Appendix
A.l Fourier Series [Benedetto 1996 B, Bracewell1986 B, Carslaw 1930 B]
A .1.1 General Rules
period: x(t) = x(t + T) basic circular frequency: OJ OJ 2n / T T = 2n / OJ = 2n / OJ1 = 1 = complex Fourier series: complex coefficients: m=oo 1 T x(t) = x(t+ T) = IJmexp(-imOJt) Fm = - f x(t) exp(+imOJt)dt m=-oo T o real periodic functions: coefficients: x(t) = x(t+ T) = x*(t) Fo = Fo* = llo = Ao Fm =F~m =~(am +ibm) =.!. ~exp(iam} 2 real Fourier series I: real coefficients: x(t) = x(t+ T) = x*(t) 1 T
00 00 ao = - f x(t)dt T o =ao+ Lamcos(mOJt) + Lbmsin(mOJt) m=1 m=1 2 T am = - f x(t)cos(mOJt)dt T o 2 T bm =-f x(t)sin(mOJt)dt T o real Fourier series TI: real parameters: x(t) = x{t+ T) = x*{t) Ao =ao 00 [2 2]112 = Ao + L~cos(mOJt-am} Am = am +bm ~O m=1 am = arc tan(bm / am} 480
real even periodic functions: coefficients: x{t) = x{t + T) = x * (t) = x{ -t) Fo = Fo* = ao = Ao 1 1 Fm =F-m =Fm *=-a 2 m =-A 2 m
um =O;bm =0 real odd periodie functions: coefficients: x{t) = x{t+ T) = x*{t) = -x{-t) Fo =ao =Ao = 0
Fm =-F-m =-Fm *=..!..ib 2 m
um = ±tr 12;am = 0
bm = Am sign( Um)
A .1. 2 Real Periodie Functions
Norrnalized parameters: Cü =Cü, = 1 and T =2tr a) Square wave:
x{t) = x{t + 2tr) = sign(sin t) = i[sint + ..!..sin 3t + ..!..sin 5t + ... ] tr 3 5
• • 1 -T .2'Tr -2T .1T t -1 b) Asymmetrie sawtooth:
x{t)=x{t+2tr)=xltr for _tr:s;x+tr=~(sint _ sin2t + sin3t _+ ... ) tr 1 2 3 481 e) Symmetrie sawtooth:
2X / n for - n / 2 ~ x ~ + n / 2 } x{t) = x{t + 2n) = { 2 - (2x / n) for + n /2 ~ x ~ +3n /2
=-28(.1'3 smt-zsm t+zsm1 '5 t-+ ... ) n 3 5
I
d) Modulus of sine:
2 4 (COS2t cos4t cos6t ) x{t)=x{t+2n)= Ismtl=--- . --+--+--+ ... n n 1·3 3·5 5·7
I
e) Reetified sine:
. ). 1 1. 2 (cos 2 t cos 4 t cos 6 t ) x{t)=x{t+2n)=H( smt ·smt=-+-smt-- --+--+--+... n 2 n 1·3 3·5 5·7 482
f) Rectangle:
I für Itl < 'C I 2} x( t) = x( t + T) = II( t I 'C) = {1/ 2 for Itl = 'C I 2 o für Itl > 'C I 2
=~+3.. fm-1sin(m'C/2)cosmt with 'C<2n 2n n m;1
x
g) Simple Fourier se ries
a) Sums valid für 0 < t < 2n:
~LJ m -I.sm mt = -1 ( n - t ) m;1 2
fm-1cos mt = -..!.Cn[2(1- cost)] m;1 2 483
ß) Sums valid for r2 < 1:
i rmsinmt = rSint[ 1-2reost+ r2 rl m=\
I m-Irmsinmt = are tan{r sint(1- reos trI} m=1
= -I ~>meos mt = (1- reos t)[ 1-2reos t + r2 ] m=1
Im-1rmeos mt = -2. en{ 1-2reost+ r2 } m=l 2 484
A • 2 Fourier Transformation [Benedetto 1996 B, Bracewell 1986 B, Campbell & Foster 1948 B, Carslaw 1930 B, Champeney 1973 B, Erdelyi et al. 1954 B, Poularikas 1995 B, Zayed 1996 B]
A .2. 1 General Rules
function: Fourier transforrn: 1 +~ +~ x( t) = - f F( w) exp( -irot )dw F(w) = f x(t)exp(+irot)dt 27r _~ -~ real function: x(t) = x *(t) F( -w) = F * (w) even function: 1 ~ ~ x(t) = x( -t) = - f F( w) cos wt· dw F( w) = F( - w) = 2 f x( t) cos wt . dt 7r o 0 odd function:
