A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

H. Kötz Institut für Mathematische Physik, Technische Universität Braunschweig, Braunschweig, FRG

Z. Naturforsch. 48a, 535-550 (1993); received December 24, 1992

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group G are an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra 'S of the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of 'S vary in case of 3 < s < r, depending on the solvability of 'S. If the r-dimensional Lie algebra 'S of the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of 'S one has to determine the optimal subsystems of semisimple subalgebras of 'S in order to construct the full optimal systems of s-dimensional subalgebras of 'S with 3 < s

Key words: Group theory; Statistical plasma physics.

Introduction connected subset M of the Euclidean space JR" + m which leave the system & invariant and map (graphs Similarity analysis is a powerful tool for obtaining of) solutions to (graphs of)solutions. The required the- exact similarity solutions of (nonlinear) partial differ- ory and description of the techniques to determine this ential equations (PDEs). In this paper we assume that connected local transformation group G can be found the reader is familiar with most of the required theory in the mentioned books (see especially Ovsiannikov of these applications of Lie groups to PDEs, which are [1] and Olver [4]). systematically and well described in the textbooks of In what follows, we assume that G is a known /--pa- Ovsiannikov [1], Ibragimov [2], Bluman and Cole [3], rameter symmetry group admitted by 2F, where r is a Olver [4], Bluman and Kumei [5], and Stephani [6]. natural number. A similarity solution of the s-param- The inclusion of integrodifferential equations (IPDEs) eter subgroup H of G (H-invariant solution of IF) is a in the Lie group method was carried out by some solution of whose graph is invariant relative to the authors, especially Taranov [7], Marsden [8], Tajiri [9], elements of H and may be obtained, under additional and Roberts [10], and we refer the reader to the men- regularity assumptions on the action of H on M, by tioned papers for detailed information in this case. solving a reduced system of PDEs (IPDEs) with n — s As mentioned in my previous paper [11] (further independent and m dependent variables (see e.g. Olver referred as I), the main aims of the similarity analysis [4]). It is not usually feasible to list all possible similar- of a given system J5" of PDEs (or IPDEs) in n indepen- ity solutions of all the s-parameter subgroups, since dent and m dependent real variables are to calculate the number of these subgroups is almost always in- and to classify the similarity solutions of First one finite, and to each s-parameter subgroup there will has to determine the maximal Lie point symmetry correspond a family of group-invariant solutions. group G admitted by This group G consists of all Therefore, one desires to minimize the search for sim- the real valued transformations acting on an open and ilarity solutions by listing the essentially different group-invariant solutions, which leads to the concept Reprint requests to Dipl.-Phys. H. Kötz, Institut für Mathe- of an optimal system of similarity solutions from matische Physik, Mendelssohnstr. 3, W-3300 Braunschweig. which every other such solution can be derived. The

