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On the inverse scattering transform, the cylindrical Korteweg-De Vries equation and similarity solutions
Volume 72A, number 3
PHYSICS LETTERS
9 July 1979
ON THE INVERSE SCATTERING TRANSFORM, THE CYLINDRICAL KORTEWEG-DE VRIES EQUATION AND SIMILARITY SOLUTIONS
R.S. JOHNSON School of Mathematics, Universityof Newcastle upon Tyne, NE1 7RL: UK Received 3 May 1979
It is shown that the inverse scattering transform for the cylindrical KdV equation can be obtained directly from that for the 2-D KdV equation. Using this transform for the similarity solution, we obtain two representations of a Painlevk transcendent.
The inverse scattering transform for the twodimensional Korteweg-de Vries (KdV) equation, (4u, + 6uu, + u,,),
+ 3uyy = 0 ,
(1)
is well known: see refs [ 1,2] , for example. (Throughout, subscripts will denote partial derivatives.) Adopting this approach, we can formulate the solution of eq. (1) as @,Y,
0 = 2@/WK(x,x;y,
t) 1
(2)
where
[ =.X/(12t)“3,
z = (z + &(3t))/(3t)’
Y=(x
9
-zvO),o
wherefb,
r;y, t)
z;_Y,
oas=0 ,
(3)
x
(8)
(9)
if f=u/t,
Fxxx +Fzzz +Ft = 0,
F,, -F,,
+ Fy = 0. (4a,b)
Now, eq. (1) can be transformed under
to
, Z*, t) ,
T,, -~~~=(c;-z)%
with
X = 41/3(x t &(3t))
(7)
where G(Y) is also to be determined. Using (7) and (8) in eq. (4b) yields
m
J K(x, s;y, W(s,
‘3 ,
t) is to be chosen. Further, we set
F = (3t)-“3G(Y)7(c;
X(x, z;y, t) +%,
t
propriate inverse scattering transform for eq. (6). However, direct application of (5) is useless for it does not eliminate the variable y nor does it incorporate a suitable conjugate variable (equivalent to z). Thus we introduce
,
u = 42’3U(X, t),
(6)
after one integration in X provided U + 0 as IX I --*=: eq. (6) is the cylindrical KdV equation first given in ref. [3]. The transformation (5) should enable the solution described by eqs. (2)-(4) to be written as the ap-
G = exp(-Y/3),
(10)
and with this same choice eq. (4a) becomes 3;& + szzz
(5)
yield
u, + U/(2t) + 6UU~ + uxxx = 0 )
and
+ 3tTt - 9 = F,7[ +Z?,
(
(11)
although eq. (9) also must be employed to give eq. (11). Finally, the linear integral equation can be written as L(E, z; t) + 7(c;, z; t)
t
oa s L(~,s;r)9(s,Z;t)ds=O,
(12)
E
197
Volume 72A, number 3
PHYSICS LETTERS
where
9 July 1979
where
K = (3t)-‘13 exp(-Y/3)L(g,
2; t),
(16)
and then U([, t) = 2(12t)-2’3(a/ag)L(E
7,E. t) .
(13)
(We can note that Y = 0 on z =x i.e. on 2 = [.) Eqs. (9) (1 l)-(13) were written down in ref. [4] by direct analogy with eqs. (2)-(4) and an equivalent integration technique for eq. (6) was developed in ref. [5] by considering an inverse spectral problem with an additional linear potential. However, the equations as derived here seem a simpler representation of the solution. The solution to eqs. (9) and (11) can be expressed as
+k jM(~,p)SAj@+4)Ai(‘l+s)dqdp=0. 0
5
If we construct the similarity solution directly from eq. (6), and impose the necessary decay conditions at infinity, it follows that v2 = 2(d/dl;) ~~(5, 0
S)Ai(5 + s)ds 1
(17)
where II@) satisfies the Painleve equation u”tu3-.$J=o.
