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On the solutions in elliptic functions of integrable nonlinear equations
Volume 96A, number 7
18 July 1983
PHYSICS LETTERS
ON THE SOLUTIONS
IN ELLIPTIC FUNCTIONS
OF INTEGRABLE
NONLINEAR
EQUATIONS
V.Z. ENOL’SKII Institute for Theoretical Physics, Academy of Sciences of the UkrainianSSR, Metrologicheskajastr. 14, Kiev 252130, Kiev-l 30, USSR Received 19 April 1983
A new solution in elliptic functions for the KdV equation is constructed the author.
Solutions of one-dimensional nonlinear equations, which are important in physics,‘and are integrable within the inverse scattering method [ 11, as well as of still actual classical problems (Kovalevskaja, Neumann and Jacobi problems, etc.) [2,3], are expressed via quotients of the theta-functions [4] 8 [m](z; B) = .&
exp{ni [((n + m’), B(n + m’))
+ 2((2 + m”), (n + m’))l)
2m=(2m',2m")EZ2g,
9
(1)
z=UxtVtjW,
u, v, WEG, defined on the Riemann hyperelliptic curve .2 =
/lo (5 -
ej) ,
surface I’g = (C;,a) of the
(ej f q, if i f I) .
In eq. (1) the g X g Riemann matrix B and the g-vectors U and V manifest themselves in terms of complete hyperelliptic integrals on I’g, the g-vector W is defined by a nonspecial divisor D = (/.Q, .. .. pg) through which the initial data are introduced into the solution [ 11. In 1974 Novikov raised the problem of a superposition law determination, allowing one to express the ggap solutions in terms of the l-gap ones, being necessarily elliptic functions [5]. After the papers [6-lo] which, in particular, contained examples of 2-gap solutions expressible in terms of elliptic functions, 0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
using the method proposed by Belokolos and
Belokolos and the author reduced, in refs. [ 11,121, the problem of the superposition law determination to the classical problem of the reduction of hyperelliptic integrals to elliptic ones, which goes back to Weierstrass (see the monograph [ 131). In fact, in order that the ggap solution of the equation, integrable in terms of theta-functions (1) could manifest itself in terms of elliptic functions, it is necessary and sufficient that the theta-functions (1) reduce to the Jacobi theta-functions and the hyperelliptic integrals on rg to elliptic ones. In other words, on I’g there should be defined g coverings (which, in general, differ from each other) fi : rg 4-Q = I, .... g, such that the functions fi([) used as substitutions in g independent on I’g hyperelliptic integrals reduce them to elliptic integrals on $, j = 1, ...) g* Further we shall restrict ourselves to the case g = 2 and formulate the statements from ref. [ 131 concerning the reduction of hyperelliptic integrals to elliptic ones. i. In order that a reducible hyperelliptic integral exists on I2 it is necessary and sufficient that the Riemann matrix B satisfies the condition 41+ 4$11+
93B12 + 44B22 + 45@11&
- B&) = 05
(2) 4; + 4(4145 - 4244) = k2 T where qj, j = 1, .... 5 are integers and k is a positive integer. ii. If condition (2) is fulfilled, then on I’2 there exists such a homology basis, in which the relation 327
Volume 96A, number 7
18 July 1983
PHYSICS LETTERS
B12 = B,, = I/k is correct for the matrix 6, and for the corresponding theta-functions the decomposition
e[m;,m;;m;‘,m’il(zl,z2;a, l/k,b) k-l
=c
exp(2ni/k)(rlr2
- m;m;)
rl,rpO
X 19[kpl(rl
+ m;), km;’ + rni] (kzl lk2a)
X 0 [k-l(r2
+ m;), km’; + rn; ] (kz2 Ik2b) .