.~ ~ x(t) = -x(-tl = -~ f F( w) sin rot· dw F( w) = -F(-w) = 2i f x(t) sin rot· dt 7r o 0 x(-t) F(-w) x(t / -r) -r F(w-r) x(t- -r) exp(iw-r) F(w)
t n x(t) (_i)n d n F( w) / dwn co rIx(t) i f F( w') dw'
-~ dnx(t)/ dtn (-iw)n F(w)
t f x(t')dt' (i / w)F(w)
-~ product: convolution: +~ 27r Xl (t) X2 (t) f FI(Q)F2(w-Q)dQ
-~ convolution: product:
+~ f XI( -r)X2(t- -r)d-r FI (w ) F2(w)
-~ autocorrelation: power spectrum: +~ Wiener-Khintchine theorem x(-r)x(-r+t)d-r f IF(w)12 ~
x(t / 1")cosQt ~[F( 1"(w +Q)) + F( 1"(w - Q))] 2
x(t / -r)sinQt ;i [F( 1"(w + Q)) - F( 1"(w -Q))] 485
A. 2.2 Real Functions
funetion: Fourier transforrn: I +- +- x( t) = - f F( OJ) exp( -iOJt )dOJ F(OJ) = x(t)exp(+iOJt)dt 27r f I -- 27rÖ(OJ)-- cosnt 7r[ Ö(OJ - n) + Ö(OJ + n)] sinn-r i7r[ ö( OJ - n) - ö( OJ + n)] Dirae delta function Ö: ö(t) I ö(t --r) exp(iOJ'Z') dßö('Z')/ dtß [-iOJ t sign function: 2i - signt OJ Heavyside unit step H: i H(t) = .!.[I + signt] 7rÖ(OJ)+- 2 OJ H(t) exp(-at) [a -/OJ . r l rectangle function ll: fO'IX 1
2i sin2( OJ / 2) ~ll(t-~) - ll(t+~) OJ
ll(t)cos7rt -1[.sznc (OJ--- I) +sznc. (OJ-+- 1)] 2 27r 2 27r 2
ll(t)sin27rt ~ [sinc( 2: -1) - sinc( 2: + 1) ]
exp{ -(t / 'Z')2} 7r1/2 'Z' exp{ -( OJ'Z' / 2)2 }
t exp( -7rt2) i(27r)-1 OJexp{ _OJ 2 / 47r}
Itl- 1I2 127r / OJII/2 486
1I2 Itl- sign t I'1 2 n I m1" 2slgnm . exp(-Itl) 2[I+ m2 t exp( -I tl) sign t inm[l+ m2t exp( -Itl)sinc(x In) arctan(21 ( 2) exp(-t)H(t) [1-imrl t·exp(-t)H(t) [1-imr2 sech 1rX = [cosh 1rX t sech( m I 2) = [cosh ( ml 2) t 2 sech21rX = [cosh 1rX r m[ n sinh ( m I 2) t tanht i[ sinh ( m / 2) t ll(t)[ 1- 4t2] ;2 [( m / 2r l sin( m 12) - cos( m / 2)]
nSinc(2:)+ (~}inc(2: -1) ll(t)cos2m + n sinc( ~+ 1) 2 2n ln[ 1+ Cl' / t)2 ] 2n m-I [1- exp( -mr)] 2 2 ln[ r +t ] 2n m-I [exp( -mT) - exp( -mr)] T2 +t2 2[t2 + r2t (n / r)exp[ -Tim!] Bessel function Jo: Jo(t / r) 2ll( mr 12)[ r-2 - m2t /2 Bessel functions: }zn: Chebyshev polynomials T2n : 2 ll(mr/2)T2n (mr) J2n (t/r) [-2r -m2f2
Bessel functions Jm: Legendre polynomials Pn: r Il2J dt) in(2ntl/2 ·ll(m/2)·Pn(m) n+-2 Legendre polynomials Pn: Bessel functions Jm:
ll(t/2)Pn(t) in (2n Im )112 J I (m) n+-2 Chebyshev polynomials Tn: Bessel functions Jn: ll(t / 2) Tn (t) ninJn(m) [ I-t2f2 487
A • 3 Laplace Transformation [Abramowitz & Stegun 1965 B, Bracewell 1986 B, Doetsch 1970 B, Doetsch 1971/73 B, Erdelyi et al. 1954 B, Pöschl 1956 B, Poularikas 1995 B, Zayed 1996 B]
A.3.1 General Rules
For the definition of the convolution see (3.2 - 24a&b). If F(t) and F(t) are continuous at t = 0, then F(t = +0) and F(t = +0) can be replaced by F(O) and F(O).
