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Pseudopotentials for the general AKNS system
Volume 73A, number 2
PHYSICS LETTERS
3 September 1979
PSEUDOPOTENTIALS FOR THE GENERAL AKNS SYSTEM R. SASAKI
0 Denmark
The NielsBohr Institute, University of Copenhagen, DK-2100 Copenhagen Received 19 June 1979
A simple derivation of the pseudopotentials of Wahiquist and Estabrook is given for the general AKNS system. A relationship between some pseudopotentia~sand the infinite numbers of conservation laws is pointed out.
The 2 X 2 inverse scattering method of Zakharov and Shabat (ZS) [1] and Ablowitz, Kaup, Newell and Segur (AKNS) [2] turned out to be one of the most powerful techniques in the recent development of the soliton physics [3] in 1 + 1 dimensions. Wahiquist and Estabrook [4,5] proposed a geometric approach, that of the so-called prolongation structures to certam kinds of non-linear equations. By applying their method to the Korteweg—de Vries equation (KdV) [4]~ and the non-linear Schrodinger equation (NLS) [5], they have derived several pseudopotentials explicitly and discussed the relationship among the pseudopotentials, the inverse scattering equation and the Bäcklund transformation. The geometrical interpretation of the AKNS—ZS system was further developed by Hermann [6], Corones [7] and Crampin [8] and the importance of the SL(2, R) structure was stressed. A simpler geometrical picture of the general AKNS system was given by the present author [9,10], namely, all the AKNS equations describe a pseudospherical surface, a surface with constant negative gaussian curvature. Then the SL(2, R) structure is nat• urally understood as the isometry group of the pseudospherical surface. Morris [111 and Dodd and Gibbon [12] tried to determine the prolongation structures of the general AKNS system. These results, however, do not give all the types of pseudopotentials given in I, but only the quadratic pseudopotentials (see eq. (15)). The purpose of this letter is to derive for the general AKNS—ZS ~ This paper will be referred to as I.
system essentially all types of pseudopotentials given in I and give them a natural interpretation. In contrast to the lie algebraic method employed in refs. [3—8] and refs. [11, 12], a simple calculus of exterior differential forms [13] is used. The method was developed in refs. [10] and [14]. The form of the 2 X 2 inverse scattering method due to ZS and AKNS introduces the scattering problem (subscripts denote derivatives) ~Z \ i \/z i 1 = q (1) \z2/ \r —?7/\z2/ x ~
~,
together with the time evolution equations ~1 \ ‘A B \
I~z2~I
=
t
I‘-C
II
I,
(2)
—Aj~z2j
where s~ is a parameter and q and r are functions of the independent variables x and t. The coefficients A, B and C in eq. (2) are one-parameter (~) families of functions of x, t and q and r with their derivatives. The integrabiity (compatibility) conditions for eqs. (1) and (2), A~= qC
—
=
rB, +r
B~ 2~B+ + Z’lr
—
2Aq, (3)
t
X
give, by construction, the non-linear equation to be solved. By introducing a traceless 2 X 2 matrix of oneforms ~2by =
(~2..)= II~-~’i ~‘2 ‘~
\(~)~
W1
77
Volume 73A, number 2 =
=
PHYSICS LETTERS
s~dx÷Adt, w2 rdx
+
=
qdx + Bdt,
(4)
Cdt,
0,
ci~= dz~ ~2~z1, —
1,
/ = 1, 2
where d denotes exterior derivative. The integrability conditions (3) are expressed as the vanishing of a traceless 2 X 2 matrix of two-forms f~J(A denotes extenor product)
e
=
dlT2
~
—
~7
A
(6)
,
or in components o (O~~)(01 02 =
03
)
01
=
dw1
—
03
=
dw3
+ 2w1 A (.~)3.
W2 A W3
,
02
=
dw2
—
2w~AW2 ,(7)
Therefore, by making an appropriate choice of {w1} the AKNS non-linear equation to be solved can be simply written as or
0~=O, i=l,2,3.