(3)
iii. If condition (2) is satisfied, then on I’* are defined coverings fj:F2 -+$, j = 1,2, such that the functions fi, j = 1, 2, are rational of the order k and can be used as the substitutions to reduce the independent hyperelliptic integrals on r2 to elliptic integrals. The statements i-iii allow one to propose the following scheme of a derivation of a solution in elliptic functions for a nonlinear equation integrable in terms of theta-functions. (1) Choose the branching points ej E I’2, j = 0, . . .. 4 in such a way, that the hyperelliptic integrals on I’* reduce to elliptic ones by a certain kth order transformation. (2) Define a nonspecial divisor D = (pl, p2) and also a certain homology basis (a, b-cycles) on r2. Express the vectors U, V, W and the matrix B in terms of elliptic integrals, find the integers qj, j = 1, .... 5 in eq. (2). (3) Construct the symplectic transformation [4,13] by the integers q obtained which connects the chosen homology basis with the one, where B,, =B,, = l/k. Reduce, due to this transformation and eq. (3) the solution to elliptic functions. Let us illustrate the circumscribed algorithm by extracting a solution in elliptic functions from the 2gap solution of the KdV equation - uxxx ,
ut = 6uu,
(4)
in the case k = 2. Let r2 be the Riemann surface, whose branching points eo, .. .. e4 satisfy the conditions e4
e1e4 =e2e3
=s.
dq = (oW1 [--l@2)C;+ J(q)1 d$ , dw2 = (oW1 V(q)E - JGq)l dt , in which the following notations
%j) =
$ $, ,
J(aj>=
are used:
6 %I,
j= 1,2,
A =Z(al)J(a2) -Z(a2)J(al) ; and the 2 X 2 Riemann matrix B = llBiiII=
The Matveev-Its formula [ 141 for the 2-gap solution of eq. (4) states U(X,
t)
=
-2(a2/a_G)
X In{0 [0, 0; 0, 0] (Ux + Vt + W; B)} + C ,
where I@,)) ,
U = (2i/A)(-l(+), 4 V= 2upo
ei + @i/A) (J&I,
-J(q)>
,
= 0, (5)
Let us define on r2 the basic cycles al, a2, bl, b2 (see fig. 1). The normalized holomorphic differentials do1 and dw, are defined by the formulas 328
Fig. 1. The homology basis on r2.
j= 1,2,
wj=-~(~‘dw,t;B,)ttj, i=l m 4
C=
lgoei -
2
$ t(dal
+ da2).
(6)
The substitution t +-Y = %E •t 40
*’
functions
,
(7)
reduces hyperelliptic I(c) = -(1/2G)[L(-Q(c))
+ L(+)(y(c))]
- yl)ti
- y2)1-“2
dy,
(8)
y-':y-+y~(yL~)1/2,
the path c belongs. Using eqs. (8), we obtain for the parameters of the solution (6) i, j = 1, 2 ,
(9)
U(‘) = U, + U2 = i/&G&)
Y(+) = Fr, + V2 = (4i/&&))Cy1 w(*) = w, + w, = -[E(u,,
+ y2 T 4)
, (10) ,
Id’)) + E(u2, Id’))] /I#‘)), (11)
c = 2(y, ty2
+ 6)
- 2&w(-)/&+) ,
+ 4E(k(+))/K(k(+))
(12)
where &I
- y2)l/2 )
= K(@‘)/(Ts’/2
~‘(‘I=
~‘(~W)/(T~‘/2
(k”‘)2
= (y1 - y2)/(‘sl/2
q=$.+ts/~),
-y2)1/2
-2(a2/ad)
t 0,(~(+)l8B(+))0~(z(-)I8B(-))]
We should choose the upper signs in the formulas (8) depending on which branch of the reflection
B(‘) = a’(*)/&),
=
8B(-1) +C,
(13)
,
j=1,2.
