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A note on the wronskian form of solutions of the KdV equation
Volume 134, number
1
PHYSICS
A NOTE ON THE WRONSKIAN S. SIRIANUNPIBOON, Department
OfMathematics,
FORM
SD. HOWARD
OF SOLUTIONS
received
12 December
1988
OF THE KdV EQUATION
and SK. ROY
LaTrobe University, Bundoora,
Received 20 July 1988; revised manuscript Communicated by A.P. Fordy
LETTERS A
Victoria. Australia 3083
16 September
1988; accepted
for publication
30 September
I988
We show here that the KdV equation has solutions in the wronskian form under more general conditions than those considered previously by other authors. These conditions are used to generate new types of solutions of the KdV in the wronskian form.
1. Introduction It is well-known
that the KdV equation
(1.1)
u,+6uuX+uXXX=0 can be written DJD,
in the form
+Di)f.!-
=L*f-.fx+Lxxxf- 4f,,fx+3cf,)‘=O
3
(1.2)
In this note we generalise ( 1.5 ) and use the wronskian technique not only to verify the solutions but also to generate new solutions of the KdV equation. The method may be applied to other equations whose N-soliton solution can be written in the form of a wronskian determinant. In particular, it is applicable to equations such as the MKdV, Boussinesq and the KP equation. These extensitions will be discussed thoroughly elsewhere.
where u=2(lnfl,,
(1.3)
and D,, D, are Hirota derivatives [ 11. In the inverse scattering transformation formalism, f is the determinant of an NX N matrix (see e.g. ref. [ 2 ] ). In the Hirota formalism [ 3 1, f is an nth order polynomial in N exponentials. In the wronskian formalism of Freeman and Nimmo [4-6 J, for the N-soliton solution of ( 1.2), f can be written as f= W(S,, S2, .... S,), where W is the wronskian determinant of the N functions S,, .... S,, each of the functions being a one-soliton solution of ( 1.2), satisfying the equations X$/at=
-4a?giax3,
a*s,/a.&?ps,,
i= 1, 2, .... N.
(1.4) (1.5)
The advantage of the Freeman-Nimmo method lies in the fact that the derivatives of the wronskian determinant can be easily written down and one can then directly verify by substitution that the solution f= W(S, ....S,) satisfies ( 1.2) or its auto-Backlund transformation. 0375-9601/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
2. Sufficient conditions the KdV equation
for the wronskian
to satisfy
Theorem. If a set of functions {(@,(x, t)}, i= 1, 2, ,..) N, satisfies the partial differential equations a&/at=
-4a3g,/ax3,
(2.1)
(2.2) where the Cyijare independent &) is a solution of (1.2).
of x, then f= W( @,,....
Proof: (In the following we use the compact notation of Freeman and Nimmo [ 51 for the wronskian and its derivatives.) Writing f= W(@,, *.., @~)=(N-ll), we obtain
B.V.
from (1.2) using (2.1):
31
Volume 134, number I
PHYSICS LETTERS A
WD, +D:l_f.f
i= 1, .... N, which satisfy (2.1) and (2.2), where A, and B, are constants, we find that W( u/( r], ), w( q2), .... w( q,,,) ) with the q, all different is also a solution to ( 1.2). This N-soliton solution was first derived by Satsuma [ 71. (b) The technique of finding other types of functions satisfying (2.1) and (2.2) is to expand w in some parameter. As an illustration, we expand (3.1) as follows:
z-~(N^~)(NL~,N-~,N-~,N-~,N) +6(N-l)(N13,N,N+l) -3(N:l)(N12,
N+3)
-12(N~2,N)(N~3,N-l,N+l) +3(N-l)(N^4,N-2,N-l,N+l) +3(N-l)(Nl3,N-1,
12 December I988
N+2) u/(n+J)
+3[(NL2,N-l,N)+(N:2,N-l)]?
= mt, P,(r/)drn
>
(3.2)
(2.3)
Now using (2.2), and the general result that for any determinant IA,)
where the P, (r]) are functions of x, t and q. From the equations for vt and w,.., we get Pl(rt) = -4Pm_&rl) PO,(V) =?Po(rl)
where I&j k denotes the determinant \A,,1 with its kth row differentiated twice with respect to x and IA, (k the determinant )A,( with its kth column differentiated twice with respect to x, we deduce the following identities:
,$,ai,(t) (N- 1) =--(Nf_3,N_l,N)+(N^2,N+l),
(2.4)
+P,_,(q),
9 7
ma2.
