Malomed B.A. Soliton Management in Periodic Systems

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Index

Action functional S, 27

Amplifiers, linear, 52

Amplitude(s), 3, 96

field's,

91/

FRM, 150

modulations, 30

of oscillations, 101

of solitons, 5, 56, 59,133

threshold value of, 95, 95/ 150

Ansatz (analytical expression), 19, 37,46

adoption of, 116

Gaussian, 132-33,143

lowest order for, 108

parameters in, 27

sech, 56, 58/

substitution of, 28,122

variational, 140

Antiwaveguides (AWGs),

105.

related problems with, 49-50, 50/

sharp optimum of, 39

solitons in, 32-37, 36/ 40-45, 42/ 43/

44/

45/

strength, 34-36, 36/ 44/

time-dependent masses and, 46

two-channel, 45-50, 49f, 50f

weak, 34

Dispersion management, multidimensional

models, 131-32

numerical results for, 134-37,135/ 136/

138/

139/

three-dimensional case for, 140-41,142/

VA and, 132-34

Distributed-filtering approximation,

40-43,

42/43/

DM. See Dispersion management

Double parametric resonances (PRs), 101-2,

102/

DSs.

See Solitons, dark

Ehrenfest theorem, 100

Erbium-doped fiber, 67

Euler-Lagrange equations, 28, 123, 125,

133,

144

Feshbach-resonance (FR), 23

drive, ac, 148-59,149/ 151/ 152/ 153/

154/

156/ 157/ 158/ 160/

driven BECs, 7

technique based on, 89

Feshbach-resonance management (FRM),

23-24.89

action of, 90

amplitude, 150

analytical results for, 96

controlled dynamical state and, 92, 93/

drive, 150-53,153/

introduction to, 143

low-frequency, 143

modelA^A and, 122-24

numerical results for, 94-95, 94^ 95/

128-30, 129/ 146-47, 146/ 147/

inOL, 143-59, 146/ 147/ 149/ 151/

152/

153/ 154/ 156/ 157/ 158/

160/

one-dimensional solitons under, 155-59,

156/

157/ 158/ 160/

splitting of higher-order solitons and,

93-96,

94/ 95/

time-periodic, 116

2D solitons in BECs under, 121-30,126/

129/

FF.

See Fundamental frequency

Fiber(s). See also Photonic crystal-fibers

BG,51

dispersion-shifted, 52

Erbium-doped, 67

Kerr effect on, 22

microstructured, 16

nonlinear graded-index, 98-100

optical, 2, 52, 132

spans,

75-76

Filtering, 40-45,42/ 43/ 44/ 45/

Fixed point (FP), 117,124,133

Fourier transform, 26, 52

Four-wave-mixing (FWM), 7

FP.

See Fixed point

FR. See Feshbach-resonance

Frequencies

driving, 153-54, 154/

large, 97

low, 143,150

shifts,

47-48

FRM. See Feshbach-resonance management

INDEX

111

Function sech (hyperbolic secant), 5

Fundamental frequency (FF), 11

fields,

75-76

waves, 73

FWHM (fiill-width-at-half-maximum), 34,

FWM. See Four-wave-mixing

Galilean boosts, 4,132

Galilean invariance, 4,10

Gap solitons. See Solitons, gap

Gap types, 14

Gaussian ansatz, 132-33,143

Gaussian pulses,

21,

33, 35-36

input, 68/

unchirped, 67

width of, 69-71

Gaussian wave, 26, 76, 90, 91/

Goldstone equation, 64

Gordon equations, sine, 18

Gordon-Haus effect, 33

GPE. See Gross-Pitaevskii equation

Gross-Pitaevskii equation (GPE),

13,

97,

126/

averaging of, 127-28

corollary of, 100

corresponding, 89

in one-dimension, 89-92, 91/ 92/ 93/ 97

simulations of, 147/

in two dimensions, 16-18,121

Group-velocity mismatch (GVM), 7

coefficients, 73

zero,

Group-velocity-birefnngence effect, 7

Group-velocity-dispersion (GVD), 3,11

coefficients, 7-8,20,25,29, 73

normal, 6, 37

segments, 44

GSs.

See Solitons, gap

GVD.

