Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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470
Index
Deformation,
439
gradient tensor,
439
incremental,
448-450
surface,
464
inner,
221
outer,
221
Derivation
Diagram, Bianchi,
331
Diffeomorphism
exhibiting exotic behavior,
277
symmetric,
272
Differential inequality
first order,
5
second order,
10, 14
Diffusions, nonlinear,
382. 383
Dilatation group,
160-162
Discontinuity, weak
Dressing method,
178-180, 319, 320
propagation of,
150, 165-167
application of,
205
Backlund transformation via,
320-335
Dromions,
178. 198, 199, 340
Dym equation,
347
induced invariance,
358
Riccati type pseudopotential,
409
2
+
1
dimensional,
354
Dym hierarchy,
347-351
reciprocal invariance of,
349, 350
Dynamical system,
211
infinitesimal form,
211
E
Eigenvalue problem, iso-spectral,
181
Elastic material, incompressible.
442
Elasticity, nonlinear equations,
438-445
bifurcation,
452-456, 458, 459
boundary value problem,
445-447
existence and uniqueness,
450-452
non-uniqueness,
456, 457
strong ellipticity,
447
Energy consideration
in elasticity,
460, 461
Equations, elasticity
constitutive,
442
equilibrium,
441
mass conservation,
440
Equilibrium equations,
441
Equivariant
function,
307
group,
262-265
map,
268
Evolution equations
coupled,
346
nonlinear,
354
symmetry generator,
220
time dependent symmetry,
220,221
Exterior derivative,
229
F
Feigenbaum cascade,
289
Fields
Bi-Hamiltonian.
234-236
inverse Hamiltonian,
234-236
Hamiltonian,
234-236
Fredholm alternative,
91-94
Frequency locked,
288
G
Gardner equation,
246
Gas dynamics,
57
Euler equations,
153-155
symmetry group,
57-60
Gel'fand-Levitan-Marchenko
equation,
186
Group, one parameter of diffeomorphisms,
218
infinitesimal generator,
218
Gronwall's inequality,
376, 377
for Brusselator,
376
H
Hadamard differentiability,
2 15
Hamiltonian,
223
flow,
223
Hamiltonian systems,
212
dimension,
124
harmonic oscillator,
213
Melnikov vector,
118-125
pendulum,
214
Poisson bracket,
221-224
symmetry
of,
213
Harmonic oscillator,
213
Heat equation, nonlinear,
19
Index
471
end problem
for,
20
Henon map,
283
Heteroclinic orbit,
83, 86
Homoclinic point,
80
Homoclinic orbit,
83, 86
Hyperbolic systems,
151
transversal,
80
strictly,
157
symmetry group,
159,
160
1
Improperly (111) posed problems,
1,
2, 4
continuous dependence,
12-21
continuous dependence on Cauchy data,
13-16
continuous dependence on geometry,
16, 17
continuous dependence on modelling,
17-21
existence,
7-12
non-existence,
8-12
nonlinear,
4-6
nonlinear Cauchy problem,
15,
16
nonuniqueness,
4-6
uniqueness,
4
Injective map,
227
Integral equation, Volterra,
182
Invariant,
264
Inverse monodromic transform,
202
Inverse scattering method,
181-186, 319
Inverse transformation,
425
Ishimori equation,
359
Isomonodromic linear equation,
202
Isomorphism, local,
304, 305
Isotropic materials.
444
Isotropy,
281-284
groups of attractors,
281
J
Jacobi identity,
232
Jumbo-Miwa equation,
50
K
Kadomtsev-Petviashvili equation,
45, 419
auto-Ecklund transformation,
421
line solitons,
191
I
form,
180
Riccati type pseudopotential.
419-423
solvability,
200
symmetry group,
45, 46
I1 form,
190
form,
47
Kadomtsev-Petviashvili equation, potential
symmetry group,
47-49
symmetry reduction,
48, 49
ffios,
dynamical computer package,
268
Kneading theory, equivariant
sequence,
290
symmetry locking,
284-291
Knot-holder,
293
strange attractor,
295
van
der Pol-Duffiing oscillator,
295
bi-Hamiltonian system,
236
Hamiltonian form,
225-228
invariant solutions,
44, 45
periodic solutions,
.202-205
Riccati type pseudopotential,
408
similar solution,
201, 202
symmetry group,
44
Korteweg de Vries equation,
44, 180, 347, 405
Korteweg de Vries-sine Gordon equation,
246
Korteweg de Vries hierarchy,
351
auto B;icklund transformation,
351
Korteweg de Vries-Burgers’ equation,
generalized,
50
generalized cylindrical,
50
Kuramoto-Sivashinsky equation,
431
L
hgrangian principal axes,
440
Lax
equation,
194
Lax
pairs,
205,353
Lie algebra,
216
commutator,
216
Jacobi identity,
216
Lie derivative,
219, 227-230
Lie module,
227
homomorphism,
242
isomorphism,
242
Lift,
278, 279
Lin-Tsien equation,
59
Lipschitz continuity,
371
Logarithmic convexity,
3, 14-16
symmetry group,
59.
60
472
Index
Logistic map, 265-267
bifurcation diagram, 266
cubic, 265
quadratic, 265
M
Manifold
intersection of, 94-106
stable, 85, 94-106
tangential intersection, 112-1
18
transversal intersection, 112-118
transversality, 101-106
unstable,
85,
94-106
Arnold, 284
compatible, 239
Henon, 283
Map
toms. 273-276
Mass conservation, 440
Material symmetry, 443-445
Melnikov method
examples of, 137-147
failure of, 146, 147
Melnikov vector,
81.