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Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. 536 H. Kotz • Classification of Similarity Solutions of Nonlinear PDEs. II 536 concept is based on the following result (see Olver [4]): real (complex) finite-dimensional Lie algebra under For any g e G with g $ H a H-invariant solution is conjugation and applied it to some fundamental Lie transformed to a g • H • g~ ^invariant solution, where algebras of physics. Especially for the Poincare alge- the two subgroups H and g • H• g'1 are called conju- bra (see [13] and also [16]) these authors gave a com- gate. If H and H' denote two s-parameter subgroups plete classification of its subalgebras up to dimension of G, then a H-invariant solution and a similarity five relative to the group of complex valued inner solution of the subgroup H' are called essentially dif- automorphisms of the algebra. ferent if H and H' are not conjugate, i.e. there exists no Recently, Coggeshall and Meyer-ter-Vehn [17] de- element g e G such that H' = g • H • Thus, the termined the 14-parameter Lie point symmetry group classification problem for the similarity solutions of admitted by the three-dimensional, one-temperature s-parameter subgroups of G leads to that of separating hydrodynamic equations, including conduction and a the collection of all s-parameter subgroups of G into thermal source. (Related systems were investigated conjugacy classes of subgroups. A minimal list of non- by Ovsiannikov [1], Coggeshall and Axford [18], conjugate s-parameter subgroups (one from every Coggeshall [19]). For a seven-parameter solvable sub- equivalence class) with 1 < s < r is said to be an opti- group corresponding to two-dimensional axisymmet- mal system Of for the symmetry group G. In order to ric geometry they calculated optimal systems of one- classify the similarity solutions of one is interested and two-dimensional subalgebras of the Lie algebra of in optimal systems Of with 1 < s < min (r, n). this subgroup using the techniques described by The problem of finding an optimal system Of for G Ovsiannikov [1] and Galas [12]. is equal to that of finding an optimal system Of of In I a survey of these known techniques for the s-dimensional subalgebras for the r-dimensional real construction of optimal subsystems of solvable sub- Lie algebra ^ of the symmetry group G, where the Lie algebras for the real Lie algebra ^ of an arbitrary algebra of G is identified with the isomorphic Lie alge- finite-dimensional Lie point symmetry group is given. bra ^ of the Killing vector fields on M whose flows Furthermore, in I a modified method, which is based coincide with the actions of the one-parameter sub- on the properties of the bilinear invariant forms (rela- groups of G on M (see e.g. Olver [4]). Here, a list of tive to the local Lie group of the inner automorphisms s-dimensional subalgebra forms an optimal system of the Lie algebra ^ or relative to the groups of the Of if every s-dimensional subalgebra X of ^ is con- inner automorphisms of its subalgebras), is presented, jugate to a unique member Jf of the list under some where the calculation of these invariant bilinear (sym- element of the adjoint representation, i.e. there is an metric) forms may be done using computer-algebra inner automorphism with g&G such programs, exspecially the REDUCE 3.2 programs that Ad (g)(Jt) =Jf. OPTSYS and BINV, which are also described in I. The A detailed description of most of the known tech- knowledge of the associated "linear" and "bilinear" niques for the construction of optimal subalgebraic invariants of an arbitrary vector in ^ is helpful during systems for the Lie algebra of infinitesimal symmetries the process of classifying the solvable and nonsolvable admitted by a system of PDEs can be found in the subalgebras of since these invariants cannot be textbooks of Ovsiannikov [1], Ibragimov [2], and Olver changed by the full adjoint action. The advantage of [4], where also a few examples are given. Following this modified technique is shown in I by applying it to Ovsiannikov [1] and Ibragimov [2], Galas [12] for- the nine-dimensional solvable Lie algebra of infinites- mulated an "algorithm" for the determination of opti- imal symmetries admitted by the two-dimensional mal systems of s-dimensional subalgebras of an arbi- non-stationary ideal magnetohydrodynamic equations. trary real Lie algebra ^ of infinitesimal symmetries for In case of a solvable r-dimensional real Lie algebra 1 < s < r, where the classification process varies, de- the optimal subsystems ©f of solvable s-dimen- pending on the solvability of & in case of 3 < s < r. sional subalgebras of ^ with 1 < s < r are full opti- For a nonsolvable Lie algebra ^ the problem of clas- mal systems Of, since any subalgebra