Hence, according to the theorem proved above, WPO, pi, ..‘, PN) is a solution of ( 1.2 ) for each N. In particular, taking y(q) from (3.1) with A=B= 1, we have [m/31 [(m-$)/21
Pm(?)= 1
n=O
x (x-
=(N^&N-2,N-l,N)+(Nl3,N,N+l)
(3.3)
/=O
(_ 1 )/+n(121)f4n
(m-21-3n)!n!l!
12q*tyf--2-++n
x{exp[q(x-4q2t)]+(-l)/+n -(N-3,N-l,N+2)
Xew[
-(N-4,N-2,N-1,NSl) +(N-3,N,N+l)+(NI2,N+3).
(2.5)
Using (2.4) and (2.5) with (2.3) we find that the right-hand side of (2.3) is the Laplace expansion by NX N minors of a determinant whose value if zero, that is, S= I+‘(@,, &, .... tiN) is a solution to ( 1.2). 3. Applications (a) Starting with the one-soliton KdV equation: f=W(?,)=A,
exp[tll(x-4$t)
+B,fw[-d=--4vft)l, 32
solutions
of the
-v(x-4tt2t) I),
where [I] is the greatest integer G r. If we take r] as purely imaginary we obtain from (3.4) the solutions which have been recently reported by Sharipov [ 8 1. The solutions of the type (3.4) can also be obtained by repeated applications of the Backlund transformation keeping the eigenvalue the same at each step. (c) Taking A= -B=l in (3.1) and writing:
w(q) =
mEo QmV >
we have
I (3.1)
(3.4)
Qm,= - 4Qmxxx 5
(3.5)
Volume 134, number
PHYSICS
1
Qo,=O, (3.6)
Hence WQo,Q,, ....QN) is a solution of (1.2) for each N. The Qm are easily calculated from (3.5 ): 1(2m+l)/31
C
PI=0
(2m+
(-4)” I-3n)!n!
X2m+
I--3np
(3.7) The solutions W(Qo, Ql, . ... QN) are the rational solutions (see, e.g., ref. [9] ). This type of solutions was also obtained by Nimmo and Freeman [4] by taking the limit vi+0 of the multisoliton solution of (1.2). (d) We also note that the wronskian of the type
Q,,, f’o(tlz), f= WQo> ...> p/k(vk),
A
12 December
1988
4. Conclusion
Qmxx=Qm--l .
Q,=
LETTERS
w(%+l),
.‘.>
. . . . p12(v2),
. . . . PO(G),
... ,
w(%v))
We have shown that the wronskian determinant of a set of functions {@i}, satisfying the more general type of equations (2.1) and (2.2) than the usual Schradinger type of equations ( 1.4) and ( 1.5) satisfies the bilinear form of the KdV equation ( 1.2). Eqs. (2.1) and (2.2) can be treated as sufficient conditions on the functions {@,}to make the wronskian a solution of ( 1.2). The method provides a simple way of producing a class of new solutions of the KdV. The relationship of these solutions with the BHcklund transformation will be discussed in detail elsewhere.
References [ I] R. Hirota, in: Solitons, eds. R.K. Bullough and P.J. Caudrey
satisfies ( 1.2). The solution describes the interaction between the solutions given in (b) and (c) and the multisoliton solutions. Some examples of such solutions are as follows:
(Springer, Berlin, 1980). [2] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (Siam, Philadelphia, 198 1) p. 172. [ 31 R. Hirota, Phys. Rev. Lett. 27 ( 197 1) 1192. [4] N.C. Freeman and J.J.C. Nimmo, Phys. Lett. A 95 (1983)
[S]J.J.C. f=W(Qo,V/,(?))=2rlxsinh[?(x-412t)l
-2 cosh[q(x-4r12t)]
,
which is the accelerating soliton solution given by Au and Fung [ IO]. Similarly .f=
WQotpo(tl),P,(v)) =2q2 cosh[2q(x-4q2t)
originally
Nimmo and N.C. Freeman, J. Phys. A 17 (1984) 1415. [6] N.C. Freeman, J. Appl. Math. 32 (1984) 125. [ 7 ] J. Satsuma, J. Phys. Sot. Japan 46 ( 1979) 359. [8] R.A. Sharipov, Sov. Phys. Doki. 32 (1987) 121. [9] M.J. Ablowitz and J. Satsuma, J. Math. Phys. 19 (1978) 2180. [lo] C. Au and P.C. Fung, J. Math. Phys. 25 (1984) 1364.
]
-6$~-4~sinh[2~(~-4~~t)] is a new solution.
