- Email: [email protected]
Pseudopotentials for non-linear evolution equations in 2 + 1 dimensions
lnt J Son-Lmeur Mrchan~r. Printed I” Great Bntam.
Vol. 23. No
oO?o- 7462.88 53.M)tO.IXl Psrpamon Press plc
5 6. pp. 361-367. 1988
PSEUDOPOTENTIALS FOR NON-LINEAR EVOLUTION EQUATIONS IN 2+ 1 DIMENSIONS MARIA CLARA NUCCI* School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 U.S.A. (Received 20 July
1987; receivedfar public~~iff~ 3 May 1988)
Abstract-In this paper the concept of pseudopotential is generalized to non-linear evolution equations in 2 + 1dimensions. If the equations satisfied by the pseudopotential are of a Riccati-type in the x-variable, it is shown how to obtain both the generalized AKNS system and the autoBacklund transformation for the corresponding non-linear evolution equation. Several examples are given: Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili, (2+ 1 dim.)-Harry-Dym, and (2 + 1 dim.)-Sawada-Kotera equations.
1. INTRODUCTION
In Cl], Wahlquist and Estabrook introduced the idea of pseudopotential into the study of non-linear evolution equations in 1+ 1 dimensions. They used the differentiai forms approach to produce pseudopotentials. In [2], Corones and Testa pointed out how pseudopotentials can be computed by classicai method (the direct approach). In [3], Nucci showed that if the equations satisfied by the pseudopotential are of a Riccati-type, then one can easily obtain both the Lax equations (AKNS form [4]) and the auto-Backlund transformation for the corresponding non-linear evolution equation. In the present paper, the concept of pseudopotential is generalized for non-linear evolution equations in 2+ 1 dimensions using the direct approach. It turns. out that by imposing the equations satisfied by the pseudopotential of a Riccati-type in the x-variable one can obtain both the generalized AKNS system and the auto-BCcklund transformation. The paper is organized as follows. In the second section we review the concept of pseudopotential in 1 + 1 dimensions as given in [3] with some examples which will be useful in the following. In Section 3 the general idea in 2 + 1 dimensions is established. Several examples are given in Section 4 (Kadomtsev-Petviashvili equation), 5 (modified Kadomtsev-Petviashvili equation), 6 (2 + 1 dim.-Harry-Dym equation), and 7 (2 + 1 dim.Sawada-Kotera equation).
2. SOME
EXAMPLES
IN I+ 1 DIMENSIONS
We consider non-linear evolution equations in the form: 41= H(% 4x, 4Xx, . . -1 We assume there exists a pseudopotential
u = u(x, C) such that: (2.2a)
u, = F(n, 4) n, = G(n, 4,4x, 4,, with F and G polynomials integrability condition
(2-l)
. . .I
(2.2b)
of second order in u [3]. The system (2.2) is subject to the F&q)
= G,(n, 4,4x,4xX, . . .I
(2.3)
whenever (2.1) is satisfied. Usually, one gets a different form of (2.2). In the following examples we will look for the computationally simplest expression. Korteweg-de
Vries equation
We consider the following form [S]: 4r = 4*.Xx -
6qqr
*Permanent address: Dipartimento di Matematica, UniversitH di Perugia, 06100 Perugia, Italy. 361
(2.4)
362
M. C. Nuccr
The chosen associated pseudopotentia1
u satisfies:
u,=q-uz
(25a)
% = (4X- 2uq),
(25b)
which implies that u is a solution of the modified Korteweg-de l4,= (u,, -
Vries equation:
2u3),.
(2.6)
The AKNS system for (2.4), which is formally given by: Y’,=AY
(2.7a)
(2.7b) is easily obtained imposing the usual linearizing transformation to (2.5a). In &hiscase it corresponds to:
for the Riccati equation [6] (2.8)
u = On44, with:
*1=*x
(2.9) (2.10)
*z = I(/* An auto-Backlund transformation formation to (2.5a). For example
for (2.4) is obtained u*=
by applying a Moebius trans-
-_u
(2.1 I)
gives from (2Sa): -u,
= q*-u2
(2.12)
where q* is another solution of (2.4). Eliminating both u and u, from (2.5a) and (2.11), it gives the usual auto-Backlund transformation for (2.4). Here, and throughout this section only the spatial part of the AKNS systems and the auto-Backlund transformations will be considered. M~di~~d Korteweg-de
Vries equation
We consider the following form [S]: (2.13)
qr = qxxx- 6q2rlx which is equivalent to (2.6). The chosen associated pseudopotential
u satisfies:
ux= -2uq+u2
(2.14a)
u, = ( - 2uq, - 2uq2),
(2.14b)
which implies that u is a solution of: u*=
3 lf,,---u-r&-2
(
t43 . 2 >X
(2.15)
From (2.14a), we obtain the following linearizing transformation: U= -(In$)x which will give rise to (2.7). To obtain an auto-Backlund transformation reapply the transformation (2.11) to (2.14a), which becomes: - u, = 2uq* + u*.