of ^ is also sifying its s-dimensional subalgebras with 3 < s < r is solvable (see e.g. lemma 1 in I). Since the techniques generally harder (see below). for obtaining optimal subalgebraic systems for a solv- In a series of papers (cf. [13], [14], and [15]) Patera, able Lie algebra or optimal subsystems of solvable Sharp, Winternitz, and Zassenhaus developed another, subalgebras of an arbitrary real finite-dimensional Lie but related method of classifying the subalgebras of a algebra have been discussed in I, we restrict our atten- H. Kotz • Classification of Similarity Solutions of Nonlinear PDEs. II 537 tion in the present work to the problem of classifying a system of partial (integro-)differential equations, the 5-dimensional subalgebras of an arbitrary non- where seN and reN denote natural numbers. Here, solvable r-dimensional real Lie algebra ^ by means of we restrict most of our attention to the task of classi- full optimal subalgebraic systems @f with 3 < s < r. fying the s-dimensional subalgebras for a nonsolvable A description of the techniques for this classification Lie algebra ^ in case 3 < s < r, since the techniques process based on the investigations of Ibragimov [2] is for obtaining optimal subsystems of solvable subalge- given in Section 1. In addition to the optimal sub- bras of a nonsolvable Lie algebra ^ are summarized systems ÖF of 5-dimensional solvable subalgebras, in I. The following description of the available tech- one has to determine optimal subsystems ©f of the niques is based on the textbooks of Ovsiannikov [1] s-dimensional semisimple subalgebras of a Levi sub- and Ibragimov [2], which Galas [12] used to formulate algebra if of ^ and optimal subsystems 0f of s-di- an "algorithm" for the determination of optimal sub- mensional subalgebras with non-trivial Levi decom- algebraic systems without using the invariants of the positions. group of the inner automorphisms. These invariants The problem of determining optimal subsystems can play an important role during the construction of ©f for a semisimple Levi factor if of ^ leads to the optimal subalgebraic systems (see e.g. Sect. 1 in I). For classification theory of complex semisimple Lie alge- the proofs of the theorems stated below we refer the bras and their real forms, which is the great achieve- reader to the usual books on Lie groups and Lie alge- ment of the classical work of Cartan and Killing and bras, for example Jacobson [21], Sagle and Walde [22], is treated in most books on Lie algebras (see e.g. Varadarajan [20], or Hilgert and Neeb [23]. Varadarajan [20]. Since in course of similarity analysis In what follows, we denote the dimension of the real of PDEs (IPDEs) one frequently finds three-dimen- finite-dimensional Lie algebra ^ by d(^) := dim(^) sional real Levi subalgebras, we only investigate these = r e N and assume that a basis of & is given by cases. {»!,...,»,.}. Then any g in a neighbourhood of the In Sect. 2 the three-dimensional (3-D) Vlasov- identity element of the Lie point symmetry group G Maxwell equations (VMS) for a multi-species plasma with Lie algebra ^ may be written as in the non-relativistic case are regarded. This system g = exp(e1 px) •... • exp(er vr), (1) admits an eight-parameter real Lie point symmetry group G with a nonsolvable Lie algebra The group where • denotes the group multiplication in G and the G contains the three-dimensional Euclidean group coefficients £x,..., £r g R are called canonical coordi- E(3), which is the semidirect product of the rotation nates of ther second kind (see Hilgert and Neeb [23]). group SO (3, IR) and the group of space translations A full optical system Of of s-dimensional subalgebras T(3), the one-parameter group of time translation, of ^ with s < r is defined as the union of the represen- and one scaling group. Using the techniques described tatives of conjugate (or similar) algebra classes of given in I and in Sect. 1, we determine full optimal systems dimension s (one from every class), where two s-di- of s-dimensional subalgebras of the eight-dimensional mensional subalgebras X and Jf are called conjugate Lie algebra ^ for s = 1,..., 8. Here, the (bi-)linear in- if there exists an inner automorphism Ad(gf) e Int(^) variants of an arbitrary vector in the Lie algebra ^ (g g G) such that Ad {g)(X)=Jf. Here, the general (relative to the group of inner automorphisms of the inner automorphism in the local Lie group Int(^) Lie algebra are rather helpful to obtain optimal consisting of all inner automorphisms & -*• & may be subalgebraic systems, since with the aid of these in- given by variants it is possible to shorten the lengthy classifica- Ad(g = exp(e1 vt) •... • exp(ervr)) tion process in comparison with the usual techniques. = Ad(exp(e1»1) o ... o Ad(exp(gr»r)), (2)

