Partial integrability of the anharmonic oscillator

Journal of Nonlinear Mathematical P hysics V olume 2007, Numb er 14 (3), 454–465 Ar ticle P artial in teg rabilit y of t he anharmonic oscillator R ob ert CONTE Servic e de physique de l’ ´ etat c ondens ´ e (CNRS URA 2464, CEA–Saclay F–911 91 Gif-sur-Yvette Ce dex, F r anc e E-mail: R ob ert.Conte@c e a.fr R e c eive d January 29, 2007; A c c epte d in r evise d form April 9, 200 7 Abstract W e c o nsider the anharmonic os cillator with a n arbitrar y-degree anharmonicity , a damping term and a forcing term, all co efficients b eing time-dependent: u ′′ + g 1 ( x ) u ′ + g 2 ( x ) u + g 3 ( x ) u n + g 4 ( x ) = 0 , n real . Its ph y sical applications r ange fro m the atomic Thomas-F ermi mo del to Emden gas dynamics equilibria, the Duffing osc illa tor a nd numerous dynamical sys tems. The present work is a n ov erv iew which includes and generalizes all previously known results of par tia l integrability of this oscillator. W e give the most general t w o conditions on the co efficien ts under whic h a first int egral of a particular t ype exists. A natural int erpretation is given for the tw o co nditions. W e compare these tw o conditions with those provided by the Painlev ´ e analysis . 1 In tro du ction The harmonic oscillat or is the simplest appro ximation to a physica l oscillat or and, when p erturbation terms are tak en into accoun t, the resulting anharmonic oscil lator is go v ern ed b y the nonlinear different ial equati on E ≡ u ′′ + g 1 u ′ + g 2 u + g 3 u n + g 4 = 0 , n ( n − 1) g 3 6 = 0 , (1.1) where ′ denotes the deriv ativ e w ith resp ect to the indep endent time or space v ariable x , g 1 ( x ) a d amping factor, g 2 ( x ) a time-dep end en t frequency co efficien t, g 3 ( x ) the simplest p ossible anharm on ic term, g 4 ( x ) a f orcing term. As to the anharmonicit y exp onen t n , it can b e either real if u ( x ) is real p ositiv e, wh ic h is the case for Lane-Emden [24] gas dynamics equilibria, rational, lik e n = 3 / 2 in the Thomas and F ermi [31, 17] atomic mo del, or m ore usu ally in teger: − 3 for the Ermak ov [12] or Pinney [28] equatio n, 3 for the Duffing oscillator [10]. F or generic v alues of the co efficien ts, this equation is equiv alent to a third order au- tonomous dyn amical system, w hic h generically admits no closed form general s olution. The pu rp ose of this article is to review all the nongeneric situations for whic h there exist exact analytic results, such as a fir st in tegral or a closed form solution, either particular or general. This can only happ en w hen the co efficien ts satisfy some constrain ts. The p ap er is organized as f ollo ws. In Section 2, w e give a Lagrangian and a Hamiltonian form ulation for any v alue of the co efficien ts ( n, g i ). This generalizes all the previous particular r esults, obtained for v alues of ( n, g 1 , g 2 , g 3 , g 4 ) equal to: Copyrigh t c  2007 by Ro b ert Conte P artial in tegrabilit y of the anharmonic oscillator 455 (5; 0 , const,const , 0) [3], (5; 2 /x, 0 , 1 , 0) [26, Eq. (3.7)], ( n ; g 1 , 0 , g 3 , 0) [30, 