33
1
PHYSICS
A NOTE ON THE WRONSKIAN S. SIRIANUNPIBOON, Department
OfMathematics,
FORM
SD. HOWARD
OF SOLUTIONS
received
12 December
1988
OF THE KdV EQUATION
and SK. ROY
LaTrobe University, Bundoora,
Received 20 July 1988; revised manuscript Communicated by A.P. Fordy
LETTERS A
Victoria. Australia 3083
16 September
1988; accepted
for publication
30 September
I988
We show here that the KdV equation has solutions in the wronskian form under more general conditions than those considered previously by other authors. These conditions are used to generate new types of solutions of the KdV in the wronskian form.
1. Introduction It is well-known
that the KdV equation
(1.1)
u,+6uuX+uXXX=0 can be written DJD,
in the form
+Di)f.!-
=L*f-.fx+Lxxxf- 4f,,fx+3cf,)‘=O
3
(1.2)
In this note we generalise ( 1.5 ) and use the wronskian technique not only to verify the solutions but also to generate new solutions of the KdV equation. The method may be applied to other equations whose N-soliton solution can be written in the form of a wronskian determinant. In particular, it is applicable to equations such as the MKdV, Boussinesq and the KP equation. These extensitions will be discussed thoroughly elsewhere.
where u=2(lnfl,,
(1.3)
and D,, D, are Hirota derivatives [ 11. In the inverse scattering transformation formalism, f is the determinant of an NX N matrix (see e.g. ref. [ 2 ] ). In the Hirota formalism [ 3 1, f is an nth order polynomial in N exponentials. In the wronskian formalism of Freeman and Nimmo [4-6 J, for the N-soliton solution of ( 1.2), f can be written as f= W(S,, S2, .... S,), where W is the wronskian determinant of the N functions S,, .... S,, each of the functions being a one-soliton solution of ( 1.2), satisfying the equations X$/at=
-4a?giax3,
a*s,/a.&?ps,,
i= 1, 2, .... N.
(1.4) (1.5)
The advantage of the Freeman-Nimmo method lies in the fact that the derivatives of the wronskian determinant can be easily written down and one can then directly verify by substitution that the solution f= W(S, ....S,) satisfies ( 1.2) or its auto-Backlund transformation. 0375-9601/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
2. Sufficient conditions the KdV equation
for the wronskian
to satisfy
Theorem. If a set of functions {(@,(x, t)}, i= 1, 2, ,..) N, satisfies the partial differential equations a&/at=
-4a3g,/ax3,
(2.1)
(2.2) where the Cyijare independent &) is a solution of (1.2).
of x, then f= W( @,,....
Proof: (In the following we use the compact notation of Freeman and Nimmo [ 51 for the wronskian and its derivatives.) Writing f= W(@,, *.., @~)=(N-ll), we obtain
B.V.
from (1.2) using (2.1):
31
Volume 134, number I
PHYSICS LETTERS A
WD, +D:l_f.f
i= 1, .... N, which satisfy (2.1) and (2.2), where A, and B, are constants, we find that W( u/( r], ), w( q2), .... w( q,,,) ) with the q, all different is also a solution to ( 1.2). This N-soliton solution was first derived by Satsuma [ 71. (b) The technique of finding other types of functions satisfying (2.1) and (2.2) is to expand w in some parameter. As an illustration, we expand (3.1) as follows:
z-~(N^~)(NL~,N-~,N-~,N-~,N) +6(N-l)(N13,N,N+l) -3(N:l)(N12,
N+3)
-12(N~2,N)(N~3,N-l,N+l) +3(N-l)(N^4,N-2,N-l,N+l) +3(N-l)(Nl3,N-1,
12 December I988
N+2) u/(n+J)
+3[(NL2,N-l,N)+(N:2,N-l)]?
= mt, P,(r/)drn
>
(3.2)
(2.3)
Now using (2.2), and the general result that for any determinant IA,)
where the P, (r]) are functions of x, t and q. From the equations for vt and w,.., we get Pl(rt) = -4Pm_&rl) PO,(V) =?Po(rl)
where I&j k denotes the determinant \A,,1 with its kth row differentiated twice with respect to x and IA, (k the determinant )A,( with its kth column differentiated twice with respect to x, we deduce the following identities:
,$,ai,(t) (N- 1) =--(Nf_3,N_l,N)+(N^2,N+l),
(2.4)
+P,_,(q),
9 7
ma2.