where the linear mapping Ad(g = exp(sv)): & 1. Optimal Systems for Nonsolvable Lie Algebras maps the vector w e ^ to the Lie series oo g* ad p w In this section we describe the basic tools for the Ad (exp (£!>))( H>) = exp (£ ad (r)) (w) = £ T7 ( ( ))* ( ) k = o kl construction of full optimal systems of s-dimensional subalgebras for the real Lie algebra ^ of a given E2 = w + £ [w, r] + — [[m>, »],t>]+..., (3) r-parameter Lie point symmetry group G admitted by 538 H. Kotz • Classification of Similarity Solutions of Nonlinear PDEs. II 538 and the inner derivation ad(r) for a given oe^ is This maximal solvable ideal Mi^) (nilpotent ideal defined by .JV^S)) in y is called the radical (nilradical) of Then c= M^) and ^ ad(»)(w>) := [w, v] (4) Hence, the radical of a finite-dimensional solvable (see Ovsiannikov [1] and Olver [4]). Clearly, Of con- Lie algebra ^ is ^ itself, i.e. = and its derived sists only of the Lie algebra ^ itself. algebra := is a subalgebra of its nilradical Since all one- and two-dimensional Lie algebras are The radical of any Lie algebra is invariant solvable, the optimal subalgebraic systems Of and Of under all (inner) automorphisms of the Lie algebra for an arbitrary real finite-dimensional Lie algebra ^ (see e.g. Varadarajan [20]). consist only of solvable Lie subalgebras and can be Definition 1 (see e.g. Ibragimov [2]). A noncommuta- constructed using the techniques described e.g. in tive finite-dimensional Lie algebra , i.e. / 0, is Sect. 1 in I. Therefore 0f = &f holds for j = 1, 2. For said to be simple if it has no ideals different from the a solvable real r-dimensional Lie algebra any of its null algebra 0 and (0 consisting of the null vector and subalgebras is solvable (cf. lemma 1 in I) and an opti- $ being called the trivial ideals in A finite-dimen- mal subalgebraic system ©f with 1 < s < (r — 1) +1 sional Lie algebra $ is called semisimple if its radical may be obtained by the "method of expansion" (see reduces to zero, i.e. = 0. subsection 1.2 in I) of the representative members of a known optimal system Of, since, according to the Lie From theorem 1 and the above definition it follows theorem in I, every (s+ l)-dimensional solvable alge- that the factor algebra is semisimple for a bra contains an s-dimensional solvable subalgebra. finite-dimensional Lie algebra Since one- and two- This still holds when one looks for the solvable sub- dimensional Lie algebras are solvable, i.e. the radical algebras of an arbitrary real Lie algebra ^ in order to of such a Lie algebra is the algebra itself, there exists obtain optimal subsystems Öf of solvable subalgebras no semisimple Lie algebra ^ of dimension d(^) = 1 or for ^ (see e.g. Ibragimov [2] and I). d(^) = 2. In the course of similarity analysis of PDEs To classify subalgebras of dimensionality 2 < s < r (IPDEs) one frequently encounters three-dimensional in the case of a nonsolvable real r-dimensional semisimple subalgebras of the real Lie algebra ^ of Lie algebra one can sucessfully employ the Levi- infinitesimal symmetries (see e.g. Ibragimov [2] and Mal'cev theorem asserting the existence and unique- Ovsiannikov [1]). These semisimple real Lie algebras ness (up to conjugation) of the Levi decomposition of of dimensionality 3 must be isomorphic either to the ^ into a semidirect sum of its radical and a semisimple Lie algebra (3, IR) of the rotation group SO (3, IR) Levi factor (see e.g. Ibragimov [2]). Furthermore, the or to the Lie algebra (2, IR) of all real 2 x 2-ma- Mafcev-Harish-Chandra theorem gives an important trices with trace zero, where both algebras are real addition to the Levi-Mal'cev theorem such that most forms of the complex Lie algebra 5^(2, (C) (see e.g. subalgebras of a nonsolvable ^ can be completely Jacobson [21]). Any three-dimensional real Lie algebra classified by means of both theorems (see e.g. Ovsian- not containing two-dimensional subalgebras is iso- nikov [1] and Ibragimov [2]). In this paper, we restrict morphic to the Lie algebra of rotations Sf(9 (3, IR) with our attention to the task of classifying the s-dimen- the basis {ul, u2, u3} and the structure [ux , u2] = u3, see e sional subalgebras of a nonsolvable real r-dimensional [u2,u3] = u1, [«3,mJ = «2 ( -§- Ovsiannikov [1]). Lie algebra with a non-trivial Levi decomposition in The special linear algebra (2, IR) spanned by the case 3 < s < r. First we give some basic definitions three basis vectors w2, w3 has the structure and theorems, which are necessary to state the Levi- K, h>3] = 2w>!, [h>2, yv3] = — 2w>2, [»„ h>2] = h>3 and Mal'cev theorem and the Mal'cev-Harish-Chandra contains two-dimensional subalgebras (see Jacobson theorem and to proceed to the description of the tech- [21]). Both <9*0(3, IR) and ^(2, IR) are even simple niques available for the named classification problem. Lie algebras. Instead of the above definition one can use the following criterion of semisimplicity of a finite- Theorem 1 (see e.g. Ovsiannikov [1] and Varadarajan dimensional real Lie algebra (S. [20]). Among the solvable (nilpotent) ideals of a finite- dimensional Lie algebra there is a unique solvable Cartan's Criterion of Semisimplicity (see e.g. Ovsian- ideal ${<£) (nilpotent ideal .Ar ) of maximal dimen- nikov [1]). The r-dimensional real Lie algebra with sion, which contains any solvable (nilpotent) ideal ofS. basis {»!,..., vr} is semisimple if and only if the H. Kotz • Classification of Similarity Solutions of Nonlinear PDEs. II 539

