Adi ve kısmi diferansiyel denklemlerin tekillik analizleri ve integre edilebilirlikleri
Singularity analysis and integrability of the ordinary and partial di̇fferential equations
- Tez No: 66682
- Danışmanlar: PROF. DR. MEHMET CAN
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Diferensiyel denklemler, Tekillik, Differential equations, Singularity
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Matematik Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Üç bölümden oluşan bu tezde tekillik analizi ile onun tam ve kısmi integre edilebilirlikle olan ilgisi incelenmiştir. Biz öncelikle integre edilebilirliğin üç değişik anlamını ifade ettik: 1.Sistemlerin kuadratürlerle çözülebilİrliği, 2.Hareket denklemlerinin güzel özelliklerinden dolayı integre edilebilir oldukları kabul edilen lineeer denklem sistemlerne indirgenebilirliği 3.Sistemlerin integro-differansiyel denklemlere indirgenerek lineerleştirilebilirlikleri nedeniyle integre edilebilirlikleri. 1.Bölüm'de cebirsel integre edilebilirlik kavramı, Yoshida'nm“İntegre edilebilir sistemler için Kowalevski üssü kompleks veya irrasyonel olmamalıdır.”tanımı altında açıklandı. Tam integre edilebilirliğin hareketin kompleks analitik integrallerinin yeterli sayıda var olması demek olduğu, tam olmayan integre edilebilirliklerin kısmi ve kısıtlı integre edilebilirlik adı altında yeterli sayıda integralin olmaması ve belli şartlar altıda integre edilebilirliğin gerçekleşmesi olarak açıklandı. 2.Bölüm içerisinde; Tekillik (Painleve) analizinden faydalanılarak ADD'ler ve KDD'lerin integre edilebiliriliği araştırıldı. Bunların incelenmesinde kullanılan ARŞ Algoritması ve Weiss Metodu sunularak örnekler verildi. 3.Bölüm'de de Ziglin Teoremi'ne dayanılarak birkaç sistem için integrallerin var olmadığı ispatlandı. Ziglin yaklaşımının lineer olmayan acılımıyla integre edilemezlik kriteri olarak“çoklu-Painleve”sunuldu. Bu pratik metodun açıklanması için bazı uygulamalar yapıldı.
Özet (Çeviri)
SUMMARY This thesis work reviews papers which illustrate the connection between integrability and the singularity structure of the solutions of nonlinear dynamical systems. in the first section we have attempted to classify various aspects of integrability. We have distinguished three different situations. a)The system can be solved by quadratures. For instances, the two dimensional Ha- miltonian system H = l/2(Px2+pv2) + F(p) + G(ç>)/p2, where p = (x3 +y3\ and ç =arctan(y/x) has the second integral ı = (xpy-ypx)~+2G(<p)- This allows the equations of motion to be reduced first to a quadrature for p: p2=2H0-2F(p)-I0/p2, where HO and Io are the conserved values of H and I respectively. Önce p(t) is obtained from above, the equation for <p can also be reduced to a quadrature: r d(9) ^±r dt J V/0-2G(p) J p2 (t) ' b)The equations of motion can be reduced to a system of linear equations which are considered to be integrable because of their nice properties. in fact their solutions can be superimposed linearly and there exist global representations for thenı in terms of contour integrals. The simplest example is that of the well known Riccati equation: JT = a(t)x2 + b(t)x + c(t) which linearizes to viiy + (b)y + acy = O a through the transformation 3C = -^-. oy ' Some PDE's are also integrable through linearization, Burger's equation being the archetype: u t + 11 xx + 2uux =0. The Cole-Hopf transformation M = V.V/V reduces its solution to that of the heat equation: vl + vxx = 0. c) The system can be linearized in terms of integro-differential equations. This is, for example, the case of the Painleve transcendantal equations. There exist several types of integrability. These are: 1.Conıplete integrability means that complex analytic integrals of motion exist in sufficient number. For a system of N first order autonomous ordinary differential equations (ÖDE), sufficient number means N-1 time independent invariants (whereupon the system can be reduced to a single quadrature). 2.Algebraic integrability is investigated as a restricted notion of integrability by H. Yoshida and he introduced the necessary conditions for algebraic integrability: No Kowalevski exponent be irrational ör complex. The extension of those results to Hamiltonian systems is not straightfonvard as shown by M. Kummer, R. Churchill, and D. Rod. 3.Öne possible form of partial integrability is to have an insufficient number of integrals of motion. Partial integrability can also be associated with the existence of integrals of inadequate form. For a system of N first order differential equations the existence of N-4 time-dependent first integrals is not sufficient for complete integrability. Another type of integrability is“constrained”integrability. Öne well- known example is the fixed-energy integrability of Hamiltonian systems. There are two types of singularities: the ones termed fıxed, because their location is determined by the equation itself, and those called movable, the location of which depends on the initial conditions. Linear equations can only have fixed singularities. viii )Nonlinear equations can have both fîxed and movable singulanties. it was Painleve who first sought to determine ali first order ÖDE's. w'=f(z,w), with f irrational in w and analytic in z, the only movable singularities of which are poles.The idea was that equations having this so called Painleve property might be easier to integrate ör solve analytically. At the second, in a remarkable series of papers