29, 16], ( n ; 0 , const , ax α , 0) [1], ( n ; g 1 , 0 , 1 , 0) [25], ( n ; g 1 , g 2 , g 3 , 0) [15], [23, Sectio n 6.7 4, v ol. 1]. In Section 3, we pro vide t w o conditions on ( n , g i ) which are sufficient to ensure the existence of a first in tegral. In S ectio n 4, w e giv e a natural interpretatio n of these t w o conditions. Finally , in sect ion 5, w e p er f orm the Pa inlev ´ e analysis of (1.1). Most of this w ork h as already b een don e by P ainlev ´ e and Gam bier [19]. Indeed, the ordinary differen tial equation (ODE) (1.1) b elongs, at least for sp ecific v alues of n and ma yb e after a change u 7→ u N of the dep end ent v ariable u in case n is n ot an in teger, to the class of second order O DEs whic h they studied and classified. Ho wev er, as opp osed to these classical authors, we do not request th e fu ll P ainlev ´ e int egrabilit y of the ODE, only some partial in tegrabilit y , and this requires some add itional wo rk. In p articular, we compute the cond ition for the absence of any infin ite mo v able branching, i.e. a multiv aluedness w hic h o ccurs at a lo catio n dep ending on the initial conditions. Suc h a condition, lik e for linear ODEs, arises from an y in teger v alue of the difference of the t w o F uc hs indices, w hether p ositiv e or negativ e, and we c heck that this condition is a differen tial consequence of the tw o conditions for th e existence of a particular firs t integ ral. Th is detailed P ainlev´ e analysis of equation (1.1 ) happ ens to b e an excelle n t example for sev eral f eatures of P ainlev ´ e analysis whic h are most of the time o v erlo ok ed. F or con v enience, we use the notatio n Log G 1 ( x ) = Z x g 1 ( t )d t, γ 3 = Log g 3 , γ 4 = Log g 4 , (1.2) and th e con ven tion th at fun ction G 1 implicitly conta ins an arbitrary multiplicat iv e con- stan t; letter K , with or withou t subscript, denotes an arbitrary constan t. F unction G 1 frequent ly o ccurs, for the w ay to suppr ess term g 1 u ′ in (1.1) is to p erform the c h ange of function u → G − 1 / 2 1 u . 2 Lagrangian and Hamiltonian form ulations F or ev ery v alue of ( n, g i ), including the logarithmic case n = − 1, the anh armonic oscillato r can b e pu t in Lagrangian form  ∂ L ∂ u ′  ′ − ∂ L ∂ u = 0 , (2.1) or in Hamilt onian form q ′ = ∂ H ∂ p , p ′ = − ∂ H ∂ q , (2.2) 456 Rob ert Conte as sh o wn by the explici t expressions for L, H , q , p L ( u, u ′ , x ) = G 1  u ′ 2 − 2 g 3 Z u 0 u n d u − g 2 u 2 − 2 g 4 u  + 1 2  hu 2  ′ , (2.3) H ( q , p, x ) = G 1  u ′ 2 + 2 g 3 Z u 0 u n d u + g 2 u 2 + 2 g 4 u  − 1 2 h ′ u 2 , (2.4) q = u, p = 2 G 1 u ′ + hu, (2.5) in wh ic h h is an arbitrary gauge fu nction of x . 