Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru Abstract Ths paper deals wth three strateges of ntegraton of an n-th order ordnary dfferental equaton, whch admts the r-dmensonal Le algebra of pont symmetres. These strateges were proposed by Le but at present they are not well known. The frst and second ntegraton strateges are based on the followng man dea: to start from an n-th order dfferental equaton wth r symmetres and try to reduce t to an n -th order dfferental equaton wth r symmetres. Whether ths s possble or not depends on the structure of the Le algebra of symmetres. These two approaches use the normal forms of operators n the space of varables frst or n the space of frst ntegrals second. A dfferent way of lookng at the problem s based on the usng of dfferental nvarants of a gven Le algebra.. Introducton The experence of an ordnary dfferental equaton ODE wth one symmetry whch could be reduced n order by one and of a second order dfferental equaton wth two symmetres whch could be solved may lead us to the followng queston: Is t possble to reduce a dfferental equaton wth r symmetres n order by r? In full generalty, the answer s no. Ths paper deals wth three ntegraton strateges whch are based on the group analsys of an n-th order ordnary dfferental equaton ODE-n, n> wth r symmetres r >. These strateges were proposed by S. Le see [ ] but at present they are not well known. We studed the connecton between the structure of a Le algebra of pont symmetres and the ntegrablty condtons of a dfferental equaton. We refer readers to the lterature where these approaches are descrbed see [3], [6], [7]. Suppose we have an n-th order ordnary dfferental equaton ODE-n, n> y n = ω x, y, y,...,y n,. whch admts r pont symmetres X,X,...,X r. It s well known due to Le that r n +4. Def nton. The nfntesmal generator X = ξx, y x + ηx, y. y Present work was partally fnanced by RFBR, the grant 96 0 0997.
Group Analyss of Ordnary Dfferental Equatons of the Order n> 65 s called a pont symmetry of ODE-n. f X n ω x, y, y,...,y n η n x, y, y,...,y n mod y n = ω.3 holds; here, X n = ξx, y x + ηx, y y + η x, y, y y + +η n x, y, y,...,y n y n.4 s an extenson prolongaton X up to the n-th dervatve. Consder the dfferental operator A = x + y + + ω x, y, y,...,y n..5 y n It s not dffcult to see that A can formally be wrtten as A d dx mod yn = ω. Proposton. Dfferental equaton. admts the nfntesmal generator X = ξx, y x + ηx, y y ff [Xn,A]= Aξx, ya holds. Concept of proof. Let ϕ x, y, y,..., y n,=,n be the set of functonally ndependent frst ntegrals of ODE-n.; then {ϕ } n = are functonally ndependent solutons of the partal dfferental equaton Aϕ =0..6 It s easy to show that ODE-n. admts the nfntesmal generator. ff X n ϕ s a frst ntegral of ODE-n. for all =,n. So, on the one hand, we have that the partal dfferental equatons.6 and [ ] X n,a ϕ =0.7 are equvalent ff X s a symmetry of.. On the other hand, we have that.6 and.7 are equvalent ff [ ] X n,a = λ x, y, y,...,y n A.8 holds. Comparng the coeffcent of x on two sdes of.8 yelds λ = Aξx, y
66 L. Berkovch and S. Popov. Frst ntegraton strategy: normal forms of generators n the space of varables Take one of the generators, say, X and transform t to ts normal form X = s,.e., ntroduce new coordnates t ndependent and s dependent,where the functons tx, y, sx, y satsfy the equatons X t =0,X s =. Ths procedure allows us to transform the dfferental equaton. nto s n =Ω t, s,...,s n,. whch, n fact, s a dfferental equaton of order n we take s as a new dependent varable. Now we nterest n the followng queston: Does. really nhert r symmetres from., whch are gven by.? Y = X n ηt, s s.. The next theorem answers ths queston. Theorem. The nfntesmal generators Y = X n ODE-n-. f and only f ηt, s s are the symmetres of [X,X ]=λ X, λ = const, =,r,.3 hold. So, f we want to follow ths frst ntegraton strategy for a gven algebra of generators, we should choose a generator X at the frst step as a lnear combnaton of the gven bass, for whch we can fnd as many generators X a satsfyng.3 as possble; choose Y a and try to do everythng agan. At each step, we can reduce the order of a gven dfferental equaton by one. Example. The thrd-order ordnary dfferental equaton 4y y =8yy y 5y 3 admts the symmetres X = x ; X = x x y y ; X 3 = y y + y + y. Remark. All examples presented n ths paper only llustrate how one can use these ntegraton strateges. Note the relatons [X,X ]=X, [X,X 3 ]=0. Transform X to ts normal form by ntroducng new coordnates: t = y; s = x. Now we have the ODE- s = 3s s + 8ts s +5s 4t s wth symmetres Y = s s + s s ; Y 3 = t t s s ; [Y,Y 3 ]=0.