Painleve and his co-workers performed an exhaustive singularity analysis of ali ODE's of the form w“=F(z,w,w'), with F rational in w1, algebraic in w and analytic in z, the critical points (branch points and essantial singularities) of which are fıxed. in other words, they were able to identfy ali such equations, the only movable singularities of which are poles. Forty four of these equations were shown to be integrable in terms of elementary functions, by quadratures, ör by linearization. For the remaining six equations, new tanscendantal functions had to be introduced. The ARŞ algorithm was originally developed in order to determine whether a nonlinear ÖDE admits movable branch points, either algebraic ör logarithmic. it is important to keep in mind that this algorithm provides a necessary condition for the absence of such movable branch points. Thus, the occurrence of movable essential singularities cannot be detected by this procedure. Arş conjecture says that a system of ÖDE's wf =Fl(wl,w2,,WB;Z),i=l,....,n satisfies the necessary conditions for the Painleve property, if its solutions can be expanded in püre Laurent series near every öne of their movable singularities at Z=ZQ. in other words, following the ARŞ algorithm we must come across no algebraic branch points and no logarithmic singularities. Every ordinary differential equation obtained by an exact reduction of a nonlinear partial differential equation solvable by İST method, has the Painleve Property.Of course, transformations of variables are allowed and a given equation may pass the test only after some transformations. This conjecture, therefore, would provide a necessary condition for the integrability of a given partial differential equation. To apply the conjecture öne needs two different ingredients: first a method to obtain ali reductions of a given PDE and second a method to test a given ÖDE for the Painleve property. Sometimes however, ali the reductions öne can find too trivial to yield an interesting information. Fortunately Weiss and collaborators made a majör progress by doing away with reduction and introducing the Painleve property for the PDE's themselves. in fact, according to Weiss, a PDE will possess the Painleve ixproperty if its solutions are single-valued about any singular manifold (j)(zı,z2,A) which is noncharacteristic. Weiss presents singular manifold method which tests the Painleve property. To verify if a PDE has the Painleve property we expand a solution of a nonlinear PDE about a movable, singular manifold 0(Zj,z2,,ZB) = O. Let u=u(zı,z2,...,Zn) be a solution of the PDE and assume that 00 «=#'2>^' ]=0 where § and Uj=Uj(Zı,Z2,,Zn) are analytic functions of (zı,z2,,Zn) in a neighborhood of manifold. Subtitution of u into the PDE determines the possible values of p and defines the recursion relations for Uj, j=0,1,2,When p is a negative integer and u is valid and general expansion about the manifold ^(z,,z2,,zB) = 0, then the solution has single valued representation about ^(znz2,,zB) = 0. If this representation is valid for ali allowed singularity manifolds, then the PDE has the Painleve property. The singular manifold method can also be applied to ordinary differential equations. For instance, Bâcklund transformation for the Henon-Heiles system xtt = -Ax - 2dxy yn=-By + cy2~dx2 with d/c-l/ö has the form x=flx0+3cl y = <t>~2y0 + flyı+y2 wherey0=-tf, * = *”, y2=^(U-B-3V-3(0tt/<j>t)2and x02 = tfV, x, = --(Vt IV + ^I <t>t)V112. The variable V is V = {<t>;t} + l where -V2+-V*+(-B-2A+-K)V2+(-B2--A2)V=Q. 2 ' 23363 xThis defines V as a Weierstrass elliptic fünction and </> = -, where u ı,U2 112 are solutions of the linear equation «B=~(F + A)ıı. in the third section using Ziglin theorem the nonexistence of interagls for some equations is proved. Original form of the Zigliris theorem was that: Suppose that a given Hamiltonian H has N-1 analytic integrals, which are functionally independent together with H, and moreover that there exists a non-resonant matrix M1 (Monodromy matrix). Then it is necessary that any other monodromy matrix M either: a)commutes with M ör b) permutes the eigenspaces of M. in particular M1 is also non-resonant and it must also neccessarily commute with M. The problem which arises at this point is that of the practical applicability of Ziglin's theorem. The knowledge of the monodromy matrices is neccassary. Ziglin's method is particularly usefol in constructing proofs of nonintegrability. As in the case of Ziglin, multivaluedness is incompatible with integrability unless the“interaction”between singularities satisfies some communication properties. The“poly-Painleve”criterion, consists in checking whether the“interaction”between two of the system's singularities leads to a multivaluedness of the“dense”type. The method is asymptotic and a parameter G will be introduced, which may be taken arbitrarly small. The main idea is the following : if öne can show that the some trajectory can be characterized by two values of an integration constant, which differ by an additive quantity proportional to s“ (for some n), this will mean that the integration constant, for this trajectory, is hopelessly (i.e.”densely“) multivalued. We must point out that the ”poly-Painleve“ criterion, although intuitively appealing in its association of ”dense" multivaluedness to non-integrability, does not yet constitute a powerfül algorithm
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