~ L{F(t)} = f F(t)exp(-pt)dt o L{ aFI (t) + bF2 (t)} = aF1(p)+bF2 (p) L{F(at)} = F(p/ a) L{H(t-to) F(t-to)} = F(p) exp( -top), to > 0 L{(-tt F(t)} = dnF(p)/ dpn L{ exp( +Pot) F(t)} = F(p- Po) L{F(t)} = pF(p)-F(t=+o) L{F(t)} = p2F(p)-pF(t=+O)-F(t=+o)
LU F(t)dt} = p-1F(p)
T L{F(t) = F(t+T)} = [1- e-Tp r f F(t)e-tPdt, F(t) periodic o L{F1(t)* F2 (t)} = F1 (p) F2 (p), convolution
where F1(t < 0) = F2 (t < 0) = 0 488
A. 3.2 Heaviside and Dirac Functions
This list eontains Laplaee transforms of the Heaviside step funetion H(t), the Dirae delta funetion &,.t) and its derivatives 8n)(t). In these equations t = +0 indieates a time just after t =O.
L{c5(t-to)} exp( -to p), to > 0
L{ :r: c5(t - to)} = L{ c5(n)(t - to)} = pn exp( -top)
L{ c5(t - (+Ü))} =1 L{:r: c5(t-{+Ü))} =L{c5(n)(t-{+ü))}
L{c5{t)} =112 L{H(t-to)} =p-1exp(-top), to>O L{H{t)} =l/p
L{~{ _1)n H(t - ~nr)} 489
A.3.3 Real Functions L{(;::)!} L{exp( -at)} =(p+arl L{texp(-at)} =(p+ar2
L{ (;::)! exp (-at)} =(p+ar o L{ exp( -at) - exp( -bt)} = [(p+a)(p+b)t b-a L{sin at} = a[p2 + a2 t L{eos at} = p[p2 +a2t
L{ sin at exp( -bt)} =a [(p+b) 2+a 2]-1
L{eos at exp( -bt)} =(p+b)[(p+b)2 +a2( L{sinh at} =a[p2 - a2 rl L{eosh at} = p[p2 -a2t
Il2 -1/2 L{( 1ftr } =p
0 -(o+.!.) L{ 4 n! t(O-1I2)} =p 2 (2n)!-fii L{Jo(at)} =[p2 +a2r1/2 L{t sin at} = 2ap[p2 + a2 r2 L{t eos at} =(p2 _a2)[p2 +a2t L{t-Isin at} = are tan ( a / p)
L{sin2( ~)} =(a2 /2p)[p2 + a2 t
L{lsinatl} = a[p2 + a2 t tanh(2a /rep) L{+fn t} =-p-I[fnp-C] C=O.5772 ... = Euler s eonstant 490
A.4 Convolution (Faltung) [Bracewell1986 B, Pöschl1956 B. Schetzen 1989 B]
A .4.1 General Rules
Definition:
+~ J(t) * g(t) = fJ(s)g(t-s)ds
-~ Commutation:
+~ J(t) * g(t) = g(t) * J(t) = f g(s) J(t - s)ds
-~ Association: J(t) * [g(t) * h(t)] = [J(t) * g(t)] * h(t) = J(t) * g(t) * h(t) Distribution over addition: J(t) * [g(t) + h(t)] = J(t) * g(t) + J(t) * h(t) Differentiation: ~[J(t) * g(t)] = dJ(t) * g(t) = J(t) * dg(t) dt dt dt Convolution with Dirac delta function: 8(t)*8(t)=8(t)
8 -[J(t-O)+ J(t+O)] (t)*J(t)= 2 {'J(t) if J(t) continuous1 Convolution with Heaviside unit step: H(t)=.!..[l+sign(t)] 2 H(t) * 8 (t) = H(t) t H(t) * J(t) = f J(s)ds
-~ H(t)tm * J(t) = (m!) ff ... ff J(s)ds mfold
t H(t)exp( -at) * J(t) = exp( -at) f J(s) exp (as)ds
-~
~ t H(t)J(t) * g(t)= f J(s)g(t-s)ds= f g(s)J(t-s)ds
0 -~ t t H(t)J(t) * H(t)g(t) = f J(s)g(t - s)ds = f g(s)J(t - s)ds 0 0 491
Convolution with sampling function or shah function: III(t) +- III(t)= Lo(t-m) m=-oo +- III(t)f(t)= Lf(m)o(t-m) m=-oo +- III(t)* f(t) = Lf(t-m) m--oo 492
A.4.2 Heaviside Unit Step
Definitions: o for t < 0 H(t)=~[I+Signt]= {1/2fort=O 1 for t > 0 [H(t) ro = o(t) ; [H(t)f1 = H(t), [H(t)rm = H(t)* H(t)** * *H(t) * H(t) mfold Powers and products: t(m-I) [H(t)]*m = ·H(t) (rn-I)! [H(t) r2 = H(t) * H(t) = t· H(t) ramp function t2 [H(t) 3 = H(t) * H(t) * H(t) = -. H(t) r 2 t O H(t) =(n !)[ H(t) r(O+I)