(2.16) for (2.13), we can (2.17)
Combining (2.14a) and (2.17), it will give: q* - q = - exp( - f(q* + q)dx)
a well-known Backfund transformation
for (2.13).
(2.18)
Pseudopotentials
for non-linear
evolution
equations
363
Harry-Dym equation
We consider the following form [S]: (2.19)
3 41 =
The chosen pseudopotential
4
4xXx.
u will satisfy: a,=
-q-2-u2
(2.20a)
u, = (2q, - 4uq),.
(2.20b)
The linearizing transformation is the same as in (2.8). An auto-Backlund transformation easily obtained from (2.20a) by using (2.11). It turns out to be trivial, i.e.: q*=q.
is
(2.2 1)
Sawada-Kotera equation It corresponds to [S]:
f 5q,q,, + 5%,X + 5q2CC. 41 = 4XXXXX The system for the chosen pseudopotential
(2.22)
u is:
u,=q+u2
(2.23a) (2.23b)
n, = (4XxX+ 2%X + 44X+ aq2), which corresponds to the equation: u, = (%Xx.X - 5u, ii,, - 5u%,, - 5uu; + u5)X. The linearizing transformation
is given by (2.16). A trivial auto-Backlund u,* = _q*_u*2
4, = 4XxXxX + 5qq.X,*+ %?.&X + 5q2q,.
4
u2
-9
is:
equation [S], i.e.: (2.26)
u satisfies: (2.27a)
%= -2-1 u, =
transformation
(2.25)
obtained using (2.11) in (2.23a). An equation very similar to (2.22) is the so-called Kaup-Kupershmidt
It can be easily shown that its chosen pseudopotential
(2.24)
+;q..- 2qq,
+ uq2
>X
(2.27b)
which corresponds to (2.24). In fact u is a pseudopotential common to the Sawada-Kotera and the Kaup-Kupershmidt equation. Note that both (2.24) and the Miura transformations (2.23a) and (2.27a) were determined in [7] by factorizing the operators for (2.22) and (2.26) respectively.
3.
PSEUDOPOTENTIALS
IN 2+ 1 DIMENSIONS
We consider non-linear evolution equations in the form qr = H(q, 4x94x,, qv, qxy, jqydx, . . 4. We assume there exists a pseudopotential
(3.1)
u = u(x, y, t) such that:
U, = F(u, q, ju,dx, Judx) a, = {G(u, 4, 4x9 Ju,dx, . . . ,}, .
(3.2a) (3.2b)
With the requirement uXt= ut, when (3.1) is satisfied. If we can impose F to be a polynomial of second order in u the usual linearizing transformation for the Riccati equation could be applied to (3.2) which will give rise to the well-known generalized AKNS system in 2+ 1
M. C. Nucc~
364
dimensions [S-12]. Every time such a pseudopotential u is found, there exists another pseudopotential u* which will satisfy the same equation as u although sign changes may occur corresponding to the substitution y = - y. Then the transformation (2.11) can be applied to u* and an auto-B~cklund transformation for (3.1) could be obtained. This procedure requires very lengthy calculations. However these extensive algebraic manipulations could be almost completely eliminated by applying one of the available Computer Algebra systems. Then all routine work would be done automatically by the computer. In the following examples, we will use the results in Section 2.
4. It
KADOMTSEV-PETVIASHVILI
EQUATION
is the 2 + 1 dimensional version of (2.4), i.e. [S]: 4t = Qxxx -
A pseudopotential
6wx + 3fclvvdx.