Group Analyss of Ordnary Dfferental Equatons of the Order n> 67 Transform Y to ts normal form: v =logt; u =logs u =u + u + 9, dy x = c y / + c 3. 8 cos 3 4 log c y 3. Second ntegraton strategy: the normal form of a generator n the space of frst ntegrals We begn wth the assumptons: a r = n; b X, =,n, act transtvely n the space of frst ntegrals,.e., there s no lnear dependence between X n, =,n,and A. We ll try to answer the followng queston: Does a soluton to the system of equatons ξ ϕ =, 3. X n ϕ = x + η +...+ ηn y y n X n ϕ = ξ x + η +...+ ηn y y n ϕ =0, =,n, 3. Aϕ = x + y +...+ ω y n ϕ =0, 3.3 exst? Asystemofnhomogeneous lnear partal dfferental equatons n n + varables x,y,..., y n 3. 3.3 has a soluton f all commutators between X n, =,n,and A are lnear combnatons of the same operators. It s easy to check that these ntegrablty condtons are fulflled ff X, =,n, generate an n -dmensonal Le subalgebra n the gven Le algebra of pont symmetres. Let ϕ be a soluton to system 3. 3.3, then [ X n,x n ] ϕ = X n X n ϕ X n X n ϕ =0 3.4 necessarly holds. On the other hand, we have [ X n,x n ] ϕ = CX n ϕ+cx k n k ϕ =C,,k =,n. 3.5 3.4 and 3.5 do not contradct each other f and only f C =0, =,n. 3.6 All precedng reasonngs lead us to the necessary condton of exstence of the functon ϕ. Ths condton s also suffcent. Now we prove t. Let u const be a soluton to system 3. 3.3, then we have [ X n,x n ] u = X n X n u = X n X n u =0, =,n, X n X n u =
68 L. Berkovch and S. Popov that s, X n u s a nonzero soluton to system 3. 3.3. Hence, X n u = fu. It s du not dffcult to check that the functon s a soluton to system 3. 3.3. fu Suppose that the ntegrablty condtons for system 3. 3.3 are fulflled. Now we consder ths system as a system of lnear algebrac equatons n ϕ x, ϕ y,..., ϕ. We y n can solve ths system usng Cramer s rule: ξ η η... η n η η ξ η η... η n... η n =. ϕ 0 η η... η n...... 0; ξ n η n η n... η n n x = ; y y 0 η n η n... η n n... ω 0 y y... ω ϕ y = ;...; ξ n 0 η n... η n n 0 y... ω ξ η... η n ξ 0 η... η n = n ϕ y ξ η η... ξ η η... 0....... ξ n η n η n... 0 y y... 0. The dfferental form dx dy dy... dy n ξ η η... η n dϕ = ξ n η n η n... η n n y y... ω s a dfferental of the soluton ϕ to system 3. 3.3. Theorem. Suppose pont symmetres X, =,n, act transtvely n the space of frst ntegrals; then there exsts a soluton to the system X n ϕ =; X n ϕ =0, =,n; Aϕ =0f and only f X, =,n, generate an n -dmensonal deal n the gven Le algebra of pont symmetres. Ths soluton s as follows: dx dy dy... dy n ξ η η... η n ξ n η n η n... η n n y y... ω ϕ = ξ η η... η n. 3.7 ξ η η... η n ξ n η n η n... η n n y y... ω
Group Analyss of Ordnary Dfferental Equatons of the Order n> 69 Now we can use ϕ x, y, y,...,y n nstead of y n as a new varable. In new varables, we have y n = y n x, y, y,...,y n ; ϕ, 3.8 X n = ξ x + η y + + ηn, =,n, 3.9 y n A = x + y + + yn x, y, y,...,y n ; ϕ. 3.0 y n System 3.8 3.0 s exactly what we want to acheve. Now we can establsh an teratve procedure. Example. y y =3y. Ths equaton admts the 3-dmensonal Le algebra of pont symmetres wth the bass. X = y ; X = x x y y ; X 3 = x and commutator relatons: [X,X ]=0, [X,X 3 ]=0, [X,X 3 ]= X 3. The gven ODE-3 s equvalent to the equaton {y, x} = 0, where {y, x} =/ y s Schwarz s dervatve. = 3/4 y 0 0 0 x 0 y y 0 0 0 y y 3y y = y / 0, ϕ = dx dy dy dy x 0 y y 0 0 0 y y 3y y = y y. Now we have the ODE-: y = y ϕ, whch admts the generators X = x x y ; = x 0 y 0 0 y y y ϕ y = y ϕ exp ϕ. X 3 = x ; = y. ϕ = dx dy dy 0 0 y y y ϕ =log y ϕ y, A general soluton of the dfferental equaton s gven by the next functon: y = ax + b cx + d.