(8)
It is easy to see that the system is closed. Differentiatingeq. (6),we get d~=
~
A
—
e A ~2,
(9)
showing that the exterior derivatives of ~o~} are contained in the ring of two-forms {0~}. It should be noted that the linear scattering problem (1), (2) or (6) and the integrability conditions (3) or (8) are invariant under the following “gauge” transformation: z z’ = Az, ~2 = A ~.2A —1 + ciA A (10) -~
e
,
AeA—’ where A is an arbitrary (space—time dependent) 2 X 2 matrix of determinant unity. In other words, the “gauge” transformation of &2 does not change the solution manifold of the AKNS non-linear equation. The one-form ~2has an interpretation as a connection (gauge field), the two-form 8 as a curvature (gauge field strength) and the closure property (9) as a Bianchi identity. In the following the one-forms {w,} need not be the original AKNS form (4) but can be any “gauge” transformed forms of them. -+
78
~‘ =
defined
at=dzt+F~dx+G1dt,
(11)
ring spanned by { O~}and {a1}: 3
da1
0k +
~b~ 1
tlj/~
=
(12)
A o~.
k 1
In eqs. (11) and
(l2)F~, G1
and a~kare functions of
{z~}and the variables appearing in &2 and b,1 are oneforms. The functions z1 and z2 appearing in the linear scattering problem (5) are by definition pseudopotentials, for ~
01
...) are
provided that their derivatives da~are contained in the (5)
,
Pseudopotentials ~z1} (i = 1, 2, 3, by the following pfaffian equations: a,=O,
one immediately finds that eqs. (1) and (2) can be written as a set of pfaftian equations [13] aj =
3 September 1979
i,/l,2.
(13)
We propose to call them the linear pseudopotentials. The next pseudopotentials z3 and z4 are introduced by the transformation = z2/z1 z4 = z1/z2 . Then the pfaffian equation (5) is equivalent to ,
(14)
a3=dz3—w3+2z3w1+z~w2, a4dz4 —w2 —2z4w1 +z~w3
(15)
,
which are Riccati equations for z3 and z4. Applying d to eqs. (15), we get da3
=
2a3
A
(w1
+
z3w2)
—
03
+
2z301
+
z~02
da4 = 2a4 A (—w1 +z4w3) —02 2z40 1 +z~03,(l6) showing that z3 and z4 are pseudopotentials. These two pseudopotentials, the so-called quadratic pseudopotentials, are discussed by many authors [4—8,11, —
121 and the SL(2, R) structure and the connection interpretation are based on eq. (15). It has been shown [8] that the pseudopotential Y8 in I is related to z3 and z4 for the original AKNS choice of {w~} for the KdV equation by a “gauge” transformation and a redefinition of the parameter. The forms of the next pseudopotentials z5 and z6 are suggested by eq. (16), namely, we define
a6
=
dz5
—
(w1
+ z3w2)
=
dz6
—
(—w1 +z4w3)
.
(17)
Volume 73A, number 2
PHYSICS LETTERS
Differentiating the above expressions, we get da5 = cr3 A w2 01 z302 —
cia6 =
—
—
a4
of primitive variables in the definition of I. The information about them, however, is already contained in the pseudopotentials z5 and z6, being generators of
—
A w3 +
Oi
—
z403
(18)
.
