Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics
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Index
N-wave equations
non-Abelian, 67
¯
∂ method, 291
¯
∂ approach, 319
¯
∂ formalism, 308, 320
¯
∂ problem, 28, 30, 320, 324, 331, 342,
343
nonlocal, 30
2D-MKdV equations, 148, 158–160
Ablowitz–Ladik (AL) equation, 225,
245
analytic solution, 249
involution, 247
Jost solution, 247
Lax representation, 246
soliton, 252, 254
spectral problem, 246
auto-B¨acklund transformation, 217
auto-GDT, 163
B¨acklund transformation, 307
B¨acklund autotransformation, 150
B¨acklund transformation, 191
B¨acklund transformation (BT), 48, 199,
203, 214, 261, 264
extended, 214
Bell polynomial (BP)
binary, 59, 200
classical, 59
difference, 31
differential, 59
exponential, 200
generalized, 31, 34, 35, 40, 47, 48, 113
lattice, 48
non-Abelian, 31, 33, 35
Benjamin–Ono (BO) equation, 269,
277, 278, 280, 290
Bianchi–Lie formula, 67
Bianchi–Lie transformation, 83
bilinear Hirota method, 261
bilocal approach, 336
bilocal formalism, 336
Bloch function, 263, 265
Bloch solution, 262
Bogoyavlenskii equation, 218
Boiti–Leon–Manna–Pempinelli
(BLMP1) equation, 217
Boiti–Leon–Pempinelli (BLP) equation,
160
Boussinesq equation, 53, 56, 116, 200,
203, 204, 210, 211
Burgers equation, 39, 48, 55, 200
generalized, 39, 54
Calogero-Sutherland model, 186
Cauchy formula, 28, 272
Cauchy–Green formula, 29, 348
Cauchy-Green formula, 331, 344
chain equation, 80–82
closure relation, 344, 349
compatibility condition, 32, 43, 48, 49,
56, 58, 60, 84, 103, 164, 226–228,
255, 292, 308, 312
conservation law, 121, 122, 126
co variance principle, 113
co variance property, 149
co variance relation, 173
379
380 Index
co variance theorem, 162
creation-annihilation operators
bosonic, 192
fermionic, 192
Darboux covariance, 31, 94, 211, 212
Darboux dressing, 1
matrix, 173
Darboux integrability, 1, 13
Darboux method, 346
Darboux operator, 193
Darboux theorem, 31
Darboux theory, 141
Darboux transformation (DT), 7, 8, 10,
21, 31, 38, 40, 43, 44, 46–48, 53,
54, 61, 67, 72, 111, 114, 131, 133,
135, 136, 138, 140, 143, 149, 152,
162, 174, 177, 180, 181, 186, 187,
191, 199, 200, 218, 231, 261
classical, 68, 87, 110, 152
compound, 101, 102, 104
dressing, 223
elementary (eDT), 14, 18, 67, 70, 71,
73–75, 77, 80, 90, 101, 102, 104
Matveev generalization, 53
non-Abelian, 130
nonisospectral, 115
two-step (tDT), 76
twofold (tDT), 67, 68, 75, 79, 80,
83–85, 90, 93, 102, 149, 151
Darboux-gauge transformation, 32, 63
Davey–Stewartson (DS) equation, 141,
150, 153, 277, 306, 308, 311, 319,
341, 345, 346, 351
derivative NLS equation, 237
discrete spectrum, 313
dispersion relation, 340
nonanalytic, 319
polynomial, 329, 331, 337
singular (SDR), 320, 323, 326, 328,
330, 331, 335, 337
dispersionless nonlinear equation, 225
Drach–Dubrovin equations, 128
dressed operator, 9
dressed potential, 135
dressible operator, 219, 221, 222
dressing, 5, 7, 12, 14, 31, 65, 151, 172,
181, 186, 199, 228
dressing chain, 109, 111, 116
dressing chain equation, 113, 114, 116,
119, 124, 130, 137, 138, 164, 193
closure, 134, 140
non-Abelian, 131, 139
periodic closure, 116
solution, 139
dressing factor, 232, 233, 236, 241, 244,
251–253, 263, 266
dressing formula, 85
dressing method, 31, 93, 100, 169, 225,
261, 263, 291
formalism, 225
dressing procedure, 10, 14, 68, 178, 187
dromion, 157, 308, 316, 317
dual spectral problem, 339
Dubrovin equations, 124, 127