70 L. Berkovch and S. Popov 4. Thrd ntegraton strategy: df ferental nvarants Defnton. Dfferental nvarants of order k DI-k are functons ψ x, y, y,..., y k, ψ 0, y k that are nvarant under the acton of X,...,X r, that s, satsfy r equatons =,r: X k ψ = ξ x, y x + η x, y y + + ηk x, y, y,...,y k y k ψ =0. 4. How can one fnd dfferental nvarants? To do t, we must know two lowest order nvarants ϕ and ψ. Theorem 3. If ψ x, y, y,...,y l and ϕ x, y, y,...,y s l s are two lowest order dfferental nvarants, then s r ; Lst of all functonally ndependent dfferental nvarants s gven by the followng sequence: ψ, ϕ, dϕ dψ,..., dn ϕ dψ n,... Let X,=,r, be symmetres of., then we have the lst of functonally ndependent DI up to the n-th order: ψ, ϕ, dϕ dψ,..., dn s ϕ dψ n s. Express all dervatves y k, k s, n terms ψ, ϕ, and dervatves ϕ m begnnng wth the hghest order. That wll gve ODE-n s: H ψ, ϕ, ϕ,..., dn ϕ dψ n s =0. Unfortunately, ths equaton does not nhert any group nformaton from.. If t s possble to solve 4., then we have ODE-s ϕ x, y, y,...,y s = f ψx,y,...,y l 4. wth r symmetres. Example 3. ODE-3 yy y = y y + yy admts the -dmensonal Le algebra of pont symmetres X = x ; X = x x y y. DI-0 does not exst. DI-: ψ =, DI-: ϕ =. Now we can fnd DI-3: y y3 dϕ dψ = ϕ = y 3y y yyy y.
Group Analyss of Ordnary Dfferental Equatons of the Order n> 7 Express y,y,y n terms of ψ, ϕ, ϕ. Ths procedure leads us to ODE-: ϕ = ϕ ψ. Hence, we obtan ϕ = Cψ or y = Cyy, that s, we obtan ODE- wth symmetres, whch can be solved as dy x = Cy + a + b. In concluson, we have to note that we have dscussed only some smple strateges. A dfferent way of lookng at the problem s descrbed n [9] and based on usng both pont and nonpont symmetres. References [] Le S., Klassfcaton und Integraton von gewöhnlchen Dfferental glechungen zwschen x, y, de ene Gruppe von Transformatonen gestatten, Math. Annalen, 888, 3, 3 8; Gesamlte Abhundungen, V.5, B.G. Teubner, Lepzg, 94, 40 30. [] Le S., Vorlesungen über Dfferentalglechungen mt Bekannten Infntesmalen Transformatonen, B.G. Teubner, Lepzg, 89. [3] Chebotarev M.G., Theory of Le Groups, M.-L., Gostekhzdat, 940 n Russan. [4] Pontrjagn L.S., Contnuous Groups, M.-L., Gostekhzdat, 954 n Russan. [5] Ovsyannkov L.V., Group Analyss of Dfferental Equatons, Academc Press, N.Y., 98. [6] Stephan H., Dfferental Equatons. Ther solutons usng symmetres, Edt. M. Maccallum, Cambrdge, Cambrdge Unv. Press, 989. [7] Olver P.J., Applcatons of Le Groups to Dfferental Equatons second edton, Sprnger-Verlag, 993. [8] Ibragmov N.H., Group analyss of ordnary dfferental equatons and prncple of nvarance n mathematcal physcs, Uspekh Mat. Nauk, 99, V.47, N 4, 83 44 n Russan. [9] Berkovch L.M., The method of an exact lnearzaton of n-order ordnary dfferental equatons, J. Nonln. Math. Phys., 996, V.3, N 3 4, 34 350. [0] Berkovch L.M. and Popov S.Y., Group analyss of the ordnary dfferental equatons of the order greater than two, Vestnk SamGU, 996, N, 8 n Russan.