3 P articular first in tegral According to No ether theorem, one ca n find first in tegrals b y lo oking at the infinitesimal symmetries of the L agrangia n. F or a detailed r eview of this Lie symmetries app roac h to the anharmonic oscilla tor, the in terested reader can refer to [14]. S ince the dep end ence of ODE (1.1) in u is rather simple, let us determine under whic h conditions on parameters ( n, g i ) there exists a particular fi rst in tegral contai ning the same kind of terms than the Hamiltonian I = f 1 u ′ 2 + f 2 Z u 0 u n d u + f 3 uu ′ + f 4 u 2 + f 5 u + f 6 , (3.1) in wh ic h th e six f unctions f i of x are to be d etermined. Eliminating u ′′ b et ween I ′ and E , we obtain I ′ − (2 f 1 u ′ + f 3 u ) E ≡ f ′ 2 Z u 0 u n d u − g 3 f 3 u n +1 + ( f 2 − 2 g 3 f 1 ) u n u ′ + f ′ 6 (3.2) +( f ′ 1 + f 3 − 2 g 1 f 1 ) u ′ 2 + ( f ′ 3 + 2 f 4 − g 1 f 3 − 2 g 2 f 1 ) uu ′ +( f ′ 4 − g 2 f 3 ) u 2 + ( f 5 − 2 g 4 f 1 ) u ′ + ( f ′ 5 − g 4 f 3 ) u. Out of the n ine monomials R u 0 u n d u, u n +1 , u n u ′ , u ′ 2 , u 2 , uu ′ , u ′ , u, 1, only eigh t are lin- early ind ep enden t since n ( n − 1) 6 = 0, thus generating eight linear homogeneous differen tial equations in six un kno wn s, hen ce generically t wo conditions on ( n, g i ). Note that, ev en in the logarithmic case n = − 1, the firs t generated equation is f ′ 2 − ( n + 1) g 3 f 3 = 0. F unctions f 2 to f 6 are giv en by f 2 = 2 g 3 f 1 , (3.3) f 3 = 2 g 1 f 1 − f ′ 1 , (3.4) f 4 = ( g 2 1 + g 2 − g ′ 1 ) f 1 − 3 2 g 1 f ′ 1 + 1 2 f ′′ 1 , (3.5) f 5 = 2 g 4 f 1 , (3.6) f 6 = δ n, − 1 Z x g 3 f 3 d x, (3.7) and fu nction f 1 m u st b e a nonzero solution co mmon to the th ree linear equations  − 2( n + 1) g 1 g 3 + 2 g ′ 3  f 1 + ( n + 3) g 3 f ′ 1 = 0 , (3.8) ( − 2 g 1 g 2 + 2 g 1 g ′ 1 − g ′′ 1 + g ′ 2 ) f 1 + ( g 2 1 + 2 g 2 − 5 2 g ′ 1 ) f ′ 1 − 3 2 g 1 f ′′ 1 + 1 2 f ′′′ 1 = 0 , (3.9) ( − 2 g 1 g 4 + 2 g ′ 4 ) f 1 + 3 g 4 f ′ 1 = 0 . (3.10) P artial in tegrabilit y of the anharmonic oscillator 457 Eac h ab o ve equation can b e int egrated on ce, K 1 = f n +3 1 G − 2 n − 2 1 g 2 3 , (3.11) K 2 = f 2 1 G − 2 1 g 2 + Z h (( g 2 1 − g ′ 1 ) f 1 − 3 2 g 1 f ′ 1 + 1 2 f ′′ 1 ) ′ f 1 G − 2 1 i d x, (3.12) K 3 = f 3 1 G − 2 1 g 2 4 . (3.13) Whatev er b e ( n, g i ), the function f 1 can alw a ys b e computed from (3.9 ); dep ending on ( n, g 4 ), it is also giv en b y n 6 = − 3 : f 1 = G 2( n +1) / ( n +3) 1 g − 2 / ( n +3) 3 (3.14) g 4 6 = 0 : f 1 = G 2 / 3 1 g − 2 / 3 4 (3.15) n = − 3 : f 1 = − g − 1 3 y 2 , y = general solutio n of  g − 1 3 y 3 ( y ′′ − 1 2 g ′ 3 g 3 y ′ − g 2 y )  ′ = 0 (3.16) where the constants K 1 and K 3 ha ve b een absorb ed in th e d efinition of G 1 . The only case in whic h equation (3.16) needs to b e consid er ed is n = − 3 , g 4 = 0, and its solution can b e found in [12, 19] [22, ¶ 14.33] [28] [2, Eq. E12] [5]. Once f 1 is determined, f 2 and f 5 are giv en b y (3. 3), (3.7), an d f 3 , f 4 b y th e three follo wing expressions, corresp onding to cases (3.1 4), (3.1 5), (3.16) resp ectiv ely , f 3 f 1 =            2 n + 3  2 g 1 + γ ′ 3  2 3 (2 g 1 + γ ′ 4 ) − γ ′ 3 − f ′ 1 f 1 (3.17) f 4 f 1 =              g 2 + − 2( n − 1) g 2 1 − ( n + 3)(2 g ′ 1 + γ ′′ 3 ) − ( n − 5) g 1 γ ′ 3 + 2 γ ′ 2 3 ( n + 3) 2 g 2 + 2 9 g 2 1 − 2 3 g ′ 1 − g 1 γ ′ 4 − 2 γ ′ 2 4 − γ ′′ 4 g 2 + 1 4 γ ′ 2 3 + 1 2 γ ′′ 3 + 3 4 γ ′ 3 f ′ 1 f 1 + 1 2 f ′′ 1 f 1 . (3.18) P arameters ( n, g i ) m ust satisfy the conditions, p olynomial in n, g 1 , g 2 , γ ′ 3 , γ ′ 4 , resulting from the elimination of f 1 b et ween