Hence, according to the theorem proved above, WPO, pi, ..‘, PN) is a solution of ( 1.2 ) for each N. In particular, taking y(q) from (3.1) with A=B= 1, we have [m/31 [(m-$)/21
Pm(?)= 1
n=O
x (x-
=(N^&N-2,N-l,N)+(Nl3,N,N+l)
(3.3)
/=O
(_ 1 )/+n(121)f4n
(m-21-3n)!n!l!
12q*tyf--2-++n
x{exp[q(x-4q2t)]+(-l)/+n -(N-3,N-l,N+2)
Xew[
-(N-4,N-2,N-1,NSl) +(N-3,N,N+l)+(NI2,N+3).
(2.5)
Using (2.4) and (2.5) with (2.3) we find that the right-hand side of (2.3) is the Laplace expansion by NX N minors of a determinant whose value if zero, that is, S= I+‘(@,, &, .... tiN) is a solution to ( 1.2). 3. Applications (a) Starting with the one-soliton KdV equation: f=W(?,)=A,
exp[tll(x-4$t)
+B,fw[-d=--4vft)l, 32
solutions
of the
-v(x-4tt2t) I),
where [I] is the greatest integer G r. If we take r] as purely imaginary we obtain from (3.4) the solutions which have been recently reported by Sharipov [ 8 1. The solutions of the type (3.4) can also be obtained by repeated applications of the Backlund transformation keeping the eigenvalue the same at each step. (c) Taking A= -B=l in (3.1) and writing:
w(q) =
mEo QmV >
we have
I (3.1)
(3.4)
Qm,= - 4Qmxxx 5
(3.5)
Volume 134, number
PHYSICS
1
Qo,=O, (3.6)
Hence WQo,Q,, ....QN) is a solution of (1.2) for each N. The Qm are easily calculated from (3.5 ): 1(2m+l)/31
C
PI=0
(2m+
(-4)” I-3n)!n!
X2m+
I--3np
(3.7) The solutions W(Qo, Ql, . ... QN) are the rational solutions (see, e.g., ref. [9] ). This type of solutions was also obtained by Nimmo and Freeman [4] by taking the limit vi+0 of the multisoliton solution of (1.2). (d) We also note that the wronskian of the type
Q,,, f’o(tlz), f= WQo> ...> p/k(vk),
A
12 December
1988
4. Conclusion
Qmxx=Qm--l .
Q,=
LETTERS
w(%+l),
.‘.>
. . . . p12(v2),
. . . . PO(G),
... ,
w(%v))
We have shown that the wronskian determinant of a set of functions {@i}, satisfying the more general type of equations (2.1) and (2.2) than the usual Schradinger type of equations ( 1.4) and ( 1.5) satisfies the bilinear form of the KdV equation ( 1.2). Eqs. (2.1) and (2.2) can be treated as sufficient conditions on the functions {@,}to make the wronskian a solution of ( 1.2). The method provides a simple way of producing a class of new solutions of the KdV. The relationship of these solutions with the BHcklund transformation will be discussed in detail elsewhere.
References [ I] R. Hirota, in: Solitons, eds. R.K. Bullough and P.J. Caudrey
satisfies ( 1.2). The solution describes the interaction between the solutions given in (b) and (c) and the multisoliton solutions. Some examples of such solutions are as follows:
(Springer, Berlin, 1980). [2] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (Siam, Philadelphia, 198 1) p. 172. [ 31 R. Hirota, Phys. Rev. Lett. 27 ( 197 1) 1192. [4] N.C. Freeman and J.J.C. Nimmo, Phys. Lett. A 95 (1983)
[S]J.J.C. f=W(Qo,V/,(?))=2rlxsinh[?(x-412t)l
-2 cosh[q(x-4r12t)]
,
which is the accelerating soliton solution given by Au and Fung [ IO]. Similarly .f=
WQotpo(tl),P,(v)) =2q2 cosh[2q(x-4q2t)
originally
Nimmo and N.C. Freeman, J. Phys. A 17 (1984) 1415. [6] N.C. Freeman, J. Appl. Math. 32 (1984) 125. [ 7 ] J. Satsuma, J. Phys. Sot. Japan 46 ( 1979) 359. [8] R.A. Sharipov, Sov. Phys. Doki. 32 (1987) 121. [9] M.J. Ablowitz and J. Satsuma, J. Math. Phys. 19 (1978) 2180. [lo] C. Au and P.C. Fung, J. Math. Phys. 25 (1984) 1364.
]
-6$~-4~sinh[2~(~-4~~t)] is a new solution.
33