AMS WORDS

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 4 (1998) 785-790

785

INTEGRABILITY OF DOUBLY-PERIODIC RICCATI^EQUATION

MAUNGandGUANKE-1NG

Department

of AppliedMathematics Beijing University ofAeronauticsandAstronautics

Beijing 100053,P.R. CHINA

(Received January4, 1996andin revisedformJune6,1997)

ABSTRACT. By the structure of solvable subgroup of

SL(2, C)

(see [1]), the integrability and properties ofsolutionsofa Riccati equationwithan elliptic functioncoefficient,which isrelatedto a Fuchsianequationonthetorus

T ,

^isstudied.

KEY WORDS AND PHRASES: Integrability, doubly-periodicRiccatiequation, monodromygroup, solvablesubgroupof

SL(2, C),

invariant.

1991AMSSUBJECT CLASSIFICATION CODES:

1. INTRODUCTION

The research on the integrability theory of differential equations is obviously important and interesting bothintheoryandinapplication. Sincethe pioneering work of Abel andGaloisfor algebraic equations and the work of Liouville foraparticularRJccatiequation(see [2]),^thetheoryof integrability ofdifferential equations has beendevelopedvery fastinseveral different ways, suchasthe Liegroup theory(see [3]), differentialalgebra(see[4])and themonodromy group theorycreatedbyPoincar6and Klein (see [5,6]), the theory ofinvariants 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 unifiedandwelldevelopedGaloistheory Becauseof the complexity, thesedifferentintegrabilitytheories of differential equations have not been unified, and for each theory there exist some fundamental difficulties (see[1,6,8]). Evenindifferentialalgebra,theconceptofintegrability foraFuchsian system hasnotbeen developed quite clearly(see 1,6])

Inthispaper,wedeveloptheresults of and[9]^togiveaclearinterpretationtotheintegrabilityof adoubly periodicP.iccatiequation by the solvability ofitsmonodromygroup.

Consider theP.iccatiequation

z’=

z

,,(t),

()

wherethe parameter

A

6R

+,

t6

C, z(t) ^{e C,}

^and$,(t)is theWeierstrassellipticfunctiongiven bythe differemialequation

$,,2

45,

[$,2 1],

^{$,(0) O. (2)}

It is known that $,(t) has the following properties: (i) $,(t) is doubly periodic with periods wl 2or6

R

and_w2 iwl;(ii) p(it) $,(t), Vt6

C,

(iii)for any

andfor 6 a,

0],

$,(t)increases from 1 toO; (iv)the point

P1

^{c icis a}pole of ordertwo, and there arenoothersingularitiesexcept

P1

^intheperiod parallelogram(see 10])

Becauseofthedouble-periodicity of$,(t),the equation(1)canbetreated as a Kiccatiequationon thetorus

T

e which isformedfrom the periodparallelogramof

786 M. LINGAND G. KE-YING

As weknow (see [1,9]), by the transformation z

-u’lu,

the solution ofthe equation (1) is relatedto thefollowingFuchsianequationon the torusT

u"- _(t)u o.

(3)

Suppose

that

(Zt (g), U2(t))

^isafundamental solution system of equation(3),^{then there}exist two

(al bl) andA2=(a b2)

matrices

A1

cl

dl

c2 d,2 ^whichbelong tothe special lineargroup

SL(2, C),

suchthat

(1( + 2o), z2( + 200) (Zl(), 2())al,

(4)

(tl( + 2ioz),u2(t + 2ic)) (Ul(t),u(t))A.

(5) Thegroup Ggeneratedby

A1

^andA2,i.e.G

(A, A2),

is just themonodromy group of equation (3) (see [1,9]).

Bymeans ofthisfundamentalsolution system

(ui(t), u2(t))

of equation(3),^{each solution}

z(t)

^of

the Riccatiequation(1)canbe expressed as

z(t) i(t) +

5ul (t) + u2(t)’

⁽⁶⁾

’here

6

e

’

^CU

^{{ oo}. Moreover,}

^wehave

(t + 2) Ii(),i(t) +,4(t)

.f (6)u (t) + ^u(t)

⁽⁷⁾

(t + 2i) f(6)(t) + (t) (6)(t) + (t)’

where

fl

^and

f2

^{are two}fractional-linear(MObius)transformationscorrespondingto

A1

^and

A f (6) a6 + bl

C:

16 + all’

⁽⁹⁾

a6 + b A()

(0)

+

Thegroup

M

generatedby

f

^and

f

^{isasubgroupof thegroupofMObius}transformation,and is called themonodromy groupof theRiccatiequation

(1).