t kH(t) * t m H(t) = rn'k'.. t(k+m+1) H(t) (rn+k+I)! Exponentials: - exp(t) H(t) = I,[ H(t)r o 0=1 exp(t)H(t) * H(t) =[exp(t) -I]H(t) exp(t) H(t) * (1- t) H(t) = t H(t) [exp(t) H(t) r2 = t exp(t) H(t) *3 t2 [exp(t)H(t)] = 2exp(t)H(t)
t exp(at) H(t) * exp(bt) H(t) =-[ a - b r l [ exp(at) - exp(bt)] exp( -at)H(t) * t H(t) = ~{t +~[ exp( -at) -I]} H(t) 493 A.4.3 Special Real Functions Rectangle functiün 0: Triangle functiün A: I für lxi< 112} I-lxi für lxi $; I} O(t)= {112 fürlxl=1I2 A(t) = { o für lxi ~ 1 o für lxi> 11 2 A(t) = O(t) * O(t) +~ A(t) * LQmO(t- m) = polygon through (m,Qm) m=-oo +~ A(t)* Lo(t- m) = 1 m- ~ . () sinn t smc t =-- nt [sinc(t)r m = sinc(t) m = 1,2,3, ... sinc(t) * Jo(nt) = Jo(nt) Bessel functiün 494 A.4.4 Hilbert Transformation [Bracewelll989 B, Erdelyi et al. 11 1954, Poularikas 1995 B, Zayed 1996 B] Definition: FHi (t) =__ 1 * f(t) =..!.. Tf(s)ds 1rt 1r s-t -~ Inversion: 1 - 1rt * FHi(t)=-f(t) Transforms: 1 --*1=0 1rt 1 1 --*O(t)=-- 1rt 1rt __1 *( __1 ) = O(t) 1rt 1rt --1 * smt=cost . 1rt __1_ * sin t = cos t -1 1rt t t __1 *[I+ t2 r=-t[l+ t2 r 1rt __1 *II(t)=..!...enlt-1I21 1rt 1r t+1I2 __1_ * J1 (at)sinat = J1(at) sin at 0< a 1rt __1_ * Jm(at)sinbt = Jm(at)cosbt 0< b < a, m = 0,1,2 1rt References B. Books Abdullaev, F.K. (1994): Theory of Solitons in Inhomogeneous Media Wiley, Chichester, UK Ablowitz, MJ., Clarkson, P.A. (1991): Solitons, Nonlinear-Evolution Equations and Inverse Scattering Cambridge Univ. Press, Cambridge, UK Abramowitz, M., Stegun, I.A. (1965): Handbook of Mathematical Functions Dover, New York Agrawal, G.P. (1989): Nonlinear Fibre Optics Academic Press, San Diego Aimes, W.F. (1972): Nonlinear Partial Differential Equations in Engineering Academic Press, New York Allen, L., Eberly, J.H. (1975/1987): Optical Resonance and Two-Level Atoms WileylDover, New York Alonso, M., Finn, E.J. 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Soviet Phys., JETP 34, 62 Index lIf noise 322 asymptotic stability, theorem on 235 y-rays 338 asymptotically stable 189 7r pulse 447 atrnosphere 322 27r pulse 446 attenuator 446 attractor 189, 193,223 Abel-Liouville-Jakobi-Ostrogradskii attractor in the origin, theorem on 235 identity 250 autocorrelation 284 acceleration 179 autocorrelation function 319 acceleration field 148, 244 autonomous system 90, 148, 149 acceleration of free fall 8, 99 average 284 acoustic waves 460 average energy density 351 Airy eosine 38, 74 average intensity 351 Airy function 38, 379 average power 285 Airy oscillator 38 axially symmetrie system 224 Airy pulse 380 Airy sine 38 Bäcklund transformation 419 allowed band 401, 410 band edge 50, 51 allowed frequency range 50 beat frequency, 49 amplification 14 Bendixon theorem 232 amplification time 14 Bessel differential equation 72, 256, 459 amplifier 447 Bessel function 40, 72, 182, 373, 376, amplitude 10 459, 486 amplitude-frequency relation 135 Bessel oscillator 40 analytical function 156 Bessel source 182 angular velocity 153 BIBO stability 