The relationship between (z3, z4) and (z5 z6) is the same as that of y8 andy3 in I. The method of deriving z5 and z6 is essentially the same as that given by the present author [10, 14] for the infinite numbers of conservation laws for the general AKNS system. This result is an abstract generalization of the work of Konno et al. [15] and Wadati et al. [16]. The expansion of dz5 and ‘~6into inverse powers of~gives the infinite numbers of local conservation laws for the AKNS system [14—16]. Therefore, contrary to the assertion of I, the pseudopotentials z5 and z6 (Y3 in I) are closely related to the known set of conservation laws. The next pseudopotentials z7 and z8 are obtained by generalizing the relationship between y3 andy2 in I. Namely, we define ,
2Z5 w a7
=
dz7
—
e_
2
(19)
,
2Z6 w clz8 e_ 3 then a simple calculation shows 2Z5 a dcx7 = 2 e_ 5 A W2 e~S 02 2z6a 2z60 da82e 6 Aw3_e_ 3
8
=
—
—
,
(20)
The set of pseudopotentials z7 and z8 for the general AKNS—ZS system has never been demonstrated in any form, so far as we are aware. The physical significance of them is yet to be studied. A few comments are in order. (a) The existence of the pseudopotentials z1—z8 depends only on the fact that the non-linear equation is written as an integrabiity condition (3) or (8) and not on the specific boundary conditions necessary for the inverse scatteringmethod to work. Therefore, the present scheme also applies to, for example, the Liouville equation •
uxt
+ eu
=
0
,
3 September 1979
(21)
which cannot be solved by the inverse scattering method. (b) The present method does not give any potendais, i.e., simple conservation laws expressed in terms
infinite numbers of conservation laws. (c) The objects discussed in this paper, i.e., the linear scattering problem, the AKNS—ZS non-linear equation as an integrability condition, the pseudopotentials as well as the geometrical interpretation [9, 10] (pseudospherical surfaces) are all gauge invariant (covariant). On the other hand, a Bäcklund transformation in the familiar form for a particular non-linear equation is not gauge invariant. For that one first specifies A, B and C in (2) as functions of q and r and then uses the quadratic pseudopotentials z3 and z4 (15) in an appropriate gauge as is shown in refs. [4, 8, 17]. In this context see § 5 of ref. [10] as an attempt to generalize the Bãcklund transformation in a gauge covariant way. For the gauge transformation properties of the pseudopotentials z3, z4, z5 and z6, see ref. [14]. A practical utility of the gauge transformation is demonstrated in ref. [14] in the derivation of local and non-local conservation laws for the sine-Gordon equation. (d) The pseudopotential y5 in I does not seem to be given by this method. It corresponds to a kind of non-local conservation law [14], which will be discussed elsewhere. The author is very grateful to R.K. Bullough for valuable discussion and for critical reading of the manuscript. Thanks are due to the Danish Research Council and to the Commemorative Association of the Japan World Exposition. References [1] V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP 34 (1972) 62. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 31(1973)125.
[3] R.K. Bullough and R.K. Dodd, in: Structural stability, a symposium in honor of R. Thom, ed. W. Güttinger (Springer, Heidelberg, 1979). [4] H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1. [5] H.D. Wahiquist and F.B. Estabrook, J. Math. Phys. 17 (1976) 1293. [6] R. Hermann, Phys. Rev. Lett. 36 (1976) 835.
79
Volume 73A, number 2 [7] [8] [9] [101 [111 [12]
PHYSICS LETTERS
J. Corones, J. Math. Phys. 18(1977)163. M. Crampin, Phys. Lett. 66A (1978) 170. R. Sasaki, Phys. Lett. 71A (1979) 390. R. Sasaki, Nuci. Phys. B, to be published. H.C. Morris, J. Math. Phys. 18 (1977) 533. R.K. Dodd and J.D. Gibbon, Proc. Roy. Soc. London A359 (1978) 411. [13] See, for example, H. Flanders, Differential forms (Academic Press, New York, 1963).
80
3 September 1979
[14] R.K. Bullough and R. Sasaki, Niels Bohr Institute preprint. [15] K. Konno, H. Sanuki and Y.H. Ichikawa, Prog. Theor. Phys. 52 (1974) 886. [16] M. Wadati, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975) 419. [17] H-H. Chen, Phys. Rev. Lett. 33 (1974) 925; K. Konno and M. Wadati, Prog. Theor. Phys. 53 (1975) 1652.