the three linear equations (3.8 )-(3.1 0 ). T here are t wo suc h conditions when g 3 g 4 is nonzero, and only one when it is zero. T h e simplest c hoice of these t w o conditions is (the lab elling refers to the contributing g i ’s): g 4 6 = 0 : C 134 ≡ 2 ng 1 − 3 γ ′ 3 + ( n + 3) γ ′ 4 = 0 , (3.19) g 4 6 = 0 : C 124 ≡ 4 g 3 1 − 18 g 1 g 2 − 18 g ′′ 1 + 27 g ′ 2 + (6 g 2 1 − 36 g 2 + 27 g ′ 1 ) γ ′ 4 − 6 g 1 γ ′ 2 4 − 4 γ ′ 3 4 + 9 g 1 γ ′′ 4 + 18 γ ′ 4 γ ′′ 4 − 9 γ ′′′ 4 = 0 (3.20) uniquely d efi ned as, resp ectiv ely , the condition indep endent of g 2 and the one ind ep en- den t of ( n, g 3 ). By elimination, one obtains the condition indep en den t of g 4 and the one indep endent of g 1 , C 123 ≡ − 4  2 g 1 + γ ′ 3  3 + ( n + 3)[( n − 3)( − 4 g 3 1 − 2 g ′′ 1 − 4 g 2 γ ′ 3 − γ ′′′ 3 ) 458 Rob ert Conte + 2( n + 3)( n − 1) g 1 g 2 + n ( − 8 g 1 g ′ 1 − 6 g 1 γ ′ 2 3 ) + ( n + 3) 2 g ′ 2 + ( n − 9)( − 2 g 2 1 γ ′ 3 − g ′ 1 γ ′ 3 ) − 3( n − 1) g 1 γ ′′ 3 + 6 γ ′ 3 γ ′′ 3 ] = 0 , (3.21) g 4 6 = 0 : C 234 ≡ n 3 g ′ 2 + n 2 ( − g 2 γ ′ 3 − γ ′′′ 3 + 3 2 γ ′′ 3 γ ′ 4 + 1 2 γ ′ 3 γ ′′ 4 − 2 γ ′ 4 γ ′′ 4 + γ ′′′ 4 ) − 3 2 γ ′ 2 3 γ ′ 4 + 1 2 γ ′ 3 3 − n 2 ( n − 1) g 2 γ ′ 4 − 1 2 ( n 2 − 3) γ ′ 3 γ ′ 2 4 + 1 2 ( n 2 − 1) γ ′ 3 4 = 0 . (3.22) F or ( n + 3) g 4 6 = 0, an y tw o of the ab o ve four conditions are functionally indep end ent. F or n = − 3, one has 27 C 123 − 4 C 3 134 = 0 and ind ep enden t conditions are C 134 and C 234 . All ab ov e conditions admit an integrat ing facto r, a natural consequence of the inte grated forms (3.11)–(3.13). This is eviden t for C 134 ; for eac h of the three others, it is sufficien t to integ rate it as a first order lin ear inhomogeneous ODE in g 2 , g 4 6 = 0 : K 134 ≡ G 2 n 1 g − 3 3 g n +3 4 , (3.23) g 4 6 = 0 : K 124 ≡  g 2 − 2 9 g 2 1 − 2 3 g ′ 1 + 1 9 g 1 γ ′ 4 + 1 9 γ ′ 2 4 − 1 3 γ ′′ 4  G − 8 / 3 1 g − 4 / 3 4 , (3.24) K 123 ≡ [( n + 3) 2 g 2 − ( n + 3)(2 g ′ 1 + γ ′′ 3 ) − 2( n + 1) g 2 1 − ( n − 1) g 1 γ ′ 3 + γ ′ 2 3 ] n +3 G 2( n − 1) 1 g − 4 3 , (3.25) g 4 6 = 0 : K 234 ≡  g 2 + ( n + 2) γ ′ 3 γ ′ 4 − γ ′ 2 3 − ( n + 1) γ ′ 2 4 + 2 n ( γ ′′ 3 − γ ′′ 4 ) 2 n 2  × g − 4 / 3 3 g 4 /n 4 . (3.26) In th e Duffing case n = 3, condition C 123 has already b een giv en [15], together with its in tegrate d form K 123 [13]. 4 In terpretation of the t w o conditions A ve ry simp le in terp retatio n can b e giv en for the tw o conditions. Indeed, the f orm of equation (1.1) is in v arian t under the sim ultaneous change of dep enden t and in d ep enden t v ariables u ( x ) → U ( X ) : u = α ( x ) U, X = ξ ( x ) , (4.1) where α and ξ are t w o arbitrary ga uge functions. The transformed ODE r eads U ′′ + 1 ξ ′  g 1 + 2 α ′ α + ξ ′′ ξ ′  U ′ + 1 ξ ′ 2  g 2 + g 1 α ′ α + α ′′ α  U + α n − 1 ξ ′ 2 g 3 U n + 1 αξ ′ 2 g 4 = 0 . (4.2) Let us adj ust the tw o functions α, ξ so as to mak e t wo of the four new co efficien ts as simple as p ossible. One of the thr ee p ossible w ays is to cancel the damp ing term b y the c hoice ξ ′ = α − 2 G − 1 1 , w h ic h reduces ODE (4.2 ) to U ′′ + α 4 G 2 1  g 2 + g 1 α ′ α + α ′′ α  U + α n +3 