Killing form & x & R, which maps the pair The Levi theorem shows that any finite-dimen- r r v e< sional Lie algebra decomposes in the semidirect sum (m, v) of the two vectors u= ^ Killing form by factor. If if is a Levi subalgebra of 3, then it is also a (1) ( ( a) the following theorem. Levi factor of the derived algebra <3 of 3 and 3 = (X) \0tif3\ <3\ ©s if is a Levi decomposition of<3 . Theorem 2 (see e.g. Ovsiannikov [1]). The radical 01 ($) of the finite-dimensional Lie algebra $ is a set of The Levi decomposition is trivial for semisimple elements ue^ for which K9 («, v) = 0 for any element v Lie algebras {ß C3) = 0) and solvable Lie algebras ( of the derived algebra <§{X) := 3], (0tif3) = 3). The uniqueness (up to conjugation) of a Levi decomposition for a finite-dimensional Lie alge- Using theorem 2, the REDUCE 3.2 program OPT- bra is proved by the Mal'cev theorem, which is a SYS (see I) determines the radical of the finite-dimen- corollary of the following Mal'cev-Harish-Chandra sional real Lie algebra ^ of infinitesimal symmetries. theorem. Definition 2 (see e.g. Ibragimov [2]). Let be a Lie algebra. If J is an ideal in and J is a subalgebra Mal'cev-Harish-Chandra Theorem (see e.g. Ovsian- (ideal) of <3 with J n /= 0 (and [J, f] = 0), then nikov [1] and Varadarajan [20]). Let & be a fi- the direct sum of the subspaces J and ß of the vector nite-dimensional real Lie algebra, M{r3) its radical, ( space y forms a Lie algebra, which is called the semi- and Jf( takes Jt into if. The classification of a semisimple Lie algebra is based on the following theorem. For the sake of simplicity we stated the above Structural Theorem (see e.g. Ovsiannikov [1] or Hilgert theorem in a weak form (see e.g. Ovsiannikov [1]). In and Neeb [23]). A finite-dimensional real Lie algebra <3 addition, we gave the strong form in parentheses <...) is semisimple if and only if there exist simple ideals (see Varadarajan [20]). of <8 such that Corollary 1 (see e.g. Jacobson [21]). Any semisimple <3 = <3 ©<#2®...®%. l subalgebra of a finite-dimensional real Lie algebra <3 ( The only ideals of <3 are the direct sums J = £ ^ w^th can be imbedded in a Levi factor of 3. /