Bii = 2B(+) + 2(-l)i+iB(-) ,
t)
we obtain finally
X ln[82(z(+)1 8B(+))B,(z(-)I
f. L(“‘(y(C))],
L(*)(,v(c)) = $ [2(y T s”~)O, Y(C) yi = i(ei + s/ej) ,
U(X,
integrals on IQ to elliptic ones:
J(c) = -+ [L(-Q(c))
18 July 1983
PHYSICS LETTERS
Volume 96A, number I
,
- y2) )
i=1,2,
and K(k), K’(k), E(u, k) are elliptic integrals. Thus under condition (5) eq. (2) is realized fork = 2, 4 = (0, 1, 0, -1,O). Then expanding the theta-function in formula (6) into a sum of products of Jacobi theta-
where z(*) = u(‘)x t V(*)f t WC”),and the quantities B(“), U’@), Y(*), IV(‘) are defined by formulas (9)(12). As far as only one 2-gap elliptic solution of eq. (4) which defines the evolution of the initial data in the form of the 2-gap Lame potential u(x, 0) = -6J(ix t o I w, w’) is described in the literature [6], the obtained solution (13) is a new quasi-periodic solution in elliptic functions *‘. In conclusion we remark that in the reduction case g = 2, k = 2 the Riemann surface r2 is automorphic with respect to the transformation g + $ = s/E. Using this fact, one can derive the equality B, 1 = B2, from condition (5) (as was done in refs. [7,10]) without the hyperelliptic integral reduction. Nevertheless, at k > 2 the coverings fi, j = 1, 2, are, generally speaking, arbitrary ones [ 131, what has predetermined the necessity to develop the scheme of the multigap solution reduction to the elliptic functions making no use of the normal existence. Notice, that while the given scheme proves to be most effective for the case g = 2, in the case of high g > 2 in ref. [ IS] an approach has been developed which allows one to separate the solutions in elliptic functions from multigap ones. The author expresses his gratitude to E.D. Belokolos, who has singled out the Weierstrass reduction as the method by means of which the solutions in elliptic functions can be -extracted from multigap ones. The author also thanks B.A. Dubrovin, A.R. Its, V.B. Matveev and S.M. Natanzon for valuable discussions. *’ The author succeeded in proving that the 2-gap Lame potential can be obtained from the Matveev-Its in the reduction case k = 3 (in preparation).
formula (6)
References [l] S.P. Novikov, ed., Theory of solitons (Nauka, Moskow,
*’ The reduction case (7), (8) was known to Legendre [ 131, and Forest and MacLaughlin have actually considered it when they proved that certain classes of the “sine-Gordon” solutions under condition (5) represent the “Lamb ansatz” 17,101.
[2] [3] [4] [5]
1980). S.P. Novikov, Usp. Mat. Nauk 37 (1982) 3. B.A. Dubrovin, Usp. Mat. Nauk 37 (1981) 12. J. Igusa, Theta functions (Springer, Berlin, 1972). S.P. Novikov, Funk. Anal. 8:3 (1974) 54.
329
Volume 96A, number 7
PHYSICS LETTERS
16) B.A. Dubrovin and S.P. Novikov, Sov. Phys. JETP 67 (1974) 2131. [7] G. Forest and D. MacLaughlin, J. Math. Phys. 23 (1982) 1248. [8] I.M. Krichever, Funk. Anal. 14:4 (1980) 45. [9] B.A. Dubrovin and S.M. Natanzon, Funk. Anal. 16:l (1982) 27. [lo] E.D. Belokolos and V.Z. Enol’skii, Teor. Mat. Fiz. 53 (1982) 271. [ 111 E.D. Belokolos and V.Z. Enol’skii, Usp. Mat. Nauk 37 (1982) 89.
330
18 July 1983
] 12 J E.D. Belokolos and V.Z. Enol’skii, preprint ITP-82-36E (ITP, Kiev, 1982). [ 13 ] A. Krazer, Lehrbuch der Thetafunktionen (Teubner, Leipzig, 1903). [14] A.R. Its, V.B. Matveev, Teor. Mat. Fiz. 23 (1975) 55; V.B. Matveev, preprint No. 373 (Univ. of Wroclaw, Wroclaw, 1976). [15 ] M.V. Babich, A.I. Bobenko and V.B. Matveev, Dokl. Acad. Sci. USSR (1983) to be published.