9 = s f(sr-“3)‘4iQ -m
+ S)Ai(Z + s)ds ,
(14)
for arbitrary f(.), and all solutions are given by eq. (14) if generalised functions are allowed. For example, choosing f( .) = 7~6(.), we obtain the simplest soliton solution from
L(.g,t; t) =
(a/al;)hl(l +kt-1’3J Ajqs’ds), I
(18)
Thus eqs. (16) and (17) constitute a linearisation of the Painleve equation: this representation is to be compared with that obtained in ref. [8]. It is fairly easy to confirm that both our linearisation and that given in ref. [8] produce identical Neuman expansions for u2([), but the general relationship between the two methods is less straightforward. Let us define N(x, y) such that N(x, 2y -X) = -(2/r) M(x, y -X),
(19)
where r2 = 2k (>O), then from eq. (16)
which is discussed in refs. [4-61. Further, letting the constant k + m , we produce a special similarity solution of eq. (6),
( )
N(x, y) - rAi y
(20)
U(& t) = 2(12~)_2/3(a2jaS2)~(jA16)dr). 5 which is considered in ref. [7] in connection with a water wave problem. The more general similarity solution is constructed by takingf(.) = kH(.), and then L&Z;
r) + kj- Ai@ + s)Ai(Z + s)ds 0
after some manipulation. Eq. (20) is essentially the equation given in ref. [8 3 for their K, , with very minor changes due to the particular form of the Painleve equation used here. The main result of ref. [8] is that u(t) =N(g, C;),written in our notation. Thus, from eqs. (19) and (17), we see that (214
u(E) = - (2Ir)MC; 90) and +kjL(&s;t)
E
jAi(s+p)Ai(Z+p)dpds=O, 0
u2Q) = 2(d/d0 jM(S.
which has the solution L&Z;
t) = jM(S, 0
198
s)Ai(Z + s)ds
,
s)Ai(t + s)ds
,
@lb)
0
(15)
ifM(t, s) satisfies eq. (16) with k = r2/2. The conclusion expressed by (2 1) is rather surprising: we have shown that two quite dissimilar forms exist for the same Painleve transcendent.
PHYSICS LETTERS
Volume 72A, number 3
We now demonstrate that both the expressions given in (21) can be obtained directly from our scattering transform without reference to the results in ref. [8]. First, it follows that if L([,Z; t) satisfies eq. (12), with 9([, 2; f) satisfying eqs. (9) and (1 I), then 3tL, + LEES+ L,,,
+ (3/4) (2CLE + i+L)
-.$LLg-ZLz-L=Q,
(22)
L,,-L,,=(&Z-li)L,
(23)
where V(& t) = 2(12t)-2’3a, CC.&0 = 2@/%)LQ, k 0, and for similarity solutions we require i, L and 9 independent oft. Now from eqs. (22) and (23) it can be shown that
upon the use of Airy’s equation and assuming L is defined for arbitrary Z. Evaluating eqs. (24) and (25) on s = 0 (provided wS is finite) shows that these equations are consistent only if w(C;,0) = hC1/2. The arbitrary constant X is fixed by requiring that eqs. (24), (25) follow from the linear integral equation for w(C;,s) (essentially eq. (16)): hence X = 1. The first part of (21) is now also proved. We have indicated how the inverse scattering transform for the cylindrical KdV equation can be derived from that for the 2-D KdV equation. Using this formulation, it is possible to obtain two different representations of the Painlevi transcendent (which describes similarity solutions of the KdV equation). Finally, it is interesting to observe that if the identity (2/r2)M2Q, 0) = (d/d&!)jk&
i’=(d/d[)u.
References
Evaluating this on Z = 4 yields the second result in (21) with li = u2 where u satisfies eq. (18). To obtain the bther expression for u(t), which previously required the use of ref. [8], we again start with eqs. (22) and (23). We define L&Z)=!
s)-$(5 + s)h
(implied by (2 1)) is used in the main theorem of ref. [8], their result follows without recourse to Neumann expansions.
where
(alapajaz),i =DL,
(25)
0
+2(t-z)(Lzz+L&=O,
D=
wtt=(.ps)w-iw,
+ 3ii’i + 3iiD,i - (32 + .$)Di
D3i - 2i
9 July 1979
wQ&+(Z+s)d&
w = -(2/r)M,
0
and then eqs. (22), (23) yield WEEE+(~Li-&v~-SwSt(~~‘-l)W=O,
(24)
[ 1] V.E. Zakharov and A.B. Shabat, Functional Anal. Appl. 8 (1974) 226. [2] V.S. Dryuma, JETP Lett. 19 (1974) 387. [3] S. Maxon and J. Viecelli, Phys. Fluids 17 (1974) 1614. [4] N.C. Freeman, Adv. Appl. Mech. Vol. 20 (1979), to be published. [S] F. Calogero and A. Degasperis, Reprint No. 93, Institute di Fisica G. Marconi, Rome (1978). [6] R.S. Johnson and S. Thompson, Phys. Lett. 66A (1978) 279. [7] R.S. Johnson, submitted to J. Fluid Mech. (1979). [a] M.J. Ablowitz and H. Segur, Phys. Rev. Lett. 38 (1977) 1103.