Dubrovin transform, 130
electron-silane scattering, 182, 184
Euler–von Neumann equation, 93
factorization, 13, 15, 20, 31, 115, 167,
186
analytic, 26, 27, 227
factorization method, 3
finite-gap solution, 110
Floquet discriminant, 261
Floquet spectrum, 261, 262
double points, 262, 265
Fredholm alternative, 283, 303
Fredholm integral equation, 27, 270,
279, 286, 293
gauge equiv alence, 324, 328
gauge transformation, 8, 63, 67, 68, 141,
237, 334
gauge-Darboux transformation (GDT),
162, 163, 167
Gel’fand–Levitan–Marchenko equation,
199, 219, 221
Goursat equation, 143, 147, 149, 151,
158
Goursat transformation, 43, 149
Green function, 179, 186, 190, 269, 280,
292, 294, 295, 300, 302, 309, 343,
345, 348
H¨older condition, 23, 26
Heisenberg spin chain equation, 324,
328, 330
Index 381
Hirota equation, 199
bilinear, 138
non-Abelian, 138
Hirota method, 31, 39, 214, 278
Hirota system, 32
generalized, 60
Hirota–Satsuma equation, 82, 104, 153
homoclinic orbit, 225, 261
homoclinic solution, 261, 264, 268
hyperspherical coordinates, 10
hyperspherical harmonics, 1, 10, 11
index of pole, 305
initial-boundary value problem, 308,
311, 320, 326
intertwining operator, 71
intert wining relation, 2, 4, 6–9, 14, 39,
67, 70, 80, 82, 193
inverse spectral transform (IST), 153,
199, 215, 219, 278, 291, 307, 311,
319, 346
involution, 229–231, 239, 251, 257, 270
isospectral deformation, 178
isospectral transformation, 2, 167
Jaynes–Cummings (JC) model, 2, 5,
194
joint covariance equation, 54, 55
joint covariance principle, 31, 50, 56,
57, 63
joint covariance property, 115, 206
Jost function, 188, 189, 280, 281, 289,
290
Jost solution, 229, 230, 236, 279, 282,
331
Kadomtsev–Petviashvili (KP) equation,
116, 141, 152, 153, 155, 211, 222,
277, 290–292, 294, 296, 297, 299,
305, 319, 342
Kaup–Kupershmudt equation, 115
KdV-MKdV system, 68, 104, 108
kink, 157
Korteweg–de Vries (KdV) equation, 8,
31, 43, 49, 50, 52, 53, 110, 116,
118, 119, 124, 191, 202, 206, 209,
211, 226, 269, 289, 290
Jost solution, 270
Lax pair, 269
potential, 214–216
soliton, 274
ladder operators, 2, 4, 10, 11
Laplace equation, 9, 142, 147
Laplace invariant, 12, 147
Laplace transformation (LT), 12, 13,
142, 143
Laplace–Goursat equation, 149
Lax equation, 32, 200, 204, 288, 307,
312
Lax operator, 118
Lax pair, 32, 49, 50, 53–58, 63, 84, 85,
94–96, 103, 115, 116, 124, 146,
152, 200, 218, 219, 221, 261, 263,
278, 308, 312, 320, 324, 347
Lax representation, 20, 161, 221, 222,
225, 226, 291, 292, 346
Lax system, 56
Liouville equation, 13, 144–146
lump, 280, 291, 299, 300, 346, 351, 352
matrix factorization, 14, 18
Matveev formulas, 69
Matveev theorem, 39, 47, 113, 149
Maxwell–Bloch equation, 326, 336, 341
Maxwell–Bloch equations, 80
Miura equation, 36, 80, 135
generalized, 38
Miura link, 81, 136, 137, 164, 217
Miura map, 48, 134
Miura transformation, 47, 51, 55, 109,
111, 125
generalized, 48, 114
MKdV equation, 118, 124, 148, 191
modified NLS equation, 225, 236, 261,
319, 331, 334, 335
breather, 244
Jost solution, 237
Lax representation, 237, 264
parity condition, 237
soliton, 241, 243, 244
monodromy, 8, 9
Morse potential, 167
Moser theorem, 19
Moutard equation, 143, 152
Moutard transformation, 43, 152–154,
158, 162, 196, 217, 218
generalized, 158, 159
382 Index
multicenter potential, 181
multicenter problem, 179
multiple eigenvalue, 294
multiple pole, 277, 300, 301, 305, 346
N-wave equations, 84
Nahm equations, 32, 61, 63, 64, 162