PHYSICS LETTERS
3 September 1979
PSEUDOPOTENTIALS FOR THE GENERAL AKNS SYSTEM R. SASAKI
0 Denmark
The NielsBohr Institute, University of Copenhagen, DK-2100 Copenhagen Received 19 June 1979
A simple derivation of the pseudopotentials of Wahiquist and Estabrook is given for the general AKNS system. A relationship between some pseudopotentia~sand the infinite numbers of conservation laws is pointed out.
The 2 X 2 inverse scattering method of Zakharov and Shabat (ZS) [1] and Ablowitz, Kaup, Newell and Segur (AKNS) [2] turned out to be one of the most powerful techniques in the recent development of the soliton physics [3] in 1 + 1 dimensions. Wahiquist and Estabrook [4,5] proposed a geometric approach, that of the so-called prolongation structures to certam kinds of non-linear equations. By applying their method to the Korteweg—de Vries equation (KdV) [4]~ and the non-linear Schrodinger equation (NLS) [5], they have derived several pseudopotentials explicitly and discussed the relationship among the pseudopotentials, the inverse scattering equation and the Bäcklund transformation. The geometrical interpretation of the AKNS—ZS system was further developed by Hermann [6], Corones [7] and Crampin [8] and the importance of the SL(2, R) structure was stressed. A simpler geometrical picture of the general AKNS system was given by the present author [9,10], namely, all the AKNS equations describe a pseudospherical surface, a surface with constant negative gaussian curvature. Then the SL(2, R) structure is nat• urally understood as the isometry group of the pseudospherical surface. Morris [111 and Dodd and Gibbon [12] tried to determine the prolongation structures of the general AKNS system. These results, however, do not give all the types of pseudopotentials given in I, but only the quadratic pseudopotentials (see eq. (15)). The purpose of this letter is to derive for the general AKNS—ZS ~ This paper will be referred to as I.
system essentially all types of pseudopotentials given in I and give them a natural interpretation. In contrast to the lie algebraic method employed in refs. [3—8] and refs. [11, 12], a simple calculus of exterior differential forms [13] is used. The method was developed in refs. [10] and [14]. The form of the 2 X 2 inverse scattering method due to ZS and AKNS introduces the scattering problem (subscripts denote derivatives) ~Z \ i \/z i 1 = q (1) \z2/ \r —?7/\z2/ x ~
~,
together with the time evolution equations ~1 \ ‘A B \
I~z2~I
=
t
I‘-C
II
I,
(2)
—Aj~z2j
where s~ is a parameter and q and r are functions of the independent variables x and t. The coefficients A, B and C in eq. (2) are one-parameter (~) families of functions of x, t and q and r with their derivatives. The integrabiity (compatibility) conditions for eqs. (1) and (2), A~= qC
—
=
rB, +r
B~ 2~B+ + Z’lr
—
2Aq, (3)
t
X
give, by construction, the non-linear equation to be solved. By introducing a traceless 2 X 2 matrix of oneforms ~2by =
(~2..)= II~-~’i ~‘2 ‘~
\(~)~
W1
77
Volume 73A, number 2 =
=
PHYSICS LETTERS
s~dx÷Adt, w2 rdx
+
=
qdx + Bdt,
(4)
Cdt,
0,
ci~= dz~ ~2~z1, —
1,
/ = 1, 2
where d denotes exterior derivative. The integrability conditions (3) are expressed as the vanishing of a traceless 2 X 2 matrix of two-forms f~J(A denotes extenor product)
e
=
dlT2
~
—
~7
A
(6)
,
or in components o (O~~)(01 02 =
03
)
01
=
dw1
—
03
=
dw3
+ 2w1 A (.~)3.
W2 A W3
,
02
=
dw2
—
2w~AW2 ,(7)
Therefore, by making an appropriate choice of {w1} the AKNS non-linear equation to be solved can be simply written as or
0~=O, i=l,2,3.