199
PHYSICS LETTERS
9 July 1979
ON THE INVERSE SCATTERING TRANSFORM, THE CYLINDRICAL KORTEWEG-DE VRIES EQUATION AND SIMILARITY SOLUTIONS
R.S. JOHNSON School of Mathematics, Universityof Newcastle upon Tyne, NE1 7RL: UK Received 3 May 1979
It is shown that the inverse scattering transform for the cylindrical KdV equation can be obtained directly from that for the 2-D KdV equation. Using this transform for the similarity solution, we obtain two representations of a Painlevk transcendent.
The inverse scattering transform for the twodimensional Korteweg-de Vries (KdV) equation, (4u, + 6uu, + u,,),
+ 3uyy = 0 ,
(1)
is well known: see refs [ 1,2] , for example. (Throughout, subscripts will denote partial derivatives.) Adopting this approach, we can formulate the solution of eq. (1) as @,Y,
0 = 2@/WK(x,x;y,
t) 1
(2)
where
[ =.X/(12t)“3,
z = (z + &(3t))/(3t)’
Y=(x
9
-zvO),o
wherefb,
r;y, t)
z;_Y,
oas=0 ,
(3)
x
(8)
(9)
if f=u/t,
Fxxx +Fzzz +Ft = 0,
F,, -F,,
+ Fy = 0. (4a,b)
Now, eq. (1) can be transformed under
to
, Z*, t) ,
T,, -~~~=(c;-z)%
with
X = 41/3(x t &(3t))
(7)
where G(Y) is also to be determined. Using (7) and (8) in eq. (4b) yields
m
J K(x, s;y, W(s,
‘3 ,
t) is to be chosen. Further, we set
F = (3t)-“3G(Y)7(c;
X(x, z;y, t) +%,
t
propriate inverse scattering transform for eq. (6). However, direct application of (5) is useless for it does not eliminate the variable y nor does it incorporate a suitable conjugate variable (equivalent to z). Thus we introduce
,
u = 42’3U(X, t),
(6)
after one integration in X provided U + 0 as IX I --*=: eq. (6) is the cylindrical KdV equation first given in ref. [3]. The transformation (5) should enable the solution described by eqs. (2)-(4) to be written as the ap-
G = exp(-Y/3),
(10)
and with this same choice eq. (4a) becomes 3;& + szzz
(5)
yield
u, + U/(2t) + 6UU~ + uxxx = 0 )
and
+ 3tTt - 9 = F,7[ +Z?,
(
(11)
although eq. (9) also must be employed to give eq. (11). Finally, the linear integral equation can be written as L(E, z; t) + 7(c;, z; t)
t
oa s L(~,s;r)9(s,Z;t)ds=O,
(12)
E
197
Volume 72A, number 3
PHYSICS LETTERS
where
9 July 1979
where
K = (3t)-‘13 exp(-Y/3)L(g,
2; t),
(16)
and then U([, t) = 2(12t)-2’3(a/ag)L(E
7,E. t) .
(13)
(We can note that Y = 0 on z =x i.e. on 2 = [.) Eqs. (9) (1 l)-(13) were written down in ref. [4] by direct analogy with eqs. (2)-(4) and an equivalent integration technique for eq. (6) was developed in ref. [5] by considering an inverse spectral problem with an additional linear potential. However, the equations as derived here seem a simpler representation of the solution. The solution to eqs. (9) and (11) can be expressed as
+k jM(~,p)SAj@+4)Ai(‘l+s)dqdp=0. 0
5
If we construct the similarity solution directly from eq. (6), and impose the necessary decay conditions at infinity, it follows that v2 = 2(d/dl;) ~~(5, 0
S)Ai(5 + s)ds 1
(17)
where II@) satisfies the Painleve equation u”tu3-.$J=o.