18 July 1983
PHYSICS LETTERS
ON THE SOLUTIONS
IN ELLIPTIC FUNCTIONS
OF INTEGRABLE
NONLINEAR
EQUATIONS
V.Z. ENOL’SKII Institute for Theoretical Physics, Academy of Sciences of the UkrainianSSR, Metrologicheskajastr. 14, Kiev 252130, Kiev-l 30, USSR Received 19 April 1983
A new solution in elliptic functions for the KdV equation is constructed the author.
Solutions of one-dimensional nonlinear equations, which are important in physics,‘and are integrable within the inverse scattering method [ 11, as well as of still actual classical problems (Kovalevskaja, Neumann and Jacobi problems, etc.) [2,3], are expressed via quotients of the theta-functions [4] 8 [m](z; B) = .&
exp{ni [((n + m’), B(n + m’))
+ 2((2 + m”), (n + m’))l)
2m=(2m',2m")EZ2g,
9
(1)
z=UxtVtjW,
u, v, WEG, defined on the Riemann hyperelliptic curve .2 =
/lo (5 -
ej) ,
surface I’g = (C;,a) of the
(ej f q, if i f I) .
In eq. (1) the g X g Riemann matrix B and the g-vectors U and V manifest themselves in terms of complete hyperelliptic integrals on I’g, the g-vector W is defined by a nonspecial divisor D = (/.Q, .. .. pg) through which the initial data are introduced into the solution [ 11. In 1974 Novikov raised the problem of a superposition law determination, allowing one to express the ggap solutions in terms of the l-gap ones, being necessarily elliptic functions [5]. After the papers [6-lo] which, in particular, contained examples of 2-gap solutions expressible in terms of elliptic functions, 0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
using the method proposed by Belokolos and
Belokolos and the author reduced, in refs. [ 11,121, the problem of the superposition law determination to the classical problem of the reduction of hyperelliptic integrals to elliptic ones, which goes back to Weierstrass (see the monograph [ 131). In fact, in order that the ggap solution of the equation, integrable in terms of theta-functions (1) could manifest itself in terms of elliptic functions, it is necessary and sufficient that the theta-functions (1) reduce to the Jacobi theta-functions and the hyperelliptic integrals on rg to elliptic ones. In other words, on I’g there should be defined g coverings (which, in general, differ from each other) fi : rg 4-Q = I, .... g, such that the functions fi([) used as substitutions in g independent on I’g hyperelliptic integrals reduce them to elliptic integrals on $, j = 1, ...) g* Further we shall restrict ourselves to the case g = 2 and formulate the statements from ref. [ 131 concerning the reduction of hyperelliptic integrals to elliptic ones. i. In order that a reducible hyperelliptic integral exists on I2 it is necessary and sufficient that the Riemann matrix B satisfies the condition 41+ 4$11+
93B12 + 44B22 + 45@11&
- B&) = 05
(2) 4; + 4(4145 - 4244) = k2 T where qj, j = 1, .... 5 are integers and k is a positive integer. ii. If condition (2) is fulfilled, then on I’2 there exists such a homology basis, in which the relation 327
Volume 96A, number 7
18 July 1983
PHYSICS LETTERS
B12 = B,, = I/k is correct for the matrix 6, and for the corresponding theta-functions the decomposition
e[m;,m;;m;‘,m’il(zl,z2;a, l/k,b) k-l
=c
exp(2ni/k)(rlr2
- m;m;)
rl,rpO
X 19[kpl(rl
+ m;), km;’ + rni] (kzl lk2a)
X 0 [k-l(r2
+ m;), km’; + rn; ] (kz2 Ik2b) .