Suppose

that

(l(t), 2(t))

^isanotherfundamental solution system of equation(3),^{and that} isthe monodromy group of equation

(3)

correspondingto thisfundamental system, then thereexists a non- singularmatrix

T

such that

(l(t),(t)) (Ul(t),u(t))T.

So

( T-GT.

Thismeans,in viewof isomorphism, that the monodromy group G of equation (3) is independent ofthe choice of the fundamentalsystemofsolutions

(ul (t), us(t)),

and so is themonodromygroup

M

of equation(1).

Itiseasytocheck that the monodromy group

M

of equation(1)andthemonodromygroup Gofthe corresponding equation

(3)

have the same solvability. The structure ofa solvable subgroup G of

SL(2, C)

has been studied in 1]. Here,westatesome relevant results asfollows.

LEMMA1. If the monodromygroup Gof equation

(3)

issolvable,thenitstwogenerators

A

^and

A2

^{mustbelongtoone}ofthe following cases:

(F1)

a 0

a 1,

c#-O

A

_{d b}

INTEORABILILTY OF DOUBLY-PERIODIC RICCATI EQUATION 7 8 7

(F=)

A

a 0

a2=l

cy60

A

_{d b}

a 0

A=(O a_l), a2

o)

A2=

_{0 b}

(F)

a 0

a +i

A2

_{_b_ 0}

(f)

a

a2=

i

A

_{c d} ^bcd

^://:

^O;

(F)

Al=

a 0

-b

-

⁰

(F)

a 0

a

+

A=(O a-X), (o

A2=

_-b

-

^{0 or 0 b d#0;}

From Lemma1,itfollows that

LEMMA

2. If the monodromy group

M

ofequation(1)issolvable,then its generators mustbelong

tooneof the followingcases:

(R1)

I2 ^/( + as) aas

^0,

(correspondingtothecases

(F1)

and

(F2));

(R2)

.f a6 +

^a,

7 8 8 M. LINGAND(3.KE-YING (correspondingto thecases

(F3), (F)

and

(3)

)’1 =a16,

f2

(correspondingtothe cases

(F4)

and

(F6));

(R4)

/1 =a16,

A ^6/(o-6 + .),

(correspondingtothe cases

(F)

^and

(F8)),

f2 (6 + Crl)/(26 + 1),

O’la

#

^0,

(correspondingto thecase

(Fs)).

LEMMA3. Ifthemonodromy group

M

of theRiccatiequation(1)issolvable, thenthere exists an ellipticfunction solutionof the equation.

PROOF.

Suppose

that

M

issolvable. From Lemma2,itfollows that

Case1. Iffl,f2 belong^to

(R1)

or

(R4),

then 6 0istheM-invariant(i.e. g(0) 0,Vg

e

M) Therefore, the corresponding solution_z2

-u/u2

of equation

(1)

is single-valued andwithdouble periodswl 2a,w 2ia.

Itremains toverify thatany singularity of

z2(t)

is apole. In fact,inthe neighborhoodofanyfixed point, say,

P1

^{:t t -a-}is, there are two regular solutions ofthe 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 rootsofthealgebraic equation

r(r- 1)

A O,

1 (t)

^and

2(t)

^areholomorphicintheneighborhood oftl,and

0 Res 1,

Then inthe

neishborhood

^of

^tl,U2(:),

^{as a solution} of equation (3), can be expressed by a linear combinationof_wl

()

and

w2(t).

^{However,as wehaveindicatedabove,}

z()

issinsle-valued,thusthe point

P1

^{should beapoleof}

z(t).

^Theisolated zeros of

u(t)

^{arealsopolesof}

z2(t).

^{Besides,there is}

noother singularity for

(t).

^So

z () u ^{()/u (t)}

^{is an}elliptic functionsolutionof equation(1) Case2. If

fl

^and

f2

^areoftheform

^(R2),

^{then 6 ccis}the M-invariant. The same method as in Case forz2, works for_zl

() u ^{()/u (t)}

^{which istherefore}verified to be anelliptic functionwith periods2aand2ia.