G 2 1 g 3 U n + α 3 G 2 1 g 4 = 0 . (4.3) P artial in tegrabilit y of the anharmonic oscillator 459 Canceling the new g 2 co efficien t amoun ts to solving the general linear second order ODE for α , whic h is p ossible (from the p oin t of view of P ainlev´ e, adopted here) but d oes not lead to an exp licit v alue of α . Th is reduced form is then U ′′ + h 3 U n + h 4 = 0 , (4.4) and this means that one can freely set g 1 = g 2 = 0 in (1.1) w ithout altering its glo bal prop erties (existence of first in tegrals, P ainlev´ e prop ert y , etc). Instead of that, one can mak e constan t either the reduced g 3 co efficien t iff ( n + 3) g 3 6 = 0 b y c ho osing α n +3 = G − 2 1 g − 1 3 , or the redu ced g 4 co efficien t iff g 4 6 = 0 b y the c h oice α = G − 2 / 3 1 g − 1 / 3 4 (let us recall that G 1 implicitly co n tains an arbitrary multiplicat iv e constant) . W e are th us led to th e redu ced form s g 4 6 = 0 : g 1 7→ 0 , g 4 7→ 1 , (4.5) g 2 7→  g 2 − 2 9 g 2 1 − 2 3 g ′ 1 + 1 9 g 1 γ ′ 4 + 1 9 γ ′ 2 4 − 1 3 γ ′′ 4  × G − 8 / 3 1 g − 4 / 3 4 , g 3 7→ g 3 G − 2 n/ 3 1 g − ( n +3) / 3 4 , n 6 = − 3 : g 1 7→ 0 , g 3 7→ 1 , (4.6) g 2 7→ [ g 2 − 1 n + 3 (2 g ′ 1 + γ ′′ 3 ) + 1 ( n + 3) 2 ( − 2( n + 1) g 2 1 − ( n − 1) g 1 γ ′ 3 + γ ′ 2 3 )] G 2( n − 1) / ( n +3) 1 g − 4 / ( n +3) 3 , g 4 7→ g 4 G 2 n/ ( n +3) 1 g − 3 / ( n +3) 3 , n = − 3 , g 4 = 0 : g 1 7→ 0 , g 3 7→ g 3 G 2 1 , g 2 7→  g 2 + g 1 α ′ α + α ′′ α  α 4 G 2 1 7→ 0 . (4.7) Then the interpretation is obvio us: an y reduced co efficien t distin ct fr om 0 or 1 is the r.h.s. of one of the integ rated conditions (3.23)–(3.26). Conv ersely , an y in tegrated condition is one of the remaining co efficien ts w hen tw o co efficien ts h a v e b een made constant b y a c hoice of gauge. F or instance, K 234 is the reduced g 2 co efficien t associated to reduced co efficien ts g 3 and g 4 equal to unit y . This can also b e seen in a more elementa ry w a y . In a gauge ( α, ξ ) such that g 1 = 0 , g ′ 3 g ′ 4 = 0, an exp ression for th e first inte gral is g 1 = 0 , g ′ 3 g ′ 4 = 0 : I 0 = u ′ 2 + 2 g 3 Z u 0 u n d u + g 2 u 2 + 2 g 4 u, (4.8) and, fr om the relation I ′ 0 − 2 u ′ E ≡ 2 g ′ 3 Z u 0 u n d u + g ′ 2 u 2 + 2 g ′ 4 u, (4.9) one d educes that th e t wo other co efficien ts g 2 and g 3 or g 4 m u st b e constant. The Hamiltonian (2.4) is a fir st integ ral if and only if g 1 = 0 and all other g i ’s are constan t. 460 Rob ert Conte 5 P ainlev ´ e analysis P ainlev´ e set u p the problem of fi nding n onlinear different ial equations able to defin e func- tions, ju st lik e the fi rst order ell iptic equation u ′ 2 = 4 u 3 − g 2 u − g 3 , ( g 2 , g 3 ) complex constan ts , (5.1) defines the elliptic function of W eierstrass ℘ ( x, g 2 , g 3 ), a doubly p erio dic function whic h includes as particular cases the w ell kn o wn trigonometric and hyp er b olic fun ctions. F o r a tutorial in tro duction, see the b o oks [21, 7]. A by-pro d uct of this quest for n ew fun ctions has b een the construction of exhaustiv e lists of nonlinear differential equations, the general solution of whic h can b e m ad e