Weierstrass integrability of differential equations

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–6) Weierstrass integrability of differential equations Jaume Gin´ e, Maite Grau Departament de Matem`atica. Universitat de Lleida. Avda. Jaume II, 69. 25001 Lleida. E-mails: [email protected], [email protected]. key words: nonlinear differential equations, integrability problem. Abstract The integrability problem consists in finding the class of functions a first integral of a given system must belong to. We recall the characterization to admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non–Liouvillian integrable systems. 1 Introduction It is not always possible and sometimes not even advantageous to explicitly write the solutions of a system of differential equations in terms of elementary functions. In fact, Poincar´e begun the qualitative theory of differential equations to well–understand the behavior of the solutions of a differential system without their explicit knowledge. For Poincar´e, it is thus necessary to study the functions defined by the differential equations by themselves and without bringing them back to simpler functions. These thoughts induced Poincar´e to tackle the study of differential equations beyond an essentially different point of view from his predecessors. His study provokes a conceptual change on the understanding of differential equations. Sometimes, though, it is possible to find elementary functions that are constants on solution curves, that is, elementary first integrals. These first integrals allow to occasionally deduce properties that an explicit solution would not necessarily reveal, see for instance [10]. This thought originated the modern integrability theory of differential equations that tries to respond to the natural question: When does a system of differential equations have a first integral that can be expressed in terms of “known functions” an how does one find such an integral? The answer when the “known functions” are the elementary functions (i.e. functions expressible in terms of exponentials, logarithms and algebraic functions) was given in [10], and when the “known functions” 1 J. Gin´e, M. Grau are the Liouvillian functions (i.e. functions that are built up from rational functions using exponentiation, integration, and algebraic functions) was given in [11]. In these two cases it is given the form of an integrating factor if the system have these type of first integrals. In this paper we extend the results presented in [10] and [11]. In elementary courses on differential equations we consider systems of the form ˙x=dx dt =P(x, y),˙y=dy dt =Q(x, y),(1) where Pand Qare polynomials in C[x, y], Cbeing the complex numbers. Throughout this paper we will denote by m= max{deg P, deg Q}the degree of system (1). Obviously, we can also express system (1) as the differential equation dy dx =Q(x, y) P(x, y).(2) We learn that although we cannot always explicitly solve this system, we are occasionally able to find first integrals, that is nonconstant functions H(x, y), analytic on some nonempty open set in C2, that are constant on the solution curves in this set. To do this we consider the differential form Q(x, y)dx −P(x, y)dy = 0. If ∂P/∂x =−∂Q/∂y, then H(x, y) = RQdx −Pdy will be a first integral. If ∂P/∂x 6=−∂Q/∂y, we are taught ad hoc methods to find an integrating factor, that is a function R(x, y) such that ∂(RP)/∂x =−∂(RQ)/∂y. In case we can find a function R,H(x, y) = RRQdx −RPdy will be a first integral. For example, if (∂Q/∂x +∂P/∂y)/P is independent of y, then R= exp(R(∂Q/∂x +∂P/∂y)/Pdx) will be an integrating factor. 2 Integrability problem We recall that the integrability problem consists in finding the class of functions a first integral of a given system (1) must belong to. For instance in [9], Poincar´e stated the problem of determining when a system (1) has a rational first integral. The works of [10] and [11] go in this direction since they give a characterization of when a polynomial system (1) has a elementary or a Liouvillian first integral. A precise definition of these classes of functions is given in [10, 11]. An important fact of their results is that invariant algebraic curves and exponential factors play a distinguished role in this characterization. Moreover, this characterization is expressed in terms of the inverse integrating factor. Now, we state some results related to integration of a system (1) by means of elementary and Liouvillian functions. Theorem 1 [10] If system (1) has an elementary first integral, then there exists ω0, ω1, . . . , ωn algebraic over the field C(x, y)and c1, c2, . . . , cnin Csuch that the elementary function ˜ H=ω0+ n X i=1 ciln(wi),(3) is a first integral of system (1). 2 Weierstrass integrability of differential equations The existence of an elementary first integral is intimately related to the existence of an algebraic inverse integrating factor, as the following result shows. Theorem 2 [10] If system (1) has an elementary first integral, then there is an inverse integrating factor of the form V=µA(x, y) B(x, y)¶1/N , where A,B∈C[x, y]and Nis a nonnegative integer number. In the work [2], the systems (1) with a (generalized) Darboux first integral, that is, with a first integral of the form H=fλ1 1fλ1 2· · · fλr rµexp µh1 gn1 1¶¶µ1µexp µh2 gn2 2¶¶µ2 · · · µexp µh` gn` `¶¶µ` (4) are studied and the following result is accomplished. Theorem 3 [2] If system (1) has a (generalized) Darboux first integral of the form (4), then there is a rational inverse integrating factor, that is, an inverse integrating factor of the form V=A(x, y) B(x, y), where A,B∈C[x, y]. Unfortunately, not all the elementary functions of the form (3) are of (generalized) Darboux type. That’s why we can find systems with an elementary first integral and without a rational inverse integrating factor, see for instance [7]. We remark that both Theorems 2 and 3 give necessary conditions to have an elementary or (generalized) Darbouxian, respectively, first integral. The reciprocal to the statement of Theorem 2 is not necessarily true. But the reciprocal to the statement of Theorem 3 is true as we will see later. The following Theorem 4 ensures that given a (generalized) Darboux inverse integrating factor, there is a Liouvillian first integral. The Liouvillian class of functions contains the rational, Darboux and elementary classes of functions. Singer gives in [11] the characterization of the existence of a Liouvillian first integral for a system (1) by means of an integrating factor. Theorem 4 [11] System (1) has a Liouvillian first integral if, and only if, there is an inverse integrating factor of the form V= exp nR(x,y) (x0,y0)ηo, where ηis a rational 1-form such that dη ≡0. Taking into account Theorem 4, Christopher in [3] gives the following result, which makes precise the form of the inverse integrating factor. 