Case3. If

fl

^{and fo_areoftheform}

^(R3)

.h ^=o2/, o:

#-0, thenit iseasytoverify that for any 6E

C,

INTEGRABILILTY OF DOUBLY-PERIODIC RICCATIEQUATION 789

where

fl

denotes theinverse transformation off,. This means thatZl

Ul/UZ

^andz2

u/u2

are both single-valued solutions with periods 2a and 4ic. Further, with the same verification as in Casel,we getthat

z: (t)

^andz2

(t)

arebothelliptic functions.

Case4. If

fl

^and

f2

^areof the form

(Rs)

/: -,

Y ⁽ + )/(6 + ), a #

^0,

then it iseasytocheck thatfor any EC

f() ,

and that

f2(,) ,,

1,2,

where

I , ^{2 i}

^{Thismeansthat}

u’+u’

_{and z_=-}

Z+

lUl +

^U2

areboth ellipticfunctionsolutionswithperiods 4a and2ic. Thiscompletes the proof.

Wenotethe following facts:

(a) IftheRiccati equation(1)has anellipticfunction solution, say,

zl(t),

thenitsgeneralsolution canbeexpressedas

Z()

^Z

(;) +

^y(),

where

y(t)

satisfies thefollowingBernoulli’s equation.

y,() y2($)+

2Zl()y(f;).

Co) Anyellipticfunction withdouble periods

w

^and

cv

^isof the form

Rl[p(t)] + R2 [p(t)]pt(),

wherep(t)istheWeierstrassellipticfunction withthesamedouble periods, p’(t)is itsderivative, and

R ^{(x), R2(x)}

^{arerationalfunctions}in x with constantcoefficients(see [11]).

By Lemma3and the factsabove,weobtainimmediately:

TREOREM. If the monodromy group of theRiccatiequation

(1)

issolvable,thenequation(1)is integrable, thatis tosay,itsgeneral solution canbeexpressedin termsofa Weierstrassellipticfunction by solving algebraic equations,differentiationandintegrationin finite terms; each solutionof equation(1) ismeromorphiconC.

By remarking the following facts: (i) for

A n(n- 1)(n

E

N),

the monodromy group G of equation

(3)

^issolvable(see[1,9]), (ii)the solvability of thegroup

M

isequivalenttothatof thegroup

G,

^wehavealso:

COROLLARY1. For A

n(n 1)(n N),

theRiccatiequation(1)isintegrable (Recentwork hasshownthatthe Riccatiequation(1)isalsointegrablewhenA

(4-)4/4-3),

^{n N. Thisnewresult}

was presented by G-nan at MathematicsToday and TomorrowInternational Conference in Florida in 1997).

2Ul +

U2

790 M.LINGANDG.KE-YING Especially,wehave:

COROLLARY2. For A 6, the Kiccati equation(1)is integrable. Its generalsolutioncan be expressedin termsof theWeierstrassellipticfunctionof order 3withperiods2cand 2ic

PROOF.

Suppose

^that

(t)

^{isthesolutionof theinitialvalue}problem

z’=z 2-6(t)

(o) o.

Underthetransformz

u’/u,

^thissolution

(t)

correspondstothesolution

fi(t)

^ofthefollowing problem

u"

6g,(t)u 0

(o) o, _’(o) o.

It has been shown

(see [9])

^that

fi(t)

has the following properties (i)

fi(t + 2c)

fi(t),and fi(g

+

^2ia)

fi(t)

(ii)the points

,,., ^{(2m 1)a} + (2n 1)ia(m,

^nE

N)

areallpoles of order2 for

();

(iii)the points

m., (2m- 1) +

^2nic

and’,,,

^2rnc

+ (2n-

1)ioarethesimple zeros of

(),

^andthat

(t)

does nothave any other singularityor zero onC. These properties imply that

() fi’()/fi(t)

isanellipticfunctionof order3 withperiods2cand2ict. Thiscompletestheproof The discussion of the particular doublyperiodicRiccati equation

(1)

can be extendedto a more general case, that is, for anyRiccati equation withelliptic functions asitscoefficients, itsintegrability definedbythe solvability ofitsmonodromy group meansthatitsgeneral solution canbeexpressed in termsofsomerelevantellipticfunctionby solving algebraic equations, differentiationandintegrationin finiteterms. However, theremaining problem,thatis,howtoconstructitsmonodromy group directly fromitscoefficients,is stillopen. (Inthe particular case,i.e. for equation(1),theexactrelationbetween themonodromy group andtheparameterAhas been obtained. This resultwaspresented byGuanat MathematicsTodayandTomorrowInternationalConferenceofMathematicsin Florida in 1997

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SL(2, C)

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T z,

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