single- v alued (in more tec hnical term s , without mo v able critical singularities, this is the so-called Painlev´ e pr op erty (PP)), wh ic h imp lies that their general solution is known in closed form. In p articular, the list of second order first degree algebraic equations, i.e. u ′′ = F ( u ′ , u, x ) , (5.2) with F rational in u ′ , algebraic in u , analytic in x , wh ic h possess the PP has b een estab- lished by P ainlev´ e and Gam bier [19]. These classical resu lts apply to our problem only for th ose v alues of n for whic h Eq. (1.1), maybe after a monomial c hange of the dep endent v ariable u = U k , k ∈ R , b elongs to th e class (5. 2). These v alues, wh ic h in clude at least all the integers, are deter- mined b elo w. Th en, the wa y those classical results can b e applied is tw ofold. 1. Require the PP for our equation or its tr ansform under u = U k . 2. Restricting to the v alues of ( n, g i ) f or whic h the first in tegral (3.1) exists, c hec k th at the t wo conditions for the existence of this first in tegral im p ly the identica l satisfac- tion of the necessary cond ition that Eq. (5.2 ) ha v e no mo v able logarithmic branch p oin ts. Indeed, this is a classical result of P oincar ´ e that the mo v able singularities (i.e. those w h ic h dep end on the initial conditions) of fir s t order algebraic ODEs can only b e algebraic, i.e. u ∼ u 0 ( x − x 0 ) p , and never log arithmic, i.e. with some Log( x − x 0 ) term. Let us do that without to o many tec hnical considerations. The ab o ve men tioned n ecessary condition that Eq. (5.2) hav e no mo v able logarith- mic b ranc h p oint s ca n only b e computed after p erform ing the follo wing steps (for the unabribged p ro cedure, see [8, section 6.6]). Step 1 . F or eac h family of movable singularities u = χ p + ∞ X j =0 u j χ j , u 0 6 = 0 , χ ′ = 1 , (5.3) determine the le ading b ehaviour ( p, u 0 ). This is ac hiev ed b y balancing the highest deriv a- tiv e u ′′ with a nonlinear term. Therefore, there exist t wo leading b eha viours, denoted “family g 3 ” (balancing of u ′′ and g 3 u n ) and “family g 4 ” (balancing of u ′′ and g 4 ) ( g 3 ) : p = − 2 n − 1 , u 0 =  − 2 n + 1 ( n − 1) 2 g 3  1 n − 1 , n 6 = − 1 , (5.4) ( g 4 ) : p = 2 , u 0 = − 1 2 g 4 , g 4 6 = 0 . (5.5) P artial in tegrabilit y of the anharmonic oscillator 461 Step 2 . F o r eac h family , compu te the F uc hs ind ices, i.e. the r oots i of the indicial equation of the linear equation obtained b y linearizing (1.1) near its leading b eha viour u ∼ u 0 χ p , and require eve ry F uc hs index to b e integ er. This linearized equation is ( g 3 ) :  d 2 d x 2 + n g 3 ( u 0 χ 2 ) n − 1  v = 0 , (5.6) ( g 4 ) :  d 2 d x 2 + 0  v = 0 , (5.7) and the F uc hs indices are obtained b y requirin g the solution v = v 0 χ p + i (1 + O ( χ )), ( g 3 ) : ( i + 1)  i − 2 − 4 n − 1  = 0 , (5.8) ( g 4 ) : ( i + 1)( i + 2) = 0 . (5 .9) The d iophan tine condition that i = 2 + 4 / ( n − 1) b e integ er has a coun table num b er of solutions sin ce we h a v e not y et pu t restrictions on n . Step 3 . F or eac h family , co mpute all the necessary