INTEGRABILITY OF DOUBLY-PERIODIC RICCATI EQUATION

By the structure of solvable subgroup of SL(2, C) (see [1]), the integrability and properties of solutions of a Riccati equation with an elliptic function coefficient, which is related to a Fuchsian equation on the torus T, is studied.


INTRODUCTION
The research on the integrability theory of differential equations is obviously important and interesting both in theory and in application.Since the pioneering work of Abel and Galois for algebraic equations and the work of Liouville for a particular RJccati equation (see [2]), the theory of integrability of differential equations has been developed very fast in several different ways, such as the Lie group theory (see [3]), differential algebra (see [4]) and the monodromy group theory created by Poincar6 and Klein (see [5,6]), the theory of invariants of the transformation group (see [7]), etc.However, the investigation into the differential equations is unlike that of algebraic equations for which there is a unified and well developed Galois theory Because ofthe complexity, these different integrability theories of differential equations have not been unified, and for each theory there exist some fundamental difficulties (see [1,6,8]).Even in differential algebra, the concept of integrability for a Fuchsian system has not been developed quite clearly (see 1,6]) In this paper, we develop the results of and [9] to give a clear interpretation to the integrability of a doubly periodic P.iccati equation by the solvability of its monodromy group.
Consider the P.iccati equation where the parameter A 6 R+, t 6 C, z(t) e C, and $,(t) is the Weierstrass elliptic function given by the differemial equation $,,2 45, [$,2 1], $,(0) O. (2) It is known that $,(t) has the following properties: (i) $,(t) is doubly periodic with periods wl 2or 6 R and w2 iwl; (ii) p(it) $,(t), Vt 6 C, (iii) for any and for 6 a, 0], $,(t) increases from 1 to O; (iv) the point P1 c ic is a pole of order two, and there are no other singularities except P1 in the period parallelogram (see 10]) Because of the double-periodicity of $,(t), the equation (1) can be treated as a Kiccati equation on the torus T e which is formed from the period parallelogram of As we know (see [1,9]), by the transformation z -u'lu, the solution of the equation ( 1) is related to the following Fuchsian equation on the torus T u"-(t)u o.
By means of this fundamental solution system (ui(t), u2(t)) of equation ( 3), each solution z(t) of the Riccati equation ( 1) can be expressed as z(t) 'here 6 e ' C U {oo}.Moreover, we have .f ( 6)u (t) + u(t) where fl and f2 are two fractional-linear (MObius) transformations corresponding to A1 and A f (6) a6 + bl C: 16 + all' A() The group M generated by f and f is a subgroup of the group of MObius transformation, and is called the monodromy group ofthe Riccati equation (1).
Suppose that (l(t), 2(t)) is another fundamental solution system of equation ( 3), and that is the monodromy group of equation ( 3) corresponding to this fundamental system, then there exists a non- singular matrix T such that (l(t),(t)) (Ul(t),u(t))T.So ( T-GT.This means, in view of isomorphism, that the monodromy group G of equation ( 3) is independent of the choice of the fundamental system of solutions (ul (t), us(t)), and so is the monodromy group M of equation (1).
It is easy to check that the monodromy group M of equation ( 1) and the monodromy group G of the corresponding equation (3) have the same solvability.The structure of a solvable subgroup G of SL(2, C) has been studied in 1].Here, we state some relevant results as follows.
It remains to verify that any singularity of z2(t) is a pole.In fact, in the neighborhood of any fixed point, say, P1 :t t -a-is, there are two regular solutions of the corresponding Fuchsian equation (3) (see [5]) Wl () ( l)r' 1 (), and W2(t) (-1)r2() +0Wl()ln( $1), where rl > O, r2 < 0 are two real roots ofthe algebraic equation r(r-1) A O, 1 (t) and 2(t) are holomorphic in the neighborhood oftl, and 0 Res 1, Then in the neishborhood of tl,U2(:), as a solution of equation ( 3), can be expressed by a linear combination of wl () and w2(t).However, as we have indicated above, z() is sinsle-valued, thus the point P1 should be a pole of z(t).The isolated zeros of u(t) are also poles of z2(t).Besides, there is no other singularity for (t).So z () u ()/u (t) is an elliptic function solution of equation (1) Case 2. If fl and f2 are of the form (R2), then 6 cc is the M-invariant.The same method as in Case for z2, works for zl () u ()/u (t) which is therefore verified to be an elliptic function with periods 2a and 2ia.Case 3. If fl and fo_ are of the form (R3) .h=o2/, o: #-0, then it is easy to verify that for any 6 E C, where fl denotes the inverse transformation of f,.This means that Zl Ul/UZ and z2 u/u2 are both single-valued solutions with periods 2a and 4ic.Further, with the same verification as in Case l, we get that z: (t) and z2 (t) are both elliptic functions.Case 4. If fl and f2 are ofthe form (Rs) /: -, Y ( + )/(6 + ), a # 0, then it is easy to check that for any E C f() , and that f2(,) ,, where I , 2 i This means that u'+u' and z_=- Z+ lUl + U2 are both elliptic function solutions with periods 4a and 2ic.This completes the proof.We note the following facts: (a) If the Riccati equation (1) has an elliptic function solution, say, zl(t), then its general solution can be expressed as Z() Z (;) + y(), where y(t) satisfies the following Bernoulli's equation.
Co) Any elliptic function with double periods w and cv is of the form Rl[p(t)] + R2 [p(t)]pt(), where p(t) is the Weierstrass elliptic function with the same double periods, p'(t) is its derivative, and R (x), R2(x) are rational functions in x with constant coefficients (see [11]).
By Lemma 3 and the facts above, we obtain immediately: TREOREM.If the monodromy group of the Riccati equation ( 1) is solvable, then equation ( 1) is integrable, that is to say, its general solution can be expressed in terms of a Weierstrass elliptic function by solving algebraic equations, differentiation and integration in finite terms; each solution of equation ( 1) is meromorphic on C.
By remarking the following facts: (i) for A n(n-1)(n E N), the monodromy group G of equation ( 3) is solvable (see [1,9]), (ii) the solvability of the group M is equivalent to that of the group G, we have also: COROLLARY 1.For A n(n 1)(n N), the Riccati equation ( 1) is integrable (Recent work has shown that the Riccati equation ( 1) is also integrable when A (4-)4/4-3), n N.This new result was presented by G-nan at Mathematics Today and Tomorrow International Conference in Florida in 1997).