(8)
It is easy to see that the system is closed. Differentiatingeq. (6),we get d~=
~
A
—
e A ~2,
(9)
showing that the exterior derivatives of ~o~} are contained in the ring of two-forms {0~}. It should be noted that the linear scattering problem (1), (2) or (6) and the integrability conditions (3) or (8) are invariant under the following “gauge” transformation: z z’ = Az, ~2 = A ~.2A —1 + ciA A (10) -~
e
,
AeA—’ where A is an arbitrary (space—time dependent) 2 X 2 matrix of determinant unity. In other words, the “gauge” transformation of &2 does not change the solution manifold of the AKNS non-linear equation. The one-form ~2has an interpretation as a connection (gauge field), the two-form 8 as a curvature (gauge field strength) and the closure property (9) as a Bianchi identity. In the following the one-forms {w,} need not be the original AKNS form (4) but can be any “gauge” transformed forms of them. -+
78
~‘ =
defined
at=dzt+F~dx+G1dt,
(11)
ring spanned by { O~}and {a1}: 3
da1
0k +
~b~ 1
tlj/~
=
(12)
A o~.
k 1
In eqs. (11) and
(l2)F~, G1
and a~kare functions of
{z~}and the variables appearing in &2 and b,1 are oneforms. The functions z1 and z2 appearing in the linear scattering problem (5) are by definition pseudopotentials, for ~
01
...) are
provided that their derivatives da~are contained in the (5)
,
Pseudopotentials ~z1} (i = 1, 2, 3, by the following pfaffian equations: a,=O,
one immediately finds that eqs. (1) and (2) can be written as a set of pfaftian equations [13] aj =
3 September 1979
i,/l,2.
(13)
We propose to call them the linear pseudopotentials. The next pseudopotentials z3 and z4 are introduced by the transformation = z2/z1 z4 = z1/z2 . Then the pfaffian equation (5) is equivalent to ,
(14)
a3=dz3—w3+2z3w1+z~w2, a4dz4 —w2 —2z4w1 +z~w3
(15)
,
which are Riccati equations for z3 and z4. Applying d to eqs. (15), we get da3
=
2a3
A
(w1
+
z3w2)
—
03
+
2z301
+
z~02
da4 = 2a4 A (—w1 +z4w3) —02 2z40 1 +z~03,(l6) showing that z3 and z4 are pseudopotentials. These two pseudopotentials, the so-called quadratic pseudopotentials, are discussed by many authors [4—8,11, —
121 and the SL(2, R) structure and the connection interpretation are based on eq. (15). It has been shown [8] that the pseudopotential Y8 in I is related to z3 and z4 for the original AKNS choice of {w~} for the KdV equation by a “gauge” transformation and a redefinition of the parameter. The forms of the next pseudopotentials z5 and z6 are suggested by eq. (16), namely, we define
a6
=
dz5
—
(w1
+ z3w2)
=
dz6
—
(—w1 +z4w3)
.
(17)
Volume 73A, number 2
PHYSICS LETTERS
Differentiating the above expressions, we get da5 = cr3 A w2 01 z302 —
cia6 =
—
—
a4
of primitive variables in the definition of I. The information about them, however, is already contained in the pseudopotentials z5 and z6, being generators of
—
A w3 +
Oi
—
z403
(18)
.