(4.1)
u is found which satisfies: u, = q -
u*-
juydx
(4.2a)
u, = (qX- 2uq - 4u, - 4r+,dx
+ 3 j q,, dx),
(4.2b)
that is: - 6u*u, - 6u,.u,dx + 3fu,,dx. ui = u.X,.X
(4.3)
From (4.2a) the corresponding linearizing transformation is (2.8). By applying (2.8) to (4.2a), we obtain the well-known generalized AKNS equation for (4.1) (see [lo, 13-153): (4.41 where: Y=
11/l $2
0
(4.5)
with $r and e2 defined in (2.9)--(2.10). Here and throughout the following sections, only the spatial part of the generalized AKNS systems and the auto-B~cklund transformations will be considered. It is easy to show that another pseudopotential u* for (4.1) can be found which satisfies: u: = q* - u** + fu,*dx
(4.6a)
u: = (q; - 2u*q* + 4~: + 4u*@:dx - 3Jq;dx),
(4.6b)
that is: u: = u:;, - 6u**u: + 6u:fu;dx
+ 3&dx
where q* is another solution of (4.1). Then by applying the transformation obtain: - u, = q* - ii2 - +,dx.
(4.7) (2.11) to (4.6a), we (4.3)
Combining (4.2a) with (4.8) and defining:
Q =j@x
(4.9a)
Q* = fq*dx
(4.9bf
gives: (Q* - Q)’ - 2(Q* + Q), - 2j(Q* - Q),dx = 0
(4.10)
the spatial-part of the well-known Backlund transformation [16] for (4.1). Note that (4.3) and (4.7) are the same equation except for the change in sign of the term 6u,ju,dx. Of course both (4.2a) and (4.6a) are Miura transfo~ations in 2 + 1 dimensions [17].
Pseudopotentials 5. MODIFIED
365
for non-linear evolution equations
KADOMTSEV-PETVIASHVILI
EQUATION
It is the 2 + 1 dimensional version of (2.13), i.e. [S]:
(5.1)
41 = 4XxX- 6q2q, + 3Jq,,dx. A pseudopotential
u is found which satisfies:
u, = - 2uq + uz - Ju,dx
(5.2a)
u, = [ - 2uq, - 2uq2 - 4u, - 4(q - u) Ju,dx - 6ufq,dx],
(52b)
that is: u,, - -;u
u, =
- ‘u; - ;
+ 3ul(Ju,,dx)u - ‘dx - 3uIu - 2u,(lu,dx)dx
1.
++P(~u,dx)2
(5.3)
X
From (5.2a) the corresponding linearizing transformation we obtain the generalized AKNS equation for (5.1):
is (2.16). Applying (2.16) to (5.2a),
(5.4) with (4.5). Another pseudopotential
u* for (5.1) is found which satisfies:
u: = - 2u*q* + Use +Ju;dx
(5.5a)
u: = [ - 2u*q: - 2u*q*’ + 4~; + 4(q* - u*)Ju;dx + 6u*{q;dx],
(5.5b)
that is (5.3). Applying the transformation
(2.11) to (5.5a), we obtain:
- u, = 2uq* + u2 -Ju,dx.
(5.6)
Combining (5.2a) with (5.6) and using (4.9), it gives the following auto-Bicklund formation for (5.1):
(Q*-Q),+wC-(Q* + Q)l+ap(Q* +Q)jtQ* +Q),expC--(Q*+ QHdx =
6. (2+ I dim.)-HARRY-DYM
o.
trans(5.7)
EQUATION
It is given by [S]: 3
qr = q qxxx + 6qyJq - ‘q,dx + 3&q - %J,dx. A pseudopotential ux= -q-2-
(6.1)
u is found which satisfies: (6.2a)
u2 - q-2Ju,dx
u, = c2q, - 4uq + 6Jq - ‘q,dx - 4qu, + (2q, + 6fq - ‘q,,dx - 4uq)ju,dx],. If we apply the transformation for (6.1):
(2.8) to (6.2a), we obtain the generalized AKNS equation
YX+(; with (4.5). Another pseudopotential
;2
)YY=
(;
-“o’>‘y
u: =
(6.3)
u* for (6.1) is found which satisfies:
_u~=_q*-f-u*~+q*-~~u,*dx -
(6.2b)
[2q; - 4u*q* + 6jq* -2q;dx
(6.4a) + 4q*u;
- (2q: + 6jq* - 2q,*dx + 4u*q*)Ju;dx],.