279 anomalous dispersion 357, 365 bifurcation 139, 300 antikink soliton 431 bifurcation diagram 304, 306, 309, 330 aperiodie modulation 37 bifurcation point 302, 319 aperiodically modulated oscilIator 129 bifurcation value 300 approximation technique 105 Bioehe wave 477 area 171 bion 433 Argand diagram 158 bistability 111, 136, 137 associated differential equation 185, 188 Bloch equations 442 associated HamiItonian system 91 Bloch function 48, 409 associative 278 Bloch theorem 409 514 Bloch wave 409 Chebyshev polynomial 486 Bloch-wave theory 400 chirp oscillator 35, 129 Boltzmann's constant 286 circuit 289 Borda model mouth 167 circular frequency 9 boundary condition 369,451 circular function 10 bounded-input bounded-output circular vortex 174 stability 279 circularly polarized 352 Bragg circular frequency 49, 406 circulation 162, 163, 266 Bragg condition 53, 65 cnoidal Sine-Gordon wave 433 Bragg effect 51 cnoidal Toda wave 449 Bragg order 57,58,63 cnoidal wave 422, 425 Bragg propagation constant 406 Cole-Hopf transformation 418, 420 Bragg reflection 403 comb potential 410 braking force 24, 79 commutative 124 breather 433 companion system 186 bright soliton 436, 439, 449 complete eIIiptic integral 94, 449 BriIIouin zone 51, 393, 397,402,410 complex dispersion relation 346 Bromwich-Wagner integral 123 complex harmonic excitation 116 Burgers equation 418 complex representation 10, 11,344 compressible fluid 262 canonical form 199 Compton propagation constant 427 canonical matrix 186 Compton wavelength 427 capiIIary wave 364 condition for stability 291 carrier wave 357, 359 conformal mapping 158 carrying capacity 239 conservation law 414, 421 Cartesian coordinates 152 conservation of the electric charge 384 catastrophe condition 313 conservative 243 catastrophe manifold 313 conservative field of force 233 catastrophe position 313 conservative system 92 catastrophe theory 300, 313 conserved density 414 Cauchy method 379 constant restoring force 84 Cauchy-Riemann conditions 157, 158 continuity equation 261, 263 causa! 278 control circuit 290 causal system 112, 122, 125, 282, 293 convective time derivative 262 causality 278, 280 convolution 124,277,382,490 cavity 460 convolution calculus 383 chain 391 Coulomb friction 79 change of polarity 78 Coulomb function 41 chaos 300 Coulomb oscillator 4 I chaos, criteria for 319 coupled-wave system of equations 407 chaos, route to 320 coupled-wave theory 406 chaotic response 140 coupling constant 404 characteristic circular frequency 6 critical 18, 113, 122,375 characteristic matrix 149, 183 critical point 149, 150, 190 characteristic time 20, 21, 27 crosscorrelation 285, 287 515 crystallattice 391 distributed feedback 403 cubic Duffing oscillator 97 distributed-feedback laser 399 divergence 152, 162,261 d'Alembert system 150, 183, 274 down-chirp 35 d'Alembert systems, equivalent 186 drag 24,82 d'Alembert's law 369 drag coefficient 82 damping 14 