conditions for the abs ence of m ov- able logarithms (in short, no-log conditions), w hic h m ight o ccur when one computes the successiv e co efficien ts u j of (5.3). One can c hec k that the family g 4 can never generate suc h no-log conditions. Th ese conditions n eed not b e compu ted on th e original equatio n (1.1), they can b e computed on an y algebraic transform if this prov es more conv enien t (indeed, mov able logarithms are not affected b y an algebraic transform on u ), suc h as u = U k : k U U ′′ + k ( k − 1) U ′ 2 + k g 1 U U ′ + g 2 U 2 + g 3 U 2+( n − 1) k + g 4 U 2 − k = 0 . (5.10) The transformed p o w er s p are p 3 = − 2 / (( n − 1) k ) , p 4 = 2 /k , and th e F uc hs ind ices are unc hanged. The computatio n of the no-log conditions is imp ossible unless there exists a k making all the p o wers of U in (5.10) at least r atio nal. In order to av oid the tec hn ical complications of dealing with rational v alues of the leading exponent p , we restrict to those v alues of n for wh ic h there exists a k making 2 + ( n − 1) k and , if g 4 is nonzero, 2 − k integ er. Th e useful tr an s forms are u = v 2 / ( n − 1) : 2 n − 1 v v ′′ − 2 n − 3 ( n − 1) 2 v ′ 2 + 2 n − 1 g 1 v v ′ + g 2 v 2 + g 3 v 4 + g 4 v 2 − 2 / ( n − 1) = 0 , (5.11) u = w 1 / ( n − 1) : 1 n − 1 ww ′′ − n − 2 ( n − 1) 2 w ′ 2 + 1 n − 1 g 1 ww ′ + g 2 w 2 + g 3 w 3 + g 4 w 2 − 1 / ( n − 1) = 0 , (5.12) u = V − 2 : − 2 V V ′′ + 6 V ′ 2 − 2 g 1 V V ′ + g 2 V 2 + g 3 V 4 − 2 n + g 4 V 4 = 0 , (5.13) u = W − 1 : − W W ′′ + 2 W ′ 2 − g 1 W W ′ + g 2 W 2 + g 3 W 3 − n + g 4 W 3 = 0 , (5.14) 462 Rob ert Conte whic h are p olynomial if and only if (5.11) : g 4 = 0 or ( g 4 6 = 0 and 2 n − 1 ∈ Z ) , (5.15) (5.12) : g 4 = 0 or ( g 4 6 = 0 and 1 n − 1 ∈ Z ) , (5.16) (5.13) : 2 n ∈ Z , (5.17) (5.14) : n ∈ Z . (5.18) The original ODE (1.1) is iden tical to (5.11) for n = 3 and to (5.12) for n = 2. T o su mmarize, let u s compu te the no-log condition Q i = 0 on the ODE for v (5.1 1). Unfortunately , one do es not kno w ho w to obtain the d ep endence of Q i on n , sin ce n m u st first b e giv en a n u merical v alue b efore Q i is computed; this makes u neasy the comparison with conditions (3.19)–(3.2 0 ), which dep end on n . T o fix the ideas, a list of useful v alues of ( n, i ) is display ed in T able 1. T able 1. V alues o f ( i, n ) for i in teger ∈ [ − 4 , 10]. 2 + 4 n − 1 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 n 1 /3 1/5 0 -1/3 -1 -3 ∞ 5 3 7/3 2 9/5 5/3 11/7 3/2 The computation of Q i for p ositiv e v alues of i is classical [19, 2]. Denoting for shortness C 1 = C 123 , C 2 = C 134 , one finds the follo wing expressions Q i for the in dicated v alues of ( n, g 4 ), ( − 3 , 0) : Q 1 = C 1 , (5.19) (5 , 0) : Q 3 = C 1 , (5.20) (3 , g 4 ) : Q 4 = ± 1 864 [ − 2 g 3 ] − 1 / 2 [( γ ′ 3 − g 1 ) C 1 − C ′ 1 ] + 1 6 C 2 , (5.21) ( 7 3 , 0) : Q 5 = ( .g 2 1 + .g 2 + .g ′ 1 + .g 1 γ ′ 3 + .γ ′ 2 3 ) C 1 + ( .g 1 + .