For simplicity the equations
(6.4b)
satisfied by u and u* are not given here. Applying the
366
M. C. Nuccr
transformation
(2.11) to (6.4a), we obtain: u, = -q*-2-u2-q*-2Ju,dx.
(6.5)
Combining (6.2a) and (6.5), gives a trivial auto-Backlund
7. (2+1
dim.)-SAWADA-KOTERA
transformation
for (6.1).
EQUATION
It is given by [S]: 41 = qxxxxx+ 5qxq.U+ 5qqxx.X + 5q2q, + 5qxxy (7.1)
- 5!qy,dx + 5qq, + 5q,jq,dx. A pseudopotential u, =
u is found which satisfies:
q + u2 -
exp(Iudx)J(Ju,dx)exp(
(7.2a)
- Judx)dx
ur = {qxxx+ 2nq,, + nq2 + qqx + 4q, + 5nl‘q,dx - 3qSn,dx + 3q,exp( ludx)j(ju,dx)exp( - 9 exp@Wjjju,,dx
- [udx)dx
- (ju,dx)2]exp( -Judx)dx},
(7.2b)
If we apply the linearizing transformation: n = - (ln II/,),
(7.3)
to (7.2a), we obtain the generalized AKNS equation for (7.1):
(7.4)
where:
Another pseudopotential u* for (7.1) is found which satisfies the same equation with the expected sign changes. Applying the transformation (2.11) to: -g=q*+u*2
+ exp (- Iu* dx)f(Juy* dx) exp(lu* dx)dx
(7.6)
-exp(Judx)J(Ju,dx)exp(
(7.7)
we obtain: u,=q*+l2
-Judx)dx.
If we combine (7.7) with (7.2a), a trivial auto-Backlund transformation for (7.1) is found. Note that for each of these examples, the function $ is the spectral function for the corresponding L operator as given in [S].
REFERENCES 1. H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations. J. math. Phys. 16, 1 (1975). 2. J. Corones and F. J. Testa, Pseudopotentials and their applications. Lecture Notes in Math. 515. Springer, Berlin (1976). 3. M. C. Nucci, Pseudopotentials, Lax equations and Backlund transformations for nonlinear evolution equations. J. Physics AZI, 73 (1988). 4. M. J. Ablowitz and H. Segur, Solitons and rhe Inverse Scattering Transform. SIAM, Philadelphia (1981). 5. B. G. Konopelchenko and V. G. Dubrovsky, Some new integrable nonlinear evolution equations in 2+ 1 dimensions. Phys. Lett. lOtA, 15 (1984). 6. E. Hille, Ordinary Differential Equations in the Complex Domnin. J. Wiley, New York (1976). 7. P. Fordy and J. Gibbons, Some remarkable nonlinear transformations. Phys. Lett. 75A, 325 (1980).
Pseudopotentials
for non-linear evolution equations
367
8. A. S. Fokas and M. J. Ablowit& The inverse scattering transform for multidimensional (2+ 1) problems. Lecrure Notes in Phys. 189. Springer, Berlin (1982). 9. R. Beals and R. R. Coifman, Multidimensional inverse scattering and non-linear P.D.E. Proc. Symp. Pure Math. Vol. 43, AMS, Providence (1985). IO. B. G. Konopelchenko and V. G. Dubrovsky, Bicklund-Calogero group and general form ofintegral equations for the two-dimensional Gelfand-Dikij-Zakharov-Shabat problem. Bilocal approach. Physica 16D, 79 (1985). II. A. S. Fokas and P. M. Santini, The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrodinger operator. Stud. appl. Math. 75, 179 (1986). 12. B. G. Konopelchenko, Nonlinear Integrable Equations. Springer, Berlin (1987). 13. S. V. Manakov, The inverse scattering transform for the time-dependent Schriidinger equation and Kadomtsev-Petviashvili equation. Physica 3D, 420 (1981). 14. B. G. Konopelchenko, On the general struture of nonlinear evolution equations integrable by the twodimensional-matrix spectral problem. Commun. math. Phys. 87, 105 (1982). 15. M. J. Ablowitz. D. Bar Yaacov and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation. Stud. appl. Math. 69, I35 (1983). 16. H. H. Chen, A. Backlund transformation in two dimensions. J. math. Phys. 16, 2382 (1975). 17. B. G. Konopelchenko, On the gauge-invariant description of the evolution equations integrable by Gelfand-Dikij spectral problems. Phys. Lett. 92A, 323 (1982).
