Some Tests For Differential Equations To Have Painlevé Property

It is an old and difficult problem to determinate a given equation whether ithas Painlevé property in the field of studying about differential equations. For along time, many mathematicians and physicians attached important to it. In theinitiate development of differential equation theory, one always made use of theformal integration to get the solution of equations. In other words, one made itpossible to transform some differential type of solution questions into makingintegral type. That is Elementary Integrals Method. However, people very soonnoticed that hardly equations can be solved by Elementary Integrals Method.In the following ages, mathematicians and physicians gradually discoveredan effective method, that is so called Painlevé singularity analysis method,which can use to study the integrability of differential equations. The basicthought of this method is to determinate global property with the equationssolution by considering the singularity of the differential equations in thecomplex plane. Painlevé shows that the nonlinear ordinary equations generallyhave singularity in complex time plane and points out the remarkableproperty ,i.e., the Painlevé Property [2, 3], which means the general solution ofequation only has pole. It is a key point of the Painlevé Analysis to distinguishwhether the given differential equations have Painlevé property.In first chapter of this paper, we introduce Painlevé equations and thebackground of the Painlevé property. At the same time, we summarize sometests for differential equations to have Painlevé property, such as IST Method,ARS Method, WTC Method and WK Method.In the second chapter, we particularly introduce the basic idea of theARS algorithm and the application of it in the Hamilton system. The ARSalgorithm consists in three steps. First, a study of the non-null first term of theequation must be done .The whole equation must be expressed as a Laurentseries for that. Then, one calculates the resonances, that is, those powers towhich the arbitrary constants of the solutions in the Laurent development appear.Lately, the consistency of the arbitrary constants must be checked, in order todiscard some extra cases which lead to false conclusions.We will illustrate the ARS method as following example. Considering theHamilton system(),2()121 332221302221H = Px2 +Py2+ω x+ωy+ax+axy+axy+ay (1)The famous Hénon-Heiles system[31](),21 3321H = Px2 +Py2+ax2+by2+axy+ay (2)is the particular case of Hamilton system (1). According to the ARS method, weknow system (1) is integral in the following two cases:(I) When the resonances are r =?1,1,4,6,then we obtain the first groupof conditions which verifies the Painlevé property. Hamilton (1) can be changedinto()()2()121 332221302221H = Px2 +Py2+ω x+ωy+bcx+cxy+cxy+cy, (3)Where the co-efficients b and ci are given byb = ?( p2 +q2)?2,c0 = p( 2p2+q2),c1=q(4p2+q2),c 2 = p( p2+4q2),c3=q(p2+2q2).Considering by combining adequately p and q with the co-efficientsα i.The sufficient condition for the co-efficients iα to verify the Painlevéproperty is:7 a 02 ? 2a12?28a22?32a32+65a1a3=0,? 32 a 02 ?28a12?2a22+7a32+65a0a2=0. (4)Besides, parameters 1ω and 2ω must accomplish one of the equalitiesI a: ω1 =ω2,,4:2131321aaIb ωω =? a+?a (3 a1 ? 5a3)(a1?4a3)≥0.,2:4131321aaIc ωω =?a+?a (3 a1 ? 5a3)(a1?4a3)≤0.In the limit situation 3a 1 ? 5a3=0,I band I c coincide. Let us note that,due to the restrictions given by (4),the quotient between 1ω and 2ω isalways finite and negative. If a1 = 4a3 and the conditions (4) are verified,H reduces to the Greene system, an integrable subcase of the Hénon andHeiles family given before.For the subcase I a,the rotationx = qpu 2? +pqv2, y = ?ppu 2 ++qqv2, (5)Converts Hamilton (3) into the Greene's type system()(2),2()12H = 1 Pu 2 +Pv2+ω u2+v2+du2v+v3where 21d = ( p2 +q2)?.Hence, subcase I a corresponds to integrableHamiltons. Subcases I b and I c are transformed into integrable systems bymeans of the rotation (5) followed by the translation,12[22(23)]2222442qu = s+ω p+qp+p+q (6)Concretely, Hamilton system is transformed into the Greene case()(2),2()121 232221H = Ps 2 +Pt2+vs+vt+bs+stWhere12( 212322),v =ω +ω 22 12(62 12322).v =?ω +ω+ωThis implied that, indeed, the Greene case is a particular subcase of I band I c.The second integral of motion, that is I G, going back by a rotation or by atranslation plus a rotation to the original variables. From this analysis we canconclude that all the cased I correspond to integrable systems.(II) The resonances attach to the Laurent power series in this occasion are:r =?1,2,3,6( 12ω ≡ ω).The corresponding Hamilton system is()()2()121 33222130H = Px2 +Py2+ω x2+y2+efx+fxy+fxy+fy, (7)where the co-efficients e and f i are given bye = [ p3 (p2+q2)]?1,f 0 = ?2 p4+2p2q2?a3p2q3?2q4?a3q5,f 1 = 3 pq(?2p2+a3p2q+2q2+a3q3),f 2 = ?3 p2q(a3p2+2q+a3q2), f 3 = a3.Now, the relations among the co-efficients iα which give movable polesis performed by combining p , q and iα .The relation is3( a 0 a2+ a1a3)?a12?a22=0. (8)With this, the Painlevé condition is satisfied and it remains to prove theintegrability of the system. We can notice that Hamilton system (1)whoseco-efficients iα satisfy(8)and 12ω = ω is transformed into the separablecase()(2),2()12H = 1 Pu 2 +Pv2+ω u2+v2+egu3+v3with g = a1 p4+2 a1p2q2+2q3+a1q4.The transformation is merely therotation of co-ordinatesx = ?qpu 2? +pqv2, y = ppu 2 ++qqv2,and the corresponding rotation of moments. The second integral, which isinvolution with H , is()(2).2()12I = 1 ω u2 ?v2+Pu 2?Pv2+egu3?v3Form I we can deduce easily the second integral of motion for theoriginal system corresponding to H , by rotating back I .In the third chapter, we expound the ideas and processes of the IST method.This method was first discovered by Gardner, Greene, Kruskal and Miura(GGKM,1968) and applied by them to the Korteweg de Vries (KdV,1895)equation in a pioneering work[14].It has many features in common with themethod of Fourier transforms and may be considered an extension of Fourieranalysis to nonlinear problems[37].The method is analogous to the Fouriertransform method of linear problem: namely, one maps the initial data into thescattering data ,follows the evolution of the set of scattering data and at anydesired time inverted the mapping with∫K ( x,y,t)+ B(x+y,t)+x∞ K(x,z,t)B(y+z,t)dz=0. (9)Thereby recovering the solution u ( x,t) to the partial differential equationu t + uuxx+uxxx=0, (10)Associated with (10)the linear eigenvalue problem,vxx+ ( 2 +16u(x,t))v=0ζ , ? ∞