See Group-velocify-dispersion

GVM. See Group-velocity mismatch

Hamiltonian representations, 4, 10

averagingof,

127-28,129

conservation of, 125

value of, 6

yield, 96

Harmonic modulation model

NLS and, 27-29

numerical results for, 30^ 31-32, 32/

soliton dynamics in, 29-32, 30/ 31/ 32/

variational equations for, 27-29, 29/'

Harmonic-oscillator length, 13

Hirota representation, 24

Hotspot(HS), 111-12, 113/

Inverse scattering transform (1ST), 1,4

application of, 5,24

effects of, 75, 78

eigenvalues, 96

suppression, 71

Inverse-velocity shift, 5

1ST. See Inverse scattering transform

Jacobian multiplicators, 118

Josephson junction, 1

KdV. See Korteweg - de Vries equation

Kerreffect, 105,106

coefficient, 115

on fibers, 22

negative, 22,74

nonlinearity of, 11,115

ordinary, 131

self-focusing sign of, 2,21

Korteweg - de Vries equation (KdV), 1

kz.

See Cos

Lagrangian, 4, 19

density, 27

effective, 27-28,122,133,144

technique, 46

yields, 123,125,133

Laplacian (kinetic-energy operator), 140144

Lattices. See also Optical lattice

low-dimensional, 15, 17/

quasi-one-dimensional,

144-Al,

I46f,

147/

two-dimensional, 148-59,149/ 151/

152/

153/ 154/ 156/ 157/ 158/

160/

virtual photonic, 16-18

Light, standing, 9-10

Longitudinal modulations, 20

Lorentz invariance, 10

Lumped-filtering systems, 43^5,44/ 45/

Manakov's system, 7-8

Maps,

20,26, 33,109,140

Massive Thirring model, 9

Mathieu equation (ME), 100,101,148

ME.

See Mathieu equation

Modulations. See also Cross-phase

modulations; Self-phase modulations

amplitude, 30

longitudinal, 20

periodic, 140

178

INDEX

rapid, 97-98

transverse, 19-20

Multiplicators, 109,118

Newtonian particle, 28

NLM. See Nonlinearity management

Nonlinear SchrOdinger equation (NLS), 1,45

average counterpart of, 63

generalizations of,

4—5,

harmonic modulation model and, 27-29

layered, 482

modified, 40

properties of, 4

solitons and, 3-8

usual (2+1 )-dimensional, 115

variant of, 131

versions of, 12, 52

Nonlinearity management (NLM), 21-22

for BG solitons, 73-74, 80-88, 83/ 84/

86/

87/ 88/

co-propagation of pair of pulses, 78,19f

for cubic solitons/DM, 73,74-78,77/ 79/

formulation of model, 80-81

model, 75-76

periodic, 83/ 85, 86/ 87/

89-92,91/

92/

93/

for quadratic solitons/DM, 73,74-78, 77/

79/

OL.

See Optical lattice

Optical beams, 98

Optical fibers, 2, 52, 132

Optical lattice (OL), 13-14

dimensions of, 15-16,16/ 130

FRM in, 143-59,146/ 147/ 149/ 151/

152/

153/ 154/ 156/ 157/ 158/

160/

potential, 14-15,15/157

Optical telecommunications

SSM and, 52

WDM and,

45,

Optics, 13-18,15/ 16/ 17/

Oscillations

amplitude of, 101

of solitons, 98-103, 99/ 102/ 103/

PAD.

See Path-average dispersion

Parametric resonances (PRs), 98-103,99/

102/

103/ See also Double parametric

resonances

Partial differential equations (PDEs), 1,

101-2,146

Path-average dispersion (PAD), 20-21

decrease of, 39

equal, 48-49, 50f

normal, 26, 35, 36/ 39

treatment of, 33-34

PCFs.

See Photonic crystal-fibers

Periodic heterogeneous systems, 21

Periodic modulations, 140

Photonic crystal-fibers (PCFs), 16,67

Poincar6 section, 126/

Poynting vectors, 8-9

PRs.

See Parametric resonances

Pulse(s). See also Gaussian pulses;

Retum-to-zero pulses

co-propagation of pair of, 78,79/

input, 138/

isolated, 67-69, 68/

paired/single, 66-71,68/ 70/

pairs,

69-71, 70/71/

propagation, 38-39,39/

reshaping of, 75

self-compression of, 67

splitting of initial, 61/ 62

transmission of

single,

76-78, 77/

Quasi-phase-matching technique (QPM), 73

Rapid modulations, 97-98

Refractive index (RI), 2,140

Resonance. See also Feshbach-resonance

fundamental, 30

second-order, 30

subharmonic, 30

Retum-to-zero pulses (RZ), 20, 22, 25, 33,

RI.

See Refractive index

Russell, John Scott, 1-2

RZ.

See Retum-to-zero pulses

Saturation, 60

Second-harmonic waves. See Waves,

second-harmonic

Second-harmonic-generating (SHG), 11-12,

modules, 75-76

periodic heterogeneous structures in, 74

spatial-domain, 73

Self-phase modulations (SPMs), 3

coefficient, 9

effects of, 46

ratio of, 7

self-focusing, 80

SH.