3 J. Gin´e, M. Grau Theorem 5 [3] If the system (1) has an inverse integrating factor of the form V= exp nR(x,y) (x0,y0)ηo, where ηis a rational 1-form such that dη ≡0, then there exists an inverse integrating factor of system (1) of the form V= exp{D/E}YCli i,(5) where D,E, and the Ciare polynomials in xand yand li∈C. We notice that Ci= 0 and D= 0 are invariant algebraic curves and exp{D/E}is an exponential factor for system (1), see for instance [7]. Theorem 5 states that the search of Liouvillian first integrals can be reduced to the search of invariant algebraic curves and exponential factors. A result to clarify the easiest functional class of the first integral once we know the inverse integrating factor is a straightforward consequence of Theorem 5 and is the reciprocal of Theorem 3. Corollary 6 If system (1) has a rational inverse integrating factor, then the system has a (generalized) Darboux first integral. The proof is based in that if Vis a rational inverse integrating factor of system (1) then, η= (Q(x, y)dx −P(x, y)dy)/V (x, y) is a rational 1-form such that dη ≡0. Since H= exp nR(x,y) (x0,y0)ηois a first integral of system (1), by Theorem 5 His a (generalized) Darboux first integral. In [8], Painlev´e proved the following result, see also [4] and references therein. Theorem 7 [8] A differential system (1) has a first integral of the form I(x, y) = (y−g1(x))α1(y−g2(x))α2. . . (y−g`(x))α`h(x),(6) where gj(x)are unknown particular solutions of (4), h(x)is an unknown function of xand the αiare unknown constants such that Q` i=1 αi6= 0, if and only if it has an integrating factor of the form M(x, y) = α(x)S(x, y) (y−g1(x))(y−g2(x)) . . . (y−g`(x)) ,(7) where S(x, y)is polynomial in the variable yof degree `−m−1. Moreover, (a) if the system has two different integrating factors M1and M2of the form (7) with M2/M1nonconstant, then there exists a change of variable that is rational in the variable ywhich transforms the equation (4) into a Riccati equation. (b) if the differential system has only one integrating factor of the form (7), then the particular solutions gi(x)from the ansatz (6) are calculated algebraically and h(x)is given by a logarithmic quadrature. 4 Weierstrass integrability of differential equations As usual we define C[[x]] the set of the formal power series in the variable xwith coefficients in Cand C[y] the set of the polynomials in the variable ywith coefficients in C. A polynomial of the form n X 0 ai(x)yi∈C[[x]][y] is called a formal Weierstrass polynomial in yof degree nif and only if an(x) = 1 and ai(0) = 0 for i<n. A formal Weierstrass polynomial whose coefficients are convergent is called Weierstrass polynomial, see [1]. In a natural generalization we call Weierstrass rational function a function which is a quotient of sums of Weierstrass polynomials. We say that a system (1) admits a Weierstrass integrating factor or a Weierstrass first integral if the integrating factor or the first integral is a Weierstrass rational function. Hence, we define Weierstrass integrability when a system (1) admits a Weierstrass rational integrating factor or a Weierstrass rational first integral. In this sense, the systems which are Darboux integrable are included in the set of systems which are Weierstrass integrable. However, the systems which are Liouvillian integrable are not all included in the set of systems which are Weierstrass integrable. Moreover, there are systems which are Weierstrass integrable which are not Liouvillian integrable, see Example 2 in [4]. The systems with a first integral of the form (6) are Weierstrass integrable because they have an integrating factor of the form (7). The following theorem determines which Weierstrass integrable systems are Liouvillian integrable. Theorem 8 If a differential system (1) has a first integral of the form (6), and at least one algebraic solution, then it has a Liovillian first integral and therefore a (generalized) Darboux inverse integrating factor of the form (5). Proof. If the system has a first integral of the form (6), then by Theorem 7 either there exists a rational Weierstrass change which transforms the system of differential equations (1) into a Riccati equation (see statement (a)) or all the particular solutions gi(x) from the ansatz (6) are algebraically calculated and h(x) is given by a logarithmic quadrature (see statement (b)). In the first case, as we have an algebraic solution, the Riccati equation can be solved by quadratures and the system has a Liouvillian first integral. In the second case the system all the curves y−gi(x) = 0 are algebraic curves and h(x) is given by a logarithmic quadrature, which implies that the inverse integrating factor (7) either is a rational integrating factor if α(x) is a polynomial or is a (generalized) Darboux integrating factor if α(x) is not a polynomial. In both cases by Corollary 6 or Theorem 5, the system has a Liouvillian first integral. Acknowledgements The authors are partially supported by a DGICYT grant number MTM2005-06098-C0202. The first author is also partially supported by a CICYT grant number 2005SGR 00550, and by DURSI of Government of Catalonia “Distinci´o de la Generalitat de Catalunya per a la promoci´o de la recerca universit`aria” 5 J. Gin´e, M. Grau Bibliography [1] E. Casas-Alvero, Singularities of Plane Curves. London Mathematical Society Lecture Note Series 276 Cambridge University Press, Cambridge, 2000. [2] J. Chavarriga, H. Giacomini, J. Gin´e, and J. 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