100
dimension, 107
first approximation, 112-118, 133, 135
for completely integrable Hamiltonian,
118-125
Method
Broomhead-King, 309
Packard-Takens, 307-309
Minimal surface equation, Cauchy problem,
19
Miura transformation, 49. 351
Modified Korteweg de Vries equation, 246
Mobius group, 417
Mobius transformation, 357, 401
N
Nanson’s formula, 440
Navier Stokes equations, 52
group actions, 54
group invariances, 53, 54
symmetry group. 53
symmetry reductions, 55-57
symmetry group, 62
Neo-Hookean equations, 61
symmetry reductions, 63, 64
Noether’s theorems, 223
N-forms, 229
Non-constant state, 171
Nonlinear superposition principle, 331-334,
338
N-soliton solutions, 250
N-wave interaction equation, 187
0
Operator
hereditary, 237
implectic, 223, 233, 240
inverse symplectic, 233
symplectic, 233
group, 262, 263
heteroclinic.
83,
86
homoclinic,
83.
86,
138,
140
pseudo, 277
attractor, 298
bifurcation sequence, 300, 301
Orbit
Oscillations, coupled identical, 296-299
Oscillator. van der Pol-Duffing. 291-294
P
Packard-Takens method, 307-309
Painleve type equations, 201
Pendulum equation, 214
phase space orbits, 224
Perturbation
Hamiltonian, 119
non-Hamiltonian, 119
Phase space, equivariant, reconstruction,
Pitchfork bifurcation
305-310
example, 290
symmetry breaking, 287
Poisson bracket, 223
Potential sine-Gordon equation, 246
Products, compatible, 238-240
Pseudopotential, 400
by computer algebra, 401
for Harry Dym equation, 409
for Korteweg de Vries equation, 408
for nonlinear Schrodinger equation, 411
for sine-Gordon equation, 410
Index
473
Lax
equations for, 404
Riccati-type. 400-403, 419-423
Riccati-type and non-integrability, 431. 432
singularity manifold equation, 405
Pseudoorbit, 277
Pull-back. 219
R
Reaction-diffusion systems, 363, 364
exponential growth. 370
finite blow up time, 365, 366
global existence, 364, 365
non-triangular, 379, 380
open problems, 378
polynomial growth, 369-375
semigroups, 390
triangular, 376, 377
weak solutions, 366, 367
without polynomial growth, 384-385
Reciprocal transformations, 341-347
in 2
+
1
dimensions, 352-359
Riccati chain, 412
and Schwarzian transformation, 413
Riccati equation, 401
invariant under Mobius group, 401
properties of, 403
Riemann-Hilbert problem, 178, 182-184
S
Saddle node bifurcation, 290
Sawada-Kotera equation, 430
singularity manifold, 430
Scattering
data, 184
higher order, 413-417
second order, 404-410
Schrodinger equation, nonlinear, 321
Riccati type pseudopotential, 411
singularity manifold equation, 429
Schwarzian derivative, 356, 402
Shadowing, 277-280
lemma, 277
lifting, 278
Shimuzi-Wadati equation, 359
Similarity solution, 150
hyperbolic systems, 163-165
sine-Gordon equation, 246
Riccati type pseudopotential, 410
singularity manifold equation, 428
Singularity manifold equations, 357
reciprocal auto-%cklund transformation,
424-426
Solitons, 177, 246
integrability, 246, 247
Soliton interactions, for Korteweg de Vries
equation, 253
Soliton solutions, for Davey-Stewartson
equation, 339-341
Splitting index, 106-111
Stability inequality, Holder, 15
Strain energy function
neo-Hookean, 461
region of stability, 466
evolution of discontinuity, 36, 37
symmetry group for, 35-37
symmetry reduction, 36, 37
evolution
of
weak discontinuity, 39
symmetry group,
38
Sweeping parameter, 136
Symmetry breaking, 258
Symmetry, dihedral, 267, 268
Symmetry groups, 31-67,263
calcultions by computer algebra, 45, 66, 67
for wave equation, 34-43
invariance condition, 161
relation between equivalent systems, 42, 43
relation to solution, 43
Symmetry. in dynamics, 262-264
Symmetry locking, 288-291
String oscillations
String, travelling
'
equivalent kneading theory, 284-291
saddle node, 289
Symmetry, scaling, 249
Symmetry systems, chaotic dynamics in, 265
T
Tangent
bundle, 216
space, 216
Taylor vortices, 310-312
Tensor
474
Index
Biot stress, 443
Cauchy stress, 441, 443
closed covariant, 230
contravariant, 228
covariant, 228
deformation gradient, 439
left stretch, 439
invariant, 230
normal stress, 441
Poisson, 233
right stretch, 439
Time series analysis, 299, 302-305
Tori, invariant, 125-137
Torus maps, 273-276
heteroclinic orbit
for,
125
bifurcating sequence, 274
equivariant, 274-276
symmetry locking, 286
Transformation, Miura, 49
Transversal intersection, necessary and
sufficient condition for,
115
V
van der Pol-Duffing oscillator, 291-294
von Karman elastic plate equations, 64
quintic, 293, 297
symmetry group, 65
symmetry reduction, 65, 66
Vortices, 259-261
W
Wave, 149
discontinuity, 154-158
exceptional,
158
Wave breakdown, 157. 158
Wave equation, quasilinear
symmetry group, 34-43
with arbitrary constitutive law, 34, 35
Hadamard relation, 155
normal vector, 155
speed
of
propagation,
155
wave front,
155
Weak solution, 390
Well posed problem,
1.
2
WKI method, 340
link with
ZS-AKNS,
346
Z
Zakharov equation, 57
ZS
(Zakharov, Shabat) spectral problem, 321,
322, 343, 344
Mathematics in Science and Engineering
Edited by William
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Georgia Institute
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