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

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A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems 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-)differen- tial 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 subal- gebras of 'S with 3 < s <r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra 'S of the eight-dimensional Lie point symmetry group G admitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case. 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 0932-0784 / 93 / 0300-0497 $ 01.30/0. - Please order a reprint rather than making your own copy. 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.
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