Vol. 23. No
oO?o- 7462.88 53.M)tO.IXl Psrpamon Press plc
5 6. pp. 361-367. 1988
PSEUDOPOTENTIALS FOR NON-LINEAR EVOLUTION EQUATIONS IN 2+ 1 DIMENSIONS MARIA CLARA NUCCI* School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 U.S.A. (Received 20 July
1987; receivedfar public~~iff~ 3 May 1988)
Abstract-In this paper the concept of pseudopotential is generalized to non-linear evolution equations in 2 + 1dimensions. If the equations satisfied by the pseudopotential are of a Riccati-type in the x-variable, it is shown how to obtain both the generalized AKNS system and the autoBacklund transformation for the corresponding non-linear evolution equation. Several examples are given: Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili, (2+ 1 dim.)-Harry-Dym, and (2 + 1 dim.)-Sawada-Kotera equations.
1. INTRODUCTION
In Cl], Wahlquist and Estabrook introduced the idea of pseudopotential into the study of non-linear evolution equations in 1+ 1 dimensions. They used the differentiai forms approach to produce pseudopotentials. In [2], Corones and Testa pointed out how pseudopotentials can be computed by classicai method (the direct approach). In [3], Nucci showed that if the equations satisfied by the pseudopotential are of a Riccati-type, then one can easily obtain both the Lax equations (AKNS form [4]) and the auto-Backlund transformation for the corresponding non-linear evolution equation. In the present paper, the concept of pseudopotential is generalized for non-linear evolution equations in 2+ 1 dimensions using the direct approach. It turns. out that by imposing the equations satisfied by the pseudopotential of a Riccati-type in the x-variable one can obtain both the generalized AKNS system and the auto-BCcklund transformation. The paper is organized as follows. In the second section we review the concept of pseudopotential in 1 + 1 dimensions as given in [3] with some examples which will be useful in the following. In Section 3 the general idea in 2 + 1 dimensions is established. Several examples are given in Section 4 (Kadomtsev-Petviashvili equation), 5 (modified Kadomtsev-Petviashvili equation), 6 (2 + 1 dim.-Harry-Dym equation), and 7 (2 + 1 dim.Sawada-Kotera equation).
2. SOME
EXAMPLES
IN I+ 1 DIMENSIONS
We consider non-linear evolution equations in the form: 41= H(% 4x, 4Xx, . . -1 We assume there exists a pseudopotential
u = u(x, C) such that: (2.2a)
u, = F(n, 4) n, = G(n, 4,4x, 4,, with F and G polynomials integrability condition
(2-l)
. . .I
(2.2b)
of second order in u [3]. The system (2.2) is subject to the F&q)
= G,(n, 4,4x,4xX, . . .I
(2.3)
whenever (2.1) is satisfied. Usually, one gets a different form of (2.2). In the following examples we will look for the computationally simplest expression. Korteweg-de
Vries equation
We consider the following form [S]: 4r = 4*.Xx -
6qqr
*Permanent address: Dipartimento di Matematica, UniversitH di Perugia, 06100 Perugia, Italy. 361
(2.4)
362
M. C. Nuccr
The chosen associated pseudopotentia1
u satisfies:
u,=q-uz
(25a)
% = (4X- 2uq),
(25b)
which implies that u is a solution of the modified Korteweg-de l4,= (u,, -
Vries equation:
2u3),.
(2.6)
The AKNS system for (2.4), which is formally given by: Y’,=AY
(2.7a)
(2.7b) is easily obtained imposing the usual linearizing transformation to (2.5a). In &hiscase it corresponds to:
for the Riccati equation [6] (2.8)
u = On44, with:
*1=*x
(2.9) (2.10)
*z = I(/* An auto-Backlund transformation formation to (2.5a). For example
for (2.4) is obtained u*=
by applying a Moebius trans-
-_u
(2.1 I)
gives from (2Sa): -u,
= q*-u2
(2.12)
where q* is another solution of (2.4). Eliminating both u and u, from (2.5a) and (2.11), it gives the usual auto-Backlund transformation for (2.4). Here, and throughout this section only the spatial part of the AKNS systems and the auto-Backlund transformations will be considered. M~di~~d Korteweg-de
Vries equation
We consider the following form [S]: (2.13)
qr = qxxx- 6q2rlx which is equivalent to (2.6). The chosen associated pseudopotential
u satisfies:
ux= -2uq+u2
(2.14a)
u, = ( - 2uq, - 2uq2),
(2.14b)
which implies that u is a solution of: u*=
3 lf,,---u-r&-2
(
t43 . 2 >X
(2.15)
From (2.14a), we obtain the following linearizing transformation: U= -(In$)x which will give rise to (2.7). To obtain an auto-Backlund transformation reapply the transformation (2.11) to (2.14a), which becomes: - u, = 2uq* + u*.