The relationship between (z3, z4) and (z5 z6) is the same as that of y8 andy3 in I. The method of deriving z5 and z6 is essentially the same as that given by the present author [10, 14] for the infinite numbers of conservation laws for the general AKNS system. This result is an abstract generalization of the work of Konno et al. [15] and Wadati et al. [16]. The expansion of dz5 and ‘~6into inverse powers of~gives the infinite numbers of local conservation laws for the AKNS system [14—16]. Therefore, contrary to the assertion of I, the pseudopotentials z5 and z6 (Y3 in I) are closely related to the known set of conservation laws. The next pseudopotentials z7 and z8 are obtained by generalizing the relationship between y3 andy2 in I. Namely, we define ,
2Z5 w a7
=
dz7
—
e_
2
(19)
,
2Z6 w clz8 e_ 3 then a simple calculation shows 2Z5 a dcx7 = 2 e_ 5 A W2 e~S 02 2z6a 2z60 da82e 6 Aw3_e_ 3
8
=
—
—
,
(20)
The set of pseudopotentials z7 and z8 for the general AKNS—ZS system has never been demonstrated in any form, so far as we are aware. The physical significance of them is yet to be studied. A few comments are in order. (a) The existence of the pseudopotentials z1—z8 depends only on the fact that the non-linear equation is written as an integrabiity condition (3) or (8) and not on the specific boundary conditions necessary for the inverse scatteringmethod to work. Therefore, the present scheme also applies to, for example, the Liouville equation •
uxt
+ eu
=
0
,
3 September 1979
(21)
which cannot be solved by the inverse scattering method. (b) The present method does not give any potendais, i.e., simple conservation laws expressed in terms
infinite numbers of conservation laws. (c) The objects discussed in this paper, i.e., the linear scattering problem, the AKNS—ZS non-linear equation as an integrability condition, the pseudopotentials as well as the geometrical interpretation [9, 10] (pseudospherical surfaces) are all gauge invariant (covariant). On the other hand, a Bäcklund transformation in the familiar form for a particular non-linear equation is not gauge invariant. For that one first specifies A, B and C in (2) as functions of q and r and then uses the quadratic pseudopotentials z3 and z4 (15) in an appropriate gauge as is shown in refs. [4, 8, 17]. In this context see § 5 of ref. [10] as an attempt to generalize the Bãcklund transformation in a gauge covariant way. For the gauge transformation properties of the pseudopotentials z3, z4, z5 and z6, see ref. [14]. A practical utility of the gauge transformation is demonstrated in ref. [14] in the derivation of local and non-local conservation laws for the sine-Gordon equation. (d) The pseudopotential y5 in I does not seem to be given by this method. It corresponds to a kind of non-local conservation law [14], which will be discussed elsewhere. The author is very grateful to R.K. Bullough for valuable discussion and for critical reading of the manuscript. Thanks are due to the Danish Research Council and to the Commemorative Association of the Japan World Exposition. References [1] V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP 34 (1972) 62. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 31(1973)125.
[3] R.K. Bullough and R.K. Dodd, in: Structural stability, a symposium in honor of R. Thom, ed. W. Güttinger (Springer, Heidelberg, 1979). [4] H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1. [5] H.D. Wahiquist and F.B. Estabrook, J. Math. Phys. 17 (1976) 1293. [6] R. Hermann, Phys. Rev. Lett. 36 (1976) 835.
79
Volume 73A, number 2 [7] [8] [9] [101 [111 [12]
PHYSICS LETTERS
J. Corones, J. Math. Phys. 18(1977)163. M. Crampin, Phys. Lett. 66A (1978) 170. R. Sasaki, Phys. Lett. 71A (1979) 390. R. Sasaki, Nuci. Phys. B, to be published. H.C. Morris, J. Math. Phys. 18 (1977) 533. R.K. Dodd and J.D. Gibbon, Proc. Roy. Soc. London A359 (1978) 411. [13] See, for example, H. Flanders, Differential forms (Academic Press, New York, 1963).
80
3 September 1979
[14] R.K. Bullough and R. Sasaki, Niels Bohr Institute preprint. [15] K. Konno, H. Sanuki and Y.H. Ichikawa, Prog. Theor. Phys. 52 (1974) 886. [16] M. Wadati, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975) 419. [17] H-H. Chen, Phys. Rev. Lett. 33 (1974) 925; K. Konno and M. Wadati, Prog. Theor. Phys. 53 (1975) 1652.