9 = s f(sr-“3)‘4iQ -m
+ S)Ai(Z + s)ds ,
(14)
for arbitrary f(.), and all solutions are given by eq. (14) if generalised functions are allowed. For example, choosing f( .) = 7~6(.), we obtain the simplest soliton solution from
L(.g,t; t) =
(a/al;)hl(l +kt-1’3J Ajqs’ds), I
(18)
Thus eqs. (16) and (17) constitute a linearisation of the Painleve equation: this representation is to be compared with that obtained in ref. [8]. It is fairly easy to confirm that both our linearisation and that given in ref. [8] produce identical Neuman expansions for u2([), but the general relationship between the two methods is less straightforward. Let us define N(x, y) such that N(x, 2y -X) = -(2/r) M(x, y -X),
(19)
where r2 = 2k (>O), then from eq. (16)
which is discussed in refs. [4-61. Further, letting the constant k + m , we produce a special similarity solution of eq. (6),
( )
N(x, y) - rAi y
(20)
U(& t) = 2(12~)_2/3(a2jaS2)~(jA16)dr). 5 which is considered in ref. [7] in connection with a water wave problem. The more general similarity solution is constructed by takingf(.) = kH(.), and then L&Z;
r) + kj- Ai@ + s)Ai(Z + s)ds 0
after some manipulation. Eq. (20) is essentially the equation given in ref. [8 3 for their K, , with very minor changes due to the particular form of the Painleve equation used here. The main result of ref. [8] is that u(t) =N(g, C;),written in our notation. Thus, from eqs. (19) and (17), we see that (214
u(E) = - (2Ir)MC; 90) and +kjL(&s;t)
E
jAi(s+p)Ai(Z+p)dpds=O, 0
u2Q) = 2(d/d0 jM(S.
which has the solution L&Z;
t) = jM(S, 0
198
s)Ai(Z + s)ds
,
s)Ai(t + s)ds
,
@lb)
0
(15)
ifM(t, s) satisfies eq. (16) with k = r2/2. The conclusion expressed by (2 1) is rather surprising: we have shown that two quite dissimilar forms exist for the same Painleve transcendent.
PHYSICS LETTERS
Volume 72A, number 3
We now demonstrate that both the expressions given in (21) can be obtained directly from our scattering transform without reference to the results in ref. [8]. First, it follows that if L([,Z; t) satisfies eq. (12), with 9([, 2; f) satisfying eqs. (9) and (1 I), then 3tL, + LEES+ L,,,
+ (3/4) (2CLE + i+L)
-.$LLg-ZLz-L=Q,
(22)
L,,-L,,=(&Z-li)L,
(23)
where V(& t) = 2(12t)-2’3a, CC.&0 = 2@/%)LQ, k 0, and for similarity solutions we require i, L and 9 independent oft. Now from eqs. (22) and (23) it can be shown that
upon the use of Airy’s equation and assuming L is defined for arbitrary Z. Evaluating eqs. (24) and (25) on s = 0 (provided wS is finite) shows that these equations are consistent only if w(C;,0) = hC1/2. The arbitrary constant X is fixed by requiring that eqs. (24), (25) follow from the linear integral equation for w(C;,s) (essentially eq. (16)): hence X = 1. The first part of (21) is now also proved. We have indicated how the inverse scattering transform for the cylindrical KdV equation can be derived from that for the 2-D KdV equation. Using this formulation, it is possible to obtain two different representations of the Painlevi transcendent (which describes similarity solutions of the KdV equation). Finally, it is interesting to observe that if the identity (2/r2)M2Q, 0) = (d/d&!)jk&
i’=(d/d[)u.
References
Evaluating this on Z = 4 yields the second result in (21) with li = u2 where u satisfies eq. (18). To obtain the bther expression for u(t), which previously required the use of ref. [8], we again start with eqs. (22) and (23). We define L&Z)=!
s)-$(5 + s)h
(implied by (2 1)) is used in the main theorem of ref. [8], their result follows without recourse to Neumann expansions.
where
(alapajaz),i =DL,
(25)
0
+2(t-z)(Lzz+L&=O,
D=
wtt=(.ps)w-iw,
+ 3ii’i + 3iiD,i - (32 + .$)Di
D3i - 2i
9 July 1979
wQ&+(Z+s)d&
w = -(2/r)M,
0
and then eqs. (22), (23) yield WEEE+(~Li-&v~-SwSt(~~‘-l)W=O,
(24)
[ 1] V.E. Zakharov and A.B. Shabat, Functional Anal. Appl. 8 (1974) 226. [2] V.S. Dryuma, JETP Lett. 19 (1974) 387. [3] S. Maxon and J. Viecelli, Phys. Fluids 17 (1974) 1614. [4] N.C. Freeman, Adv. Appl. Mech. Vol. 20 (1979), to be published. [S] F. Calogero and A. Degasperis, Reprint No. 93, Institute di Fisica G. Marconi, Rome (1978). [6] R.S. Johnson and S. Thompson, Phys. Lett. 66A (1978) 279. [7] R.S. Johnson, submitted to J. Fluid Mech. (1979). [a] M.J. Ablowitz and H. Segur, Phys. Rev. Lett. 38 (1977) 1103.
199