(2.16) for (2.13), we can (2.17)
Combining (2.14a) and (2.17), it will give: q* - q = - exp( - f(q* + q)dx)
a well-known Backfund transformation
for (2.13).
(2.18)
Pseudopotentials
for non-linear
evolution
equations
363
Harry-Dym equation
We consider the following form [S]: (2.19)
3 41 =
The chosen pseudopotential
4
4xXx.
u will satisfy: a,=
-q-2-u2
(2.20a)
u, = (2q, - 4uq),.
(2.20b)
The linearizing transformation is the same as in (2.8). An auto-Backlund transformation easily obtained from (2.20a) by using (2.11). It turns out to be trivial, i.e.: q*=q.
is
(2.2 1)
Sawada-Kotera equation It corresponds to [S]:
f 5q,q,, + 5%,X + 5q2CC. 41 = 4XXXXX The system for the chosen pseudopotential
(2.22)
u is:
u,=q+u2
(2.23a) (2.23b)
n, = (4XxX+ 2%X + 44X+ aq2), which corresponds to the equation: u, = (%Xx.X - 5u, ii,, - 5u%,, - 5uu; + u5)X. The linearizing transformation
is given by (2.16). A trivial auto-Backlund u,* = _q*_u*2
4, = 4XxXxX + 5qq.X,*+ %?.&X + 5q2q,.
4
u2
-9
is:
equation [S], i.e.: (2.26)
u satisfies: (2.27a)
%= -2-1 u, =
transformation
(2.25)
obtained using (2.11) in (2.23a). An equation very similar to (2.22) is the so-called Kaup-Kupershmidt
It can be easily shown that its chosen pseudopotential
(2.24)
+;q..- 2qq,
+ uq2
>X
(2.27b)
which corresponds to (2.24). In fact u is a pseudopotential common to the Sawada-Kotera and the Kaup-Kupershmidt equation. Note that both (2.24) and the Miura transformations (2.23a) and (2.27a) were determined in [7] by factorizing the operators for (2.22) and (2.26) respectively.
3.
PSEUDOPOTENTIALS
IN 2+ 1 DIMENSIONS
We consider non-linear evolution equations in the form qr = H(q, 4x94x,, qv, qxy, jqydx, . . 4. We assume there exists a pseudopotential
(3.1)
u = u(x, y, t) such that:
U, = F(u, q, ju,dx, Judx) a, = {G(u, 4, 4x9 Ju,dx, . . . ,}, .
(3.2a) (3.2b)
With the requirement uXt= ut, when (3.1) is satisfied. If we can impose F to be a polynomial of second order in u the usual linearizing transformation for the Riccati equation could be applied to (3.2) which will give rise to the well-known generalized AKNS system in 2+ 1
M. C. Nucc~
364
dimensions [S-12]. Every time such a pseudopotential u is found, there exists another pseudopotential u* which will satisfy the same equation as u although sign changes may occur corresponding to the substitution y = - y. Then the transformation (2.11) can be applied to u* and an auto-B~cklund transformation for (3.1) could be obtained. This procedure requires very lengthy calculations. However these extensive algebraic manipulations could be almost completely eliminated by applying one of the available Computer Algebra systems. Then all routine work would be done automatically by the computer. In the following examples, we will use the results in Section 2.
4. It
KADOMTSEV-PETVIASHVILI
EQUATION
is the 2 + 1 dimensional version of (2.4), i.e. [S]: 4t = Qxxx -
A pseudopotential
6wx + 3fclvvdx.