Levi factor if' of Jf has to be conjugate to a semi- automorphism Ad(gf) e Int(^) given in (2), where < simple subalgebra in any Levi subalgebra of S. g = exp(£x • ... • exp(ervr) with real parameters

£r. The values of these parameters should be Mal'cev Theorem (see e.g. Jacobson [21] or Ibragimov choosen to achieve the maximum possible "simplifi- [2]). Let $ be a finite-dimensional real Lie algebra and cation" of the real coordinates ,..., ßr of w. This M (&) its radical. If there are two Levi decompositions permits the choice of the simplest representative of the <$ = £{<$) ®s ©s if' of

2. The Three-Dimensional Vlasov-Maxwell Equations plasma with a fixed and homogeneous ion background through a formal approach to the infinite sys- We consider three-dimensional motions of a colli- tem of PDEs for the moments of the electron distribu- sionless plasma of a species without a background or tion function. In 1985 Roberts [10] derived the full Lie external fields in the non-relativistic case {a e N). In point symmetry group admitted by the 1-D VMS in what follows, the particles of species a have charge q*, case of a one-species plasma with a background and mass m*, and the three-dimensional distribution func- a multi-species plasma with q^/m^ ^ qß/mß for a ^ ß tion /"* =f f (x*, vv*, t*), where x* = x*e1 + y* e2+ z* e3 (a, ß = 1,..., a), where also a homogeneous back- is the space vector, vv* denotes the velocity vector, ground was assumed. and t* is the time. The distribution functions ff If one looks at a plasma consisting of o > 1 species,

(oc = 1,..., a) for the a species, the selfconsistent elec- where qjma # qß/mß for any ß / a (a, ß = 1,..., o) tric field E* = £*(x*, t*), and the selfconsistent mag- holds, the real Lie algebra & of the full Lie point netic field B* = B* (x*, t*) satisfy the following three- symmetry group G = exp(^) admitted by (6) is of di- dimensional (3-D) Vlasov-Maxwell system (VMS): mension r = d(^) = dim(^) = 8 and may be spanned by the following infinitesimal generators written in terms of Cartesian coordinates: ^ + vv-Vf/« + ^[£ + vvxB]-V*/« = 0 at mx for any a = 1,o , 6 6 a a a E c 0X ' e? e? dt' Vfxß = -+ Z qa J w/^x, vv, t) dw, a a a a a y v5 — y — x ——(- ——3- — w* - + E x X 3 dx a y dw a E Vx E= I qx j f*{x, vv, t) d w , (6) a=l R3 a a a - EX + BY BX -—, dB y V;x£ = a E a BX a BY ~äT' a a a a a z z Ve • B = 0 v6= z X h w w* (- E —

z with the dimensionless variables ax a z 8w* aw a EX

vv* c a a a (7) x := vv :=—, t:= — t*. -EX + BR BX -—, c L z a EZ a BX a b 2 \q*\ L2C , y z dw a w a EY fr-= ^i ff (a = 1,...,

Table 1. The commutator table for the eight-dimensional Lie (d) Scale transformation: algebra : (x, vv, t, ff,..., f°, E, B) Vj V V »1 2 »3 ®4 »5 »6 1 »8 (x exp (e8), vv, r exp (£8), // exp (- 2 £8),... [Vi,Vj] ..., f{, exp (- 2 £g), £ exp (- e8), B exp (- e8)) »1 0 0 0 0 -»2 -®3 0 »1 »2 0 0 0 0 »1 0 ®3 "2 (£8 g R). 0 0 0 0 0 »3 »1 "2 »3 The full symmetry group G is generated by these »4 0 0 0 0 0 0 0 »4 one-parameter transformation groups. Roberts an- P5 P2 0 0 0 »7 -»6 0 nounced in [26] a manuscript on an investigation of »6 »3 0 -»1 0 0 »5 0 the Lie point symmetries admitted by the 3-D VMS r7 0 »3 -»2 0 »6 0 0 »8 -»1 -»2 -p3 -»4 0 0 0 0 and described verbally the structure of the symmetry group as given above. Three additional discrete symmetries of the 3-D VMS are given by

a Sl:(x,w,t,f1\...,f1 ,E,B) read. Obviously, the center 2Hfg) := [u e <3 \ [u, ®] = 0 V® g is the null algebra 0, and the derived algebra h+(x, -vv, -;,//,...,/*,£, -B), (1) := [3, is ^ (®1,...,®7). Here and in what (x, vv, t, fi,f", E, B) follows, Jf,...,«,) denotes the subalgebra Jf of <3 spanned by the basis vectors «!,...,«,. H-(-x, -vv.t,//,...,/^, -E,B), (8) The corresponding subgroups of symmetries of the : (x, vv, t, f1,..., f", £, B) Vlasov-Maxwell equations are: ' * ( x, vv, -f,//,...,-£, -B),