γ ′ 3 ) C ′ 1 + .C ′′ 1 , (5.22) (2 , g 4 ) : Q 6 = .C 1 + .C 2 1 + .C ′ 1 + .C ′′ 1 + .C ′′′ 1 + .C 2 + .C ′ 2 , (5.23) ( 9 5 , 0) : Q 7 = .C 1 + . . . + .C (4) 1 , (5.24) ( 5 3 , 0) : Q 8 = .C 1 + . . . + .C (5) 1 . (5.25) where d ots stand for rational n umb ers when i = 5 and p ol ynomials of g 1 , g 2 , γ 3 , γ 4 when i > 5. Similar relations hav e b een c hec k ed for i = 9 and i = 10 (Th omas-F ermi case) but are not repro duced here. Condition Q 4 = 0 con tains a ± sign arising from the tw o p ossible c hoices for v 0 and is equiv alent to the t wo conditions ( γ ′ 3 − g 1 ) C 123 − C ′ 123 = 0 , C 134 = 0 . (5.26) W e therefore c hec k the prop ert y that eac h Q i is ind eed a d ifferen tial consequence of the tw o co nditions C 123 = 0 , C 134 = 0 for the existence of a fir st in tegral (3.1) ∀ i ∈ N , ∀ g i : ( C 1 = 0 , C 2 = 0) ⇒ ( Q i = 0) . (5.27) P artial in tegrabilit y of the anharmonic oscillator 463 F or n egat iv e [18, 6] v alues of th e F uc hs index i , the results [6] are the follo wing: the family g 4 nev er generates an y no-log condition, and , for the family g 3 , a no-log condition arises from the F uchs index − 1, and this condition is a differen tial consequence of co ndi- tions (3.19)–(3.2 0 ), at least for the examples hand led ( n, r, g 4 ) = (1 / 5 , − 3 , 0) , (1 / 3 , − 4 , 0). This is also an exper im ental v erification of ∀ i ∈ Z , ∀ g i : ( C 1 = 0 , C 2 = 0) ⇒ ( Q − 1 = 0) (5.28) and th is relation cann ot b e rev ersed, as prov en b y P ainlev ´ e and Gam b ier. F or instance, in the case of the Duffing oscillator ( n, i, g 4 ) = (3 , 4 , g 4 ), condition Q 4 = 0 implies the reducibilit y of v to the second P ainlev ´ e transcendent whereas the stronger conditions C 1 = 0 , C 2 = 0 imply the reducibilit y of v to an elli ptic function. R emark . When one in cludes th e con tribution of the S c hw arzian in the defin ition of the gradien t of the expansion v ariable χ , as d on e in the in v arian t Pai nlev ´ e analysis [4], χ ′ = 1 + S 2 χ 2 , (5.29) all the computed no-log conditions Q i = 0, equatio ns (5.19)–(5.2 5 ), are indep endent of this Sc h warzian S , as opp osed e.g. to the Lorenz model [9]. This certainly ind icates some hierarc hy b etw een the lev el of nonintegrabilit y of these t w o dynamical systems. R emark . F or some small v alues of | i | , there is equ iv alence b et w een the no-log condi- tion an d (3.19)–(3.20). T his nongeneric situation o ccur s only for th e follo w ing v alues of ( n, i, g 4 ), ( − 3 , 1 , 0), i.e. th e Ermako v-Pinney equatio n [12, 28], (5 , 3 , 0), i.e. an equatio n considered b y L an e and Emden [24, 11], Chand rasekhar [3] and Logan [26, p. 52], (1 / 5 , − 3 , 0), an equation whic h could deserve more stud y . 6 Conclusion This w ork generalizes all p revious r esu lts on the p artial integrabilit y of the anharmonic oscillato r. It giv es a natural in terpretation of the t wo conditions for th e existence of a particular firs t int egral, in terms of reduced co efficien ts. Finally , th is sy s tem is an excellen t example to study seve ral features of Painlev ´ e analysis. A go o d, r ecen t bibliograph y can b e foun d in R ef. [20]. Ac kno wledgmen ts The author wish es to thank M. 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