- Email: [email protected]
Stationary solitary waves of the fifth order KdV-type equations
21 March 1994
N
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 186 (1994) 300-302
Stationary solitary waves of the fifth order KdV-type equations V.I. Karpman Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Received 12 December 1993; accepted for publication 29 December 1993 Communicated by V.M. Agranovich
Abstract
Stationary solitary waves ("bright" and "dark" solitons), as well as kinks, satisfying the fifth order KdV-type equations, are considered. Necessary conditions for their existence are derived.
We study solitary waves and kinks of the fifth order KdV-type equations,
Ou -at
p Ou
O3u
OSu
sgn(yfl)e2F (Iv) ( ~) + F " ( ~) + sgn(afl___~) [ F p + l ( ~ ) _ Cp+l ] p+l
_ (i)
+
with integer p, 1 ~ p ~ 3. Our approach is a generalization o f the one developed for the fifth order K d V ( p = 1 ) and third order nonlinear SchrSdinger ( N S ) equations [ 1,2]. F o r p > 3 and y = 0 , the soliton solutions o f Eq. (1) are unstable [ 3 - 5 ] and the role o f the last term in ( 1 ) will be investigated separately. A stationary soliton solution o f Eq. (1) can be written as
u = la/otl ' / P F ( ~ ) ,
(2)
- sgn(pa) [F(~) - C] = 0 , where C = F ( - ~
),
e= l ya/flZ l 1/2.
(5)
For odd p we can assume that a f t > 0 because the opposite case can be reduced to the first one. For even p (in this paper p = 2 ) the sign o f otfl is essential. Assume, first, that F ( - ~ ) = F ( o o ) = C and consider the propagation of a small amplitude monochromatic wave on the background u = l a / a l t/PC, i.e. take
u(x, t ) = l a l a l l/p
where
× [C+constXexp(ikx-ig2t) ],
~= la/fll l/Z[X-Xo( t) ],
(4)
dx o
--dT- = a ,
(3)
a = c o n s t and F(~) is a dimensionless function. Substituting (2) into ( 1 ), we obtain E-mail: [email protected].
(6)
where the constant is small. Substituting (6) in the linearized equation ( 1 ), we find the dispersion equation £2=I2(k) and, respectively, the phase velocity V(k) = K2( k ) / k. The stationary soliton with velocity a can exist only if the equation
V(k) =a
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI0375-9601 (94)00068-Z
(7)
EL Karpman /Physics Letters A 186 (1994) 300-302
does not have real roots k [ 1 ]. Otherwise, it would be a resonant interaction between the soliton and the radiation with the wave number k, satisfying Eq. (7). In the soliton frame, the phase velocity of the radiation is V(k) - a. Therefore, substituting F(~) = C + const X exp (ik~)
(8)
into the linearized equation (4), we have Eq. (7) in the form
dl=sgn(afl)CV-sgn(fta). (10)
For the stationary soliton, Eq. (9) cannot have real roots, i.e. k2 - 1 1 + ( 1 - 4 d o d l ) 1/2
2
-
do
(11)
must be either complex or negative. So, one of the two following conditions must be satisfied,
4dod~> 1,
(12)
4dodl
do<0,
dl<0.
(13)
Conditions (12) and (13) may be replaced, respectively, by d~>0,
4dod~>l
(14)
(complex/~2) or do<0,
(19)
where Re x > 0
(20)
and the equation for x is (21)
Condition (20) is satisfied if x 2 is complex or positive. Comparing (21) with (9), we see that 22= - k 2 and, therefore, (14) or (15) ensure (20). On the other hand, if dido<0,
dl<0
(15)
(complex or negative 22). For aft> O, both conditions are consistent with the assumption C--O.
(16)
In this case, (14) and ( 15 ) take the forms aft>0,
~,ft>0,
fta0.5,
(17)
aft>0,
~/ft<0,
fta>0.
(18)
For p = 1, the soliton solutions of Eq. (4), satisfying these conditions, have been obtained numerically by Kawahara [ 6 ]. Therefore, in this case (17) and ( 18 ) are not only necessary, but also sufficient conditions. For p > 1, the sufficiency of (17) and (18) has not yet been proven.
(22)
Eq. (21) has both positive and negative ~2. Therefore, condition (22) is enough to satisfy (20), but in this case there is a positive k 2 = - 22, which means that the stationary soliton does not exist. In the case C = 0 and aft> 0, condition (22) means that yft< 0,
or
do>0,
F(~)=C+const×exp(-T-x~), ~ + o o ,
(9)
where do=sgn(Tft)E 2,
Let us now examine the asymptotics of F(~) for ~ o o . We find
dol~4wx2Wd] = 0 .
dok4-k2+dl = 0 ,
301
fta
(23)
fta>O.
