396
References
[I] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. 121 V.I. Karpman, 131 ES. Benilov, ( 1993) 270. [4] Y Pomeau, A. 127. [ 51 V.I. Karpman.
Phys. Lett. A 186 (1994) 303. R. Grimshaw. E.P. Kuznetsova. Physica Ramani, B. Grammaticos. Phys. Lett. A 215 (1996)
D 69
Physica D 31 ( 1988) 257.