(4.1)
u is found which satisfies: u, = q -
u*-
juydx
(4.2a)
u, = (qX- 2uq - 4u, - 4r+,dx
+ 3 j q,, dx),
(4.2b)
that is: - 6u*u, - 6u,.u,dx + 3fu,,dx. ui = u.X,.X
(4.3)
From (4.2a) the corresponding linearizing transformation is (2.8). By applying (2.8) to (4.2a), we obtain the well-known generalized AKNS equation for (4.1) (see [lo, 13-153): (4.41 where: Y=
11/l $2
0
(4.5)
with $r and e2 defined in (2.9)--(2.10). Here and throughout the following sections, only the spatial part of the generalized AKNS systems and the auto-B~cklund transformations will be considered. It is easy to show that another pseudopotential u* for (4.1) can be found which satisfies: u: = q* - u** + fu,*dx
(4.6a)
u: = (q; - 2u*q* + 4~: + 4u*@:dx - 3Jq;dx),
(4.6b)
that is: u: = u:;, - 6u**u: + 6u:fu;dx
+ 3&dx
where q* is another solution of (4.1). Then by applying the transformation obtain: - u, = q* - ii2 - +,dx.
(4.7) (2.11) to (4.6a), we (4.3)
Combining (4.2a) with (4.8) and defining:
Q =j@x
(4.9a)
Q* = fq*dx
(4.9bf
gives: (Q* - Q)’ - 2(Q* + Q), - 2j(Q* - Q),dx = 0
(4.10)
the spatial-part of the well-known Backlund transformation [16] for (4.1). Note that (4.3) and (4.7) are the same equation except for the change in sign of the term 6u,ju,dx. Of course both (4.2a) and (4.6a) are Miura transfo~ations in 2 + 1 dimensions [17].
Pseudopotentials 5. MODIFIED
365
for non-linear evolution equations
KADOMTSEV-PETVIASHVILI
EQUATION
It is the 2 + 1 dimensional version of (2.13), i.e. [S]:
(5.1)
41 = 4XxX- 6q2q, + 3Jq,,dx. A pseudopotential
u is found which satisfies:
u, = - 2uq + uz - Ju,dx
(5.2a)
u, = [ - 2uq, - 2uq2 - 4u, - 4(q - u) Ju,dx - 6ufq,dx],
(52b)
that is: u,, - -;u
u, =
- ‘u; - ;
+ 3ul(Ju,,dx)u - ‘dx - 3uIu - 2u,(lu,dx)dx
1.
++P(~u,dx)2
(5.3)
X
From (5.2a) the corresponding linearizing transformation we obtain the generalized AKNS equation for (5.1):
is (2.16). Applying (2.16) to (5.2a),
(5.4) with (4.5). Another pseudopotential
u* for (5.1) is found which satisfies:
u: = - 2u*q* + Use +Ju;dx
(5.5a)
u: = [ - 2u*q: - 2u*q*’ + 4~; + 4(q* - u*)Ju;dx + 6u*{q;dx],
(5.5b)
that is (5.3). Applying the transformation
(2.11) to (5.5a), we obtain:
- u, = 2uq* + u2 -Ju,dx.
(5.6)
Combining (5.2a) with (5.6) and using (4.9), it gives the following auto-Bicklund formation for (5.1):
(Q*-Q),+wC-(Q* + Q)l+ap(Q* +Q)jtQ* +Q),expC--(Q*+ QHdx =
6. (2+ I dim.)-HARRY-DYM
o.
trans(5.7)
EQUATION
It is given by [S]: 3
qr = q qxxx + 6qyJq - ‘q,dx + 3&q - %J,dx. A pseudopotential ux= -q-2-
(6.1)
u is found which satisfies: (6.2a)
u2 - q-2Ju,dx
u, = c2q, - 4uq + 6Jq - ‘q,dx - 4qu, + (2q, + 6fq - ‘q,,dx - 4uq)ju,dx],. If we apply the transformation for (6.1):
(2.8) to (6.2a), we obtain the generalized AKNS equation
YX+(; with (4.5). Another pseudopotential
;2
)YY=
(;
-“o’>‘y
u: =
(6.3)
u* for (6.1) is found which satisfies:
_u~=_q*-f-u*~+q*-~~u,*dx -
(6.2b)
[2q; - 4u*q* + 6jq* -2q;dx
(6.4a) + 4q*u;
- (2q: + 6jq* - 2q,*dx + 4u*q*)Ju;dx],.
For simplicity the equations
(6.4b)
satisfied by u and u* are not given here. Applying the
366
M. C. Nuccr
transformation
(2.11) to (6.4a), we obtain: u, = -q*-2-u2-q*-2Ju,dx.