(3)
iii. If condition (2) is satisfied, then on I’* are defined coverings fj:F2 -+$, j = 1,2, such that the functions fi, j = 1, 2, are rational of the order k and can be used as the substitutions to reduce the independent hyperelliptic integrals on r2 to elliptic integrals. The statements i-iii allow one to propose the following scheme of a derivation of a solution in elliptic functions for a nonlinear equation integrable in terms of theta-functions. (1) Choose the branching points ej E I’2, j = 0, . . .. 4 in such a way, that the hyperelliptic integrals on I’* reduce to elliptic ones by a certain kth order transformation. (2) Define a nonspecial divisor D = (pl, p2) and also a certain homology basis (a, b-cycles) on r2. Express the vectors U, V, W and the matrix B in terms of elliptic integrals, find the integers qj, j = 1, .... 5 in eq. (2). (3) Construct the symplectic transformation [4,13] by the integers q obtained which connects the chosen homology basis with the one, where B,, =B,, = l/k. Reduce, due to this transformation and eq. (3) the solution to elliptic functions. Let us illustrate the circumscribed algorithm by extracting a solution in elliptic functions from the 2gap solution of the KdV equation - uxxx ,
ut = 6uu,
(4)
in the case k = 2. Let r2 be the Riemann surface, whose branching points eo, .. .. e4 satisfy the conditions e4
e1e4 =e2e3
=s.
dq = (oW1 [--l@2)C;+ J(q)1 d$ , dw2 = (oW1 V(q)E - JGq)l dt , in which the following notations
%j) =
$ $, ,
J(aj>=
are used:
6 %I,
j= 1,2,
A =Z(al)J(a2) -Z(a2)J(al) ; and the 2 X 2 Riemann matrix B = llBiiII=
The Matveev-Its formula [ 141 for the 2-gap solution of eq. (4) states U(X,
t)
=
-2(a2/a_G)
X In{0 [0, 0; 0, 0] (Ux + Vt + W; B)} + C ,
where I@,)) ,
U = (2i/A)(-l(+), 4 V= 2upo
ei + @i/A) (J&I,
-J(q)>
,
= 0, (5)
Let us define on r2 the basic cycles al, a2, bl, b2 (see fig. 1). The normalized holomorphic differentials do1 and dw, are defined by the formulas 328
Fig. 1. The homology basis on r2.
j= 1,2,
wj=-~(~‘dw,t;B,)ttj, i=l m 4
C=
lgoei -
2
$ t(dal
+ da2).
(6)
The substitution t +-Y = %E •t 40
*’
functions
,
(7)
reduces hyperelliptic I(c) = -(1/2G)[L(-Q(c))
+ L(+)(y(c))]
- yl)ti
- y2)1-“2
dy,
(8)
y-':y-+y~(yL~)1/2,
the path c belongs. Using eqs. (8), we obtain for the parameters of the solution (6) i, j = 1, 2 ,
(9)
U(‘) = U, + U2 = i/&G&)
Y(+) = Fr, + V2 = (4i/&&))Cy1 w(*) = w, + w, = -[E(u,,
+ y2 T 4)
, (10) ,
Id’)) + E(u2, Id’))] /I#‘)), (11)
c = 2(y, ty2
+ 6)
- 2&w(-)/&+) ,
+ 4E(k(+))/K(k(+))
(12)
where &I
- y2)l/2 )
= K(@‘)/(Ts’/2
~‘(‘I=
~‘(~W)/(T~‘/2
(k”‘)2
= (y1 - y2)/(‘sl/2
q=$.+ts/~),
-y2)1/2
-2(a2/ad)
t 0,(~(+)l8B(+))0~(z(-)I8B(-))]
We should choose the upper signs in the formulas (8) depending on which branch of the reflection
B(‘) = a’(*)/&),
=
8B(-1) +C,
(13)
,
j=1,2.