(24)
or
7ft> 0,
This explains why for condition (23) solitons have not been found numerically for any E [6 ]. For condition (24), Kawahara [6] reported on soliton type solutions at sufficiently small ~, but they were, actually, radiating solitons [ 1 ]. Conditions (17), ( 18 ) and (23), (24) were previously obtained in Ref. [ 1 ] by means of a geometrical approach to Eq. (7). Turning to C # 0, we assume a f t < 0. This is important only for the fifth order MKDV equation with p = 2 . Thus, instead of (4) we now consider the equation sgn (Tfl) E2F °v) (~) + F " (~) - ~ [F3(~) - - C 3 ] - sgn (fta) [F(~) - C] = 0 .
(25)
The necessary conditions for the existence of stationary solitons, following from (12) and (13), are
aftO, fla ~ ( 1 - C 2 ) - 1 > 0 , (26)
302 aft
V.L Karpman / PhysicsLettersA 186 (1994) 300-302 7ft<0,
fta<0,
C2>I,
(27)
;,ft<0,
fta>0,
C2<~.
(28)
or aft<0,
There is also the possibility of a stationary kink with F ( - o o ) = C,
F(oo) = - C .
(29)
Then, in the limit ~--,oo, from (25) follows C= +x//3,
fta
(30)
This coincides with the corresponding relations for the M K D V kink at e = 0. Substituting (8) with (29) and (30) in (25) for ~-~ _ oo, we come to the following, additional to (30), conditions for the existence of stationary kinks, aft
7ft<0 •
litons, described by Eq. (1), can exist only under conditions (14) or ( 15 ). For a f t > 0 and C = 0 , (14) and (15) lead to (17) and (18). For a f t < 0 (which is essential only for p = 2 ) and F ( ~ = o o ) = F(~=-oo)=C~0, from (14) and (15) follow (26)-(28). The stationary kinks, described by Eq. (1) with p = 2 and ( 29 ), can exist only under'conditions ( 30 ) and (31).
(31)
In conclusion, we have shown that the stationary so-
References [ 1] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. [2] V.I. Karpman, Phys. Lett. A 181 (1993) 211. [3]M.J. Ablowitz and H. Segur, Solitons and the inverse scatteringtransform (SIAM, Philadelphia, PA, 1981). [4] E.A. Kuznetsov,A.M. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103. [ 5 ] J.J. Rasmussenand K. Rypdal, Phys. Scr. 33 (1986) 481. [6] T. Kawahara, J. Phys. Soc. Japan 33 (1972) 260.
N
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 186 (1994) 300-302
Stationary solitary waves of the fifth order KdV-type equations V.I. Karpman Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Received 12 December 1993; accepted for publication 29 December 1993 Communicated by V.M. Agranovich
Abstract
Stationary solitary waves ("bright" and "dark" solitons), as well as kinks, satisfying the fifth order KdV-type equations, are considered. Necessary conditions for their existence are derived.
We study solitary waves and kinks of the fifth order KdV-type equations,
Ou -at
p Ou
O3u
OSu
sgn(yfl)e2F (Iv) ( ~) + F " ( ~) + sgn(afl___~) [ F p + l ( ~ ) _ Cp+l ] p+l
_ (i)
+
with integer p, 1 ~ p ~ 3. Our approach is a generalization o f the one developed for the fifth order K d V ( p = 1 ) and third order nonlinear SchrSdinger ( N S ) equations [ 1,2]. F o r p > 3 and y = 0 , the soliton solutions o f Eq. (1) are unstable [ 3 - 5 ] and the role o f the last term in ( 1 ) will be investigated separately. A stationary soliton solution o f Eq. (1) can be written as
u = la/otl ' / P F ( ~ ) ,
(2)
- sgn(pa) [F(~) - C] = 0 , where C = F ( - ~
),
e= l ya/flZ l 1/2.
(5)
For odd p we can assume that a f t > 0 because the opposite case can be reduced to the first one. For even p (in this paper p = 2 ) the sign o f otfl is essential. Assume, first, that F ( - ~ ) = F ( o o ) = C and consider the propagation of a small amplitude monochromatic wave on the background u = l a / a l t/PC, i.e. take
u(x, t ) = l a l a l l/p
where
× [C+constXexp(ikx-ig2t) ],
~= la/fll l/Z[X-Xo( t) ],
(4)
dx o
--dT- = a ,
(3)
a = c o n s t and F(~) is a dimensionless function. Substituting (2) into ( 1 ), we obtain E-mail: [email protected].