Lax representation, 62
Nizhnik–Veselov–Novikov equation, 153
NLS–Maxwell–Bloch system, 327
Noether identity, 215
Noether theorem, 199, 214, 215
nonlinear quantum mechanics, 92
nonlinear Schr¨odinger (NLS) equation,
82, 133, 225, 228, 232, 233, 261,
319, 324
analytic solution, 230
breather, 235, 236
operator, 130
split, 133
nonlocal
¯
∂ problem, 336
normalization, 227
canonical, 26, 227, 256, 272, 320, 327,
343
noncanonical, 30, 240, 250, 331
Oika wa–Yajima equation, 82
P¨oschl–Teller potential, 166, 167, 171
parity condition, 240
parity property, 246, 331, 334
positon, 31, 43–45
QR algorithm, 15, 19
quasideterminant, 1, 9, 20–22, 31, 40,
43, 141
Ramsauer–Taunsend minimum, 184
recursion operator, 325, 328, 330, 333,
340
reduction, 260
reflection coefficient, 281, 285
reflectionless potential, 45, 186–188,
190, 219, 274, 299
RH data, 231, 255, 274
continuous, 231
discrete, 231, 236
Riemann–Hilbert (RH) problem, 1, 23,
29, 226, 227, 250, 256, 258, 278,
286, 287, 289, 291, 296, 297, 300,
301, 307, 311, 320
differential, 278
local, 225, 277
matrix, 27, 28, 231
nonlocal, 286, 297, 315
normalization, 231
regular, 228, 240, 241
regularization, 231, 236, 240
regularized, 27
scalar, 26, 27
with shift, 272
with zeros, 79, 228, 231, 235, 236,
240, 251
ring
associative differential, 67, 73, 77, 80
differential, 32, 68, 132
division, 73
non-Abelian, 1
non-Abelian differential, 31, 56
operator, 80, 135
unitary, 32
Sawada–Kotera equation, 115, 116, 199,
205, 206
scattering equation, 280, 296, 344
scattering problem, 179
Schlesinger transformation, 14, 67, 72,
74, 78, 80–82
Schr¨odinger equation, 19, 52, 92, 111,
152, 164–166, 177, 179, 195, 269,
313
matrix, 173
radial, 171, 173, 174
Schr¨odinger operator, 8, 9, 186
self-consistent source, 319
self-scattering solution, 68, 91, 93,
95–97, 100
shape-invariant potential, 161, 168, 169
algebraic deformation, 169
singular, 171
singular integral equation, 241
singular manifold metho d , 217
sinh-Gordon equation, 144, 146
Sokhotsky–Plemelj formulas, 24, 26, 27
soliton, 67, 84, 85, 90, 91, 191, 199, 228,
231, 232, 234, 236, 240, 277, 278,
280, 288–291, 299, 307, 326, 334,
335
Index 383
soliton equation, 148
soliton theory, 152
spectral curve, 20, 126, 127, 129
spectral equation, 207, 228–230, 293,
301, 309
spectral problem, 43, 64, 236, 255, 284,
286, 289, 291, 308–310, 319
adjoint, 230, 283, 284, 294
elliptic, 346
matrix, 133, 225
Wadati–Konno–Ichika wa, 237, 332
Zakharov–Shabat (ZS), 56, 67, 68,
71, 76, 79–81, 90, 135, 137, 140,
320, 321, 326, 331, 337, 340
spectral transform, 297, 333
Sturm–Liouville equation, 114, 218
Sturm-Liouville operator, 193
superpotential, 164, 165, 170, 193
supersymmetric quantum mechanics,
191, 192, 194
supersymmetry, 191–194
unbroken, 194
support function, 158, 195
Sylvester–Toda lattice, 22
three-wave equations, 225, 254, 255
adjoint spectral problem , 257
analytic solution, 257
composite soliton, 259
Jost solution, 256
Lax representation, 255
soliton, 255, 258
spectral problem, 256
three-wave system, 85, 91
Toda system, 15, 19
Tsallis q = 1 entropy, 92
Volterra integral equation, 270, 279
von Neumann equation, 91, 92, 94–96,
100
linear, 92, 100
nonlinear, 67, 95
Zakharov–Shabat (ZS) equation, 60, 71,
72, 74, 83, 87
Zakharov–Shabat (ZS) operator, 59, 69,
70, 72, 73, 77
Zakharov–Shabat dressing, 223
zero-curvature representation, 226
zero-range potential (ZRP), 174, 176,
178, 179, 186
generalized, 175–177