Bii = 2B(+) + 2(-l)i+iB(-) ,
t)
we obtain finally
X ln[82(z(+)1 8B(+))B,(z(-)I
f. L(“‘(y(C))],
L(*)(,v(c)) = $ [2(y T s”~)O, Y(C) yi = i(ei + s/ej) ,
U(X,
integrals on IQ to elliptic ones:
J(c) = -+ [L(-Q(c))
18 July 1983
PHYSICS LETTERS
Volume 96A, number I
,
- y2) )
i=1,2,
and K(k), K’(k), E(u, k) are elliptic integrals. Thus under condition (5) eq. (2) is realized fork = 2, 4 = (0, 1, 0, -1,O). Then expanding the theta-function in formula (6) into a sum of products of Jacobi theta-
where z(*) = u(‘)x t V(*)f t WC”),and the quantities B(“), U’@), Y(*), IV(‘) are defined by formulas (9)(12). As far as only one 2-gap elliptic solution of eq. (4) which defines the evolution of the initial data in the form of the 2-gap Lame potential u(x, 0) = -6J(ix t o I w, w’) is described in the literature [6], the obtained solution (13) is a new quasi-periodic solution in elliptic functions *‘. In conclusion we remark that in the reduction case g = 2, k = 2 the Riemann surface r2 is automorphic with respect to the transformation g + $ = s/E. Using this fact, one can derive the equality B, 1 = B2, from condition (5) (as was done in refs. [7,10]) without the hyperelliptic integral reduction. Nevertheless, at k > 2 the coverings fi, j = 1, 2, are, generally speaking, arbitrary ones [ 131, what has predetermined the necessity to develop the scheme of the multigap solution reduction to the elliptic functions making no use of the normal existence. Notice, that while the given scheme proves to be most effective for the case g = 2, in the case of high g > 2 in ref. [ IS] an approach has been developed which allows one to separate the solutions in elliptic functions from multigap ones. The author expresses his gratitude to E.D. Belokolos, who has singled out the Weierstrass reduction as the method by means of which the solutions in elliptic functions can be -extracted from multigap ones. The author also thanks B.A. Dubrovin, A.R. Its, V.B. Matveev and S.M. Natanzon for valuable discussions. *’ The author succeeded in proving that the 2-gap Lame potential can be obtained from the Matveev-Its in the reduction case k = 3 (in preparation).
formula (6)
References [l] S.P. Novikov, ed., Theory of solitons (Nauka, Moskow,
*’ The reduction case (7), (8) was known to Legendre [ 131, and Forest and MacLaughlin have actually considered it when they proved that certain classes of the “sine-Gordon” solutions under condition (5) represent the “Lamb ansatz” 17,101.
[2] [3] [4] [5]
1980). S.P. Novikov, Usp. Mat. Nauk 37 (1982) 3. B.A. Dubrovin, Usp. Mat. Nauk 37 (1981) 12. J. Igusa, Theta functions (Springer, Berlin, 1972). S.P. Novikov, Funk. Anal. 8:3 (1974) 54.
329
Volume 96A, number 7
PHYSICS LETTERS
16) B.A. Dubrovin and S.P. Novikov, Sov. Phys. JETP 67 (1974) 2131. [7] G. Forest and D. MacLaughlin, J. Math. Phys. 23 (1982) 1248. [8] I.M. Krichever, Funk. Anal. 14:4 (1980) 45. [9] B.A. Dubrovin and S.M. Natanzon, Funk. Anal. 16:l (1982) 27. [lo] E.D. Belokolos and V.Z. Enol’skii, Teor. Mat. Fiz. 53 (1982) 271. [ 111 E.D. Belokolos and V.Z. Enol’skii, Usp. Mat. Nauk 37 (1982) 89.
330
18 July 1983
] 12 J E.D. Belokolos and V.Z. Enol’skii, preprint ITP-82-36E (ITP, Kiev, 1982). [ 13 ] A. Krazer, Lehrbuch der Thetafunktionen (Teubner, Leipzig, 1903). [14] A.R. Its, V.B. Matveev, Teor. Mat. Fiz. 23 (1975) 55; V.B. Matveev, preprint No. 373 (Univ. of Wroclaw, Wroclaw, 1976). [15 ] M.V. Babich, A.I. Bobenko and V.B. Matveev, Dokl. Acad. Sci. USSR (1983) to be published.