(6)
where the constant is small. Substituting (6) in the linearized equation ( 1 ), we find the dispersion equation £2=I2(k) and, respectively, the phase velocity V(k) = K2( k ) / k. The stationary soliton with velocity a can exist only if the equation
V(k) =a
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI0375-9601 (94)00068-Z
(7)
EL Karpman /Physics Letters A 186 (1994) 300-302
does not have real roots k [ 1 ]. Otherwise, it would be a resonant interaction between the soliton and the radiation with the wave number k, satisfying Eq. (7). In the soliton frame, the phase velocity of the radiation is V(k) - a. Therefore, substituting F(~) = C + const X exp (ik~)
(8)
into the linearized equation (4), we have Eq. (7) in the form
dl=sgn(afl)CV-sgn(fta). (10)
For the stationary soliton, Eq. (9) cannot have real roots, i.e. k2 - 1 1 + ( 1 - 4 d o d l ) 1/2
2
-
do
(11)
must be either complex or negative. So, one of the two following conditions must be satisfied,
4dod~> 1,
(12)
4dodl
do<0,
dl<0.
(13)
Conditions (12) and (13) may be replaced, respectively, by d~>0,
4dod~>l
(14)
(complex/~2) or do<0,
(19)
where Re x > 0
(20)
and the equation for x is (21)
Condition (20) is satisfied if x 2 is complex or positive. Comparing (21) with (9), we see that 22= - k 2 and, therefore, (14) or (15) ensure (20). On the other hand, if dido<0,
dl<0
(15)
(complex or negative 22). For aft> O, both conditions are consistent with the assumption C--O.
(16)
In this case, (14) and ( 15 ) take the forms aft>0,
~,ft>0,
fta
(17)
aft>0,
~/ft<0,
fta>0.
(18)
For p = 1, the soliton solutions of Eq. (4), satisfying these conditions, have been obtained numerically by Kawahara [ 6 ]. Therefore, in this case (17) and ( 18 ) are not only necessary, but also sufficient conditions. For p > 1, the sufficiency of (17) and (18) has not yet been proven.
(22)
Eq. (21) has both positive and negative ~2. Therefore, condition (22) is enough to satisfy (20), but in this case there is a positive k 2 = - 22, which means that the stationary soliton does not exist. In the case C = 0 and aft> 0, condition (22) means that yft< 0,
or
do>0,
F(~)=C+const×exp(-T-x~), ~ + o o ,
(9)
where do=sgn(Tft)E 2,
Let us now examine the asymptotics of F(~) for ~ o o . We find
dol~4wx2Wd] = 0 .
dok4-k2+dl = 0 ,
301
fta
(23)
fta>O.
(24)
or
7ft> 0,
This explains why for condition (23) solitons have not been found numerically for any E [6 ]. For condition (24), Kawahara [6] reported on soliton type solutions at sufficiently small ~, but they were, actually, radiating solitons [ 1 ]. Conditions (17), ( 18 ) and (23), (24) were previously obtained in Ref. [ 1 ] by means of a geometrical approach to Eq. (7). Turning to C # 0, we assume a f t < 0. This is important only for the fifth order MKDV equation with p = 2 . Thus, instead of (4) we now consider the equation sgn (Tfl) E2F °v) (~) + F " (~) - ~ [F3(~) - - C 3 ] - sgn (fta) [F(~) - C] = 0 .
(25)
The necessary conditions for the existence of stationary solitons, following from (12) and (13), are
aft
302 aft
V.L Karpman / PhysicsLettersA 186 (1994) 300-302 7ft<0,
fta<0,
C2>I,
(27)
;,ft<0,
fta>0,
C2<~.
(28)
or aft<0,
There is also the possibility of a stationary kink with F ( - o o ) = C,
F(oo) = - C .
(29)
Then, in the limit ~--,oo, from (25) follows C= +x//3,
fta
(30)
This coincides with the corresponding relations for the M K D V kink at e = 0. Substituting (8) with (29) and (30) in (25) for ~-~ _ oo, we come to the following, additional to (30), conditions for the existence of stationary kinks, aft
7ft<0 •
litons, described by Eq. (1), can exist only under conditions (14) or ( 15 ). For a f t > 0 and C = 0 , (14) and (15) lead to (17) and (18). For a f t < 0 (which is essential only for p = 2 ) and F ( ~ = o o ) = F(~=-oo)=C~0, from (14) and (15) follow (26)-(28). The stationary kinks, described by Eq. (1) with p = 2 and ( 29 ), can exist only under'conditions ( 30 ) and (31).
(31)
In conclusion, we have shown that the stationary so-
References [ 1] V.I. Karpman, Phys. Rev. E 47 (1993) 2073. [2] V.I. Karpman, Phys. Lett. A 181 (1993) 211. [3]M.J. Ablowitz and H. Segur, Solitons and the inverse scatteringtransform (SIAM, Philadelphia, PA, 1981). [4] E.A. Kuznetsov,A.M. Rubenchik and V.E. Zakharov, Phys. Rep. 142 (1986) 103. [ 5 ] J.J. Rasmussenand K. Rypdal, Phys. Scr. 33 (1986) 481. [6] T. Kawahara, J. Phys. Soc. Japan 33 (1972) 260.