(a) Space translations (T(3)): 0 where S2 = S3 holds. For j = 1, 2, 3 the mapping Gy. (x, vv, r, f?,E,B) Sj ° Sj is the identity i- (x + £j ej, vv, t, ...,/*, £, B) So'- (x, vv, f, //,..., f", B) (ej g R; )= 1, 2, 3). (b) Time translation: r x v e G4: (x, vv, t, /*, £,ß) For given u = £ < k k ^ the inner derivation fc = 1 r H^(x,vv,r + £4,//,...£, B) (e4 g R). ad(n): maps w = £ ßj^e^ to ad(i/)(w) =

\->(Rx,R vv, t, ... ,/*, R£, KB)

of simultaneous rotations in the components of x, vv, £, ß with the 3 x 3 orthogonal matrix

1 ' COS £5 sin £5 ' COS £6 0 sin £6 /I 0 0

R •= — sin £5 COS £5 0 1 0 0 COS £7 sin £7 ° 1 0 0 1/ ^ —sin £6 0 COS E6I io — sin £7 COS £7

= Rs = Re =:R7

which depends on the three real parameters £,• (j = 5, 6, 7), and the one-parameter subgroups

Gy. (x, vv, f, //,..., /*, £, ß) i • (Ä, x, Rj vv, r,//,..., /*, R. £, R. B) (,' = 5, 6, 7). H. Kotz • Classification of Similarity Solutions of Nonlinear PDEs. II 544

Table 2. Adjoint representation of the Lie group G.

*J 1), V-> V A 9t Ad (gt)(vj)

01 = exP(ei ®i) »1 V2 "3 »4

92 = exp(e2 v2) "1 »2 "3 »4 93 = exp(e3 »3) »2 »3 »4 9 A - exp(e4 r4) v2 "3 »4 V 95 = exp(e5 ®5) »1 cos e5— v2 sin e5 l sin £5 + v2 cos e5 »3 "4 96 = exp(e6 v6) cos e6— ®3 sin e6 »2 "1 sin £6+ t>3 cos £6 »4 9i = exp(e7 r7) ®1 "2 cos e7 — »3 sin e7 "2 sin e7 + t>3 cos £7 "4 9s exp(e8 ®8) exp e8 v2 exp e8 i>3 exp £g "4 exp £8

vi ti6. Vi7 Va8 9i Ad(öf,) (Vj)

9i - exP(£i ®i) »5+^1 »2 »6+ßl »3 »7 »8 V £ 92 = exp(e2 v2) 5~ 2 »1 »6 P7+ £2 l>3 »8 -e2 «2

93 = exp(e3 r3) »5 ®6-«3®l r7- £3 v2 »8 -£3 "3

- 04 exp(e4 r4) »5 »6 »7 "8 -£4 t>4 95 - exp(e5 v5) »5 "6 cos £5 — r7 sin £5 v7 cos £5+ p6 sin £5 »8 96 = exp(e6 ®6) "7 sin e6+ v5 cos e6 »6 »7 COS £6— p5 sin £6 »8 07 = exp(e7 r7) »5 cos e7— v6 sin e7 "5 sin £7 + p6 cos £7 "7 »8 08 = exp(e8 t>8) »5 »6 «7 "8