See Waves, second-harmonic

SHG. See Second-harmonic-generating

INDEX

179

Solitary waves. See Solitons

Soliton(s)

alternate, 143,148,150-55,151/ 152/

153/

154/

amplitude of, 5, 56, 59,133

area of, 56

BECs/one-dimensional, 89-103,91/ 92/

93/

94/ 95/ 99/ 102/

inBECs/optics, 13-18, 15/ 16/ 17/

23-24

EG, 8-10, 73-74, 80-88, 83/ 84/ 86/

87/

88/

binding potential of, 6

bound states of, 47, 48^9

bright,localized, 6

collisions between,

45-50,49f,

50f

concept of, 1-13

degradation of, 38

in DM, 32-37,36/ 40-^5,42/ 43/ 44/

45/

dsmamics in harmonic modulation model,

29-32,30/31/32/

embedded, 10

energies, 30-32, 30/ 31/ 32/

external/intrinsic instability of, 101-2

FRM/one-dimensional, 155-59,156/

157/

158/ 160/

fundamental (splinters), 97-103, 99/

102/

104/

higher-order, 93-96, 94/ 95/

interactions between generation of

moving v., 85-88, 88/

management for channel, 105-13,107/

111/113/

matter-wave, 13

multidimensional optical, 23

NLM/DM for cubic/quadratic, 73, 74-78,

77/

79/

NLS and, 3-8

one-dimensional, 89-103, 91/ 92/ 93/

94/95/99/102/

one-dimensional optical, 20-23

optical, 2-13

oscillations of, 98-103, 99/ 102/ 103/

period, 5-6

in periodic heterogeneous media, 18-24

quasi, 25

in randomly, modulated systems, 22

semi-stable, 10

spatial, 11,106, 108-11,116-19,118/

spatially-averaged squared width of, 150

splittingof,

31-32, 32/

in SSM, 52-62, 57/ 58/ 59/ 61/ 62/ 63/

SSM/diagrams of states for, 60-64, 61/

62/

63/

stable multidimensional, 12

3-,

6, 94/ 96

3D,

144-47,146/ 147/

2-,

6, 96

2D,

121-30,126/ 129/

vectorial (two-component), 7-8

Soliton management

BSEs and, 23-24

general description of, 18-20

multidimensional optical solitons and, 23

one-dimensional optical solitons and,

20-23

Solitons, dark (DSs), 2, 6-7

Solitons, gap (GSs), 2, 9

alternate regular-, 8, 143

creation of, 8, 143

family of, 80-81

ID,

second-harmonic-generation, 11-12

temporal, 10

3D,

15, 17/

2D,

14-15, 16/

Solitons, spatiotemporal (STSs), 2, 12, 23,

131

Solitons, Townes (TSs), 115, 123, 137,145

Split-step model (SSM), 22

breathers in, 60-64, 61/ 62/ 63/

combined DM-,

66-71,

68/ 70/

constitutive equation for, 55

diagrams of states for solitons in, 60-64,

61/

62/ 63/

formulation of, 52-54

introduction to, 51-52

isolated pulses and, 67-69,68/

numerical result comparisons of, 56-60,

57/

58/ 59/

optical telecommunications and, 52

pulse pairs and, 69-71,70/ 71/

random, 53,64—66,65/

solitons in, 52-62, 57/ 58/ 59/ 61/ 62/

63/

three-step system for, 66

VA to, 54-56

SPM. See Self-phase modulations

SSM. See Split-step model

Stationary transmissions, 34

STSs.

See SoHtons, spatiotemporal

Tandem model, 22, 73-74

Termination, 60

Thomas-Fermi functions (TF), 90-92, 91/

180

INDEX

Threshold values, 95,95/ 150,159

Topological charges, 15

Townes solitons. See Solitons, Townes

Transverse modulations, 19-20

TSs.

See Solitons, Townes

Variational approximation (VA), 10,19

approach, 25-26

frameworks of, 101

model and, 122-24

multidimensional DM and, 132-34

numerical results of, 30/; 31-32, 32/

118-19,119/

predictions,

29-31,

30/ 39, 39/ 134-37,

135/

136/ 138/ 139/

spatial solitons and, 116-19,118/

to SSM, 54-56

standard, 98

in three-dimensional case, 124—27,126/

Velocities, 96

Vortices, 119

electromagnetic, 8

equation, 2-3

FF,73

fields, 252

functions, 19

Gaussian,26, 76, 90, 91/

linear, 9

ID,

one-dimensional, 2

Wave, continuous (CW), 4-5

Waveguides (WGs),

105.