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Subject Index

adjoint mapping, 17 antisymmetry property, 4 associated algebra, 17 axisymmetric motion, 54

B-defining equation, 154, 159, 173 B I-defining equation, 165 basis for a Lie algebra, 11 Bianchi Theorem, 199 bilinear equation, 230, 239 Boussinesq equation, 222, 225, 226 breather, 224

canonical invariant manifolds, 154 classification equations, 17 combination of solutions, 213 commutative Lie algebra, 5 commutator, 4 conditional invariance, 275, 340 conservation laws, 70 critical point, 210 critical value, 210 Crocco transform, 349 cylindrical Lagrangian coordinates, 51

defect of a manifold, 7 defining equation of an invariant set, 158 defining equations, 10 differential consequences, 129 differential constraints, 231 differential invariant, 9 differential invariant manifold, 10 differential reduction, 141 doubly escaping solution, 188 doubly limited solution, 189 Dubreil-J acotin equation, 220, 227

equation with an arbitrary element, 16 equations of gas dynamics, 173 equivalence transformation, 16, 37, 80, 81 essentially different solutions, 19 Eulerian coordinates, 31 exact solution, 88, 106, 108

factor system, 13, 26, 101 flow portrait, 214 four-layer sink, 199 free boundary, 69, 90

generalized separation of variables, 164 Grad-Shafranovequation, 212 gravitational heat convection, 262 group-theoretic classification, 15, 39,

45, 49, 54, 65 group-theoretic classification problem, 265

Hirota method, 227

ideal, 5 ideal incompressible fluid, 20, 29, 51 infinitesimal generator, 4 inhomogeneous fluid, 44 initial-boundary value problem, 68 inner automorphisms, 18 inner derivative, 17 interphase heat transfer coefficient, 250 invariance of the initial conditions, 39 invariant H-solution, 12 invariant manifold, 5, 128, 130, 149 invariant manifold of a group, 6 invariant mapping, 5 invariant set of a system, 129 invariant solution, 88, 108, 126

396

Jacobi identity, 4

kernel of the basic group, 16 Kolmogorov-Petrovskir-Piskunov

equation, 130 Karman solution, 326

L-defining equations, 154 Lagrangian coordinates, 35, 42, 57, 61, 90 Lie algebra of infinitesimal generators, 5 Lie derivative, 4 Lie's Theorem, 3 Lie-Backlund operators, 38 limited running solution, 188 Liouville equation, 230 local r-parameter Lie group, 2 Long equation, 216 loop path, 214

many-parameter solutions, 221 Marangoni boundary layer, 258 mass Lagrangian coordinate, 62 microconvection, 362 motion of liquid in the gravitation

field, 115 multiplicative separation of variables, 191

N avier-Stokes equations, 274 nonlinear Poisson equation, 159 nonlinear wave equation, 162 nonsingular manifold, 7, 12 nonstationary Euler equations, 68 nontrivial separating of variables, 181

Oberbeck-Boussinesq equations 262 one-dimensional motion of a gas, 61 one-parameter group, 2 one-parameter group G1 , 8 one-parameter group of transformations, 2 optimal system of one-dimensional

subalgebras, 22 optimal system of subalgebras, 17,

19, 97, 115 orbit, 2, 6

Subject Index

partially invariant H-solution, 14 planar motion of an inhomogeneous

fluid, 97 Prandtl equation, 165 Prandtl number, 262 Ptolemaic flow, 80, 82 prolongation of a group, 9 prolongation of an operator, 9 prolonged operator, 9

rank of a manifold, 7 rank of invariant solutions, 13 Rayleigh number, 262 regular stationary tornado, 375 Reynolds number, 276, 298 rotationally-symmetric motion of

a liquid, 47, 115

similar subalgebras, 19 sine-Gordon equation, 239 singular manifold, 7 solution of "breather" type, 203 solution of sink or source type, 198 specialization, 16 structure constants, 5 subalgebra of a Lie algebra, 5

t-system, 132 tangent vector field, 3 thermocapillary convection, 255 total differentiation, 8 transitive group, 6 transversal impulse, 224 trilinear equation, 230, 233 two-dimensional motion, 29

ultraparabolic equations, 350 uncoupled problem, 345

x-system, 132

Zabolotskaya-Khokhlov equation, 167