(6.5)
Combining (6.2a) and (6.5), gives a trivial auto-Backlund
7. (2+1
dim.)-SAWADA-KOTERA
transformation
for (6.1).
EQUATION
It is given by [S]: 41 = qxxxxx+ 5qxq.U+ 5qqxx.X + 5q2q, + 5qxxy (7.1)
- 5!qy,dx + 5qq, + 5q,jq,dx. A pseudopotential u, =
u is found which satisfies:
q + u2 -
exp(Iudx)J(Ju,dx)exp(
(7.2a)
- Judx)dx
ur = {qxxx+ 2nq,, + nq2 + qqx + 4q, + 5nl‘q,dx - 3qSn,dx + 3q,exp( ludx)j(ju,dx)exp( - 9 exp@Wjjju,,dx
- [udx)dx
- (ju,dx)2]exp( -Judx)dx},
(7.2b)
If we apply the linearizing transformation: n = - (ln II/,),
(7.3)
to (7.2a), we obtain the generalized AKNS equation for (7.1):
(7.4)
where:
Another pseudopotential u* for (7.1) is found which satisfies the same equation with the expected sign changes. Applying the transformation (2.11) to: -g=q*+u*2
+ exp (- Iu* dx)f(Juy* dx) exp(lu* dx)dx
(7.6)
-exp(Judx)J(Ju,dx)exp(
(7.7)
we obtain: u,=q*+l2
-Judx)dx.
If we combine (7.7) with (7.2a), a trivial auto-Backlund transformation for (7.1) is found. Note that for each of these examples, the function $ is the spectral function for the corresponding L operator as given in [S].
REFERENCES 1. H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations. J. math. Phys. 16, 1 (1975). 2. J. Corones and F. J. Testa, Pseudopotentials and their applications. Lecture Notes in Math. 515. Springer, Berlin (1976). 3. M. C. Nucci, Pseudopotentials, Lax equations and Backlund transformations for nonlinear evolution equations. J. Physics AZI, 73 (1988). 4. M. J. Ablowitz and H. Segur, Solitons and rhe Inverse Scattering Transform. SIAM, Philadelphia (1981). 5. B. G. Konopelchenko and V. G. Dubrovsky, Some new integrable nonlinear evolution equations in 2+ 1 dimensions. Phys. Lett. lOtA, 15 (1984). 6. E. Hille, Ordinary Differential Equations in the Complex Domnin. J. Wiley, New York (1976). 7. P. Fordy and J. Gibbons, Some remarkable nonlinear transformations. Phys. Lett. 75A, 325 (1980).
Pseudopotentials
for non-linear evolution equations
367
8. A. S. Fokas and M. J. Ablowit& The inverse scattering transform for multidimensional (2+ 1) problems. Lecrure Notes in Phys. 189. Springer, Berlin (1982). 9. R. Beals and R. R. Coifman, Multidimensional inverse scattering and non-linear P.D.E. Proc. Symp. Pure Math. Vol. 43, AMS, Providence (1985). IO. B. G. Konopelchenko and V. G. Dubrovsky, Bicklund-Calogero group and general form ofintegral equations for the two-dimensional Gelfand-Dikij-Zakharov-Shabat problem. Bilocal approach. Physica 16D, 79 (1985). II. A. S. Fokas and P. M. Santini, The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrodinger operator. Stud. appl. Math. 75, 179 (1986). 12. B. G. Konopelchenko, Nonlinear Integrable Equations. Springer, Berlin (1987). 13. S. V. Manakov, The inverse scattering transform for the time-dependent Schriidinger equation and Kadomtsev-Petviashvili equation. Physica 3D, 420 (1981). 14. B. G. Konopelchenko, On the general struture of nonlinear evolution equations integrable by the twodimensional-matrix spectral problem. Commun. math. Phys. 87, 105 (1982). 15. M. J. Ablowitz. D. Bar Yaacov and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation. Stud. appl. Math. 69, I35 (1983). 16. H. H. Chen, A. Backlund transformation in two dimensions. J. math. Phys. 16, 2382 (1975). 17. B. G. Konopelchenko, On the gauge-invariant description of the evolution equations integrable by Gelfand-Dikij spectral problems. Phys. Lett. 92A, 323 (1982).