adu := Qe Cjk) = can be computed as Lie series (cf. (3)) in the usual way and are listed in Table 2. The computation of the Killing form : (u, w) h-• «8 «5 a6 0 0 — a -»2 -«3 Kg(u, w) := tr(ad(a) ° ad(w)) and the Casimir poly- «5 «8 a7 0 0 ~a3 — a: nomial C9: u i—• C8). A Levi factor of the Lie alge- The maximal rank of the endomorphism ad(«): bra ^ is the semisimple (even simple) rotation algebra obtained when u ranges over the whole space is R) spanned by the basis {»5,®6,®7}. Thus, a max (rank (ad (a))) = 6. As stated in (2), the general ele- Levi decomposition of ^ is given by the semidirect U € <§ sum <§ = 9H )) r 1 1 8 8 dent scalar invariants of an arbitrary u = £ a,- Vj e 0 =1 = Ad(exp(e1r1)) ... ° Ad(exp(e8p8)) (relative to the group Int(^)) is J with real parameters ,...,e8, where the inner auto- = d(S?) - max (rank (ad (m))) morphisms

Ad (exp (EJ VJ)) := exp(e,. adty)): -

~~Table 3. (Bi-)linear invariants of a vector u in a subalgebra Jf of ^ (relative to Int(Jf)).~~

~~Case Subalgebra Jf d(Jt) /(Jf) Invariants of u~~

~~1 jf(vlt..;V 8) = $ 8 2 2 a\+~~

~~2 M«) = o 7 3 3 a4, a|+ a7, (OC 8 = 0) t*!, oc7 — a2 a6 + a3a5 a 3 M«) = o •KA'i, •• •, V4, v8) = &C&) 5 1 1 s («5 = a6 = a7 = 0)~~

~~4 M2(*) = 0 4 4 4 a1,a2,a3,a4 (a5=a6=a7=a8 = 0)~~

(see (12) in I). The general invariant bilinear symmetric definite bilinear form in the coordinates a5, a6, a7 of form u, 3#p \= {w (u) = 0} is also an ideal of ^ (see r = 8 fli *>):= X <*j $fk ßk Sect. 1 in I). ^ coincides with the radical of ^ j,k= 1 {we g\n2(w) = 0} as the intersection of the

r ideals a1 and a* 1 forms also an ideal. The abelian v e< an with w = X ßk k & d the real symmetric matrix ideal is the nilradical Jf (<$) of For these ideals k= 1 X the (bi-)linear invariants of an arbitrary element (

~~3/exx (m>) = 0} = forms an ideal in ^ w = X 0/ Vj := Ad(0 = exp(ex rx) •... • exp(8r rr))(w) (see Sect. 1 in I). Since a2 = with~~

0 and 0<2>, ») = a2= 0 (oc = a = a = 0), re- linear form, any one-dimensional subalgebra X{u) of 5 6 7 spectively. Hence, we concentrate on the four non- ^ belongs to one of the two nonintersecting algebra intersecting algebra classes classes given by 0 and 0,

~~B: oc8, = 0 and a> 0,~~

~~C: a8 ;=1 and a = 0,~~

~~D: a?,= 0 and a = 0.~~

2 (From the Casimir polynomial C& with C^ (U) = K9(U, U) = 4(-a + a8) it follows only a separation of all the 2 one-dimensional subalgebras into the three algebra classes given by C9(u)> 0 (a8 0 (a|>a ).) In addition, we investigate the vector ü = Y. j = (^d(exp(e6 r6)) 8 7=1 0 v Ad(exp(e71>7))) («) for u = £ <*,- j ^ 0 with a8 = 1 or a8 = 0. If we choose 7=1

~~6 arctan' for a5 ^ 0~~

~~y sign(a6) for a5 = 0 and a6 ^ 0 £7 =~~

~~+ arctan for a3 ^ 0 and a5 = a6 = a7 = 0~~

~~+ ysign(a2) for a2 ^ 0 and a3 = a5 = a6 = a7 = 0~~

~~(The upper (lower) sign is choosen for a4 > 0 (a4 < 0).) and~~

~~arctan for l/äf+a|>0 VÄ+~~

~~- — sign(a7) for a5=a6=0 and a7#0~~

~~— arctan for J/a2+a3>0 and a5=a6=a7 = 0 l/*2 +~~