drumskin 458 damping time 14 dry friction 79 Darboux modulation 64 Duffing approximation 138 dark soliton 436, 440 Duffing oscillator 90,91, 100, 134 de Broglie wave 348 dead zone 78, 85 effective phase 346 Debye attenuation 21 effective wavelength 346 Debye dispersion 387 eigenfunction 347, 469 Debye relaxation 20, 21 eigenvalue 12, 45, 48, 469 Debye's law 386 eigenvalue equation 48 deformation 265 eigenvalue spectrum 282, 283 degeneracy 247 elastic wave 333, 336 degenerate 199,223,245 electric circuit 6 degrees of freedom 246 electric energy 26 delay equation 295, 297 electric permittivity 338, 383 delay system 294 electric susceptibility 386, 389 delay time 294 electrical conductivity 383 delta-function input 280 electromagnetic wave 338 density wave 332 elliptic excitation 140 determinant 48 elliptic function 92 deterministic chaos 111, 238, 317 elliptical integral 422 DFB 403 elliptically polarized 352 DFB laser 399 endpoint condition 452 dielectric media 386 energy 76 dielectric polar liquid 21 energy density 350 dielectric polarization 386 energy exchange 23, 26, 407 differential operator 347, 366 energy-momentum relation 372, 410 differential operators diffusion 377 envelope 357, 359 differential operators for energy and envelope differential equation 357, 365, momentum 349 366 diffusion equation 347, 377 envelope differential operator 366 dipole moment 269 envelope dispersion relation 357, 365 dipole source 269, 273 envelope velocity 357, 359 Dirac delta function 42, 43, 124, 268, equipotentialline 159, 165, 178 280,488,490 equivalent d'Alembert system 186, 191 Dirac delta function, derivative 43 Euler down-chirp 37 discriminant 183 Euler relation 161 dispersion 360, 361, 363 Euler differential equation 67, 256 dispersion relation 344,363,401 Euler equation 340 516 Euler-Painleve equation 473 frequency 9, 134 Euler-Painleve wave 473 frequency gap 50, 51, 401, 404 evolution equation 414, 421 frequency mixing 111 exponent of the singularity 74 frequency multiplication 111 exponential chirp 37 frequency spectrum 20, 120 exponential relaxation 281, 288 Frobenius series 67, 74 exponentially stable 189, 253 external excitation 111 gain function 114, 135 gain modulation 400, 405 feedback 26, 131, 289 Galilei transformation 417 feedback-control system 290 gap 410 feedback function 131 gauge equation 270 feedback proportional to the veiocity Gauss function 43 296, 298 Gauss law 162, 181 feedback, virtual 131 Gauss theorem 263 Feigenbaum constant 321, 329 Gaussian pulse 439 fiber laser 441 geometrical-optics approximation 77 Fick equation 377 gradient 267 field 332 gradient dominated Hertz wave 471,476 field line 260 gradient field 154, 236 Floquet equation 49, 52, 53 gradient flow 155 Floquet matrix theory 400 gradient system 154, 177, 267 Floquet solution 48 gravity wave on a deep liquid 364 Floquet theorem 409 gravity wave on a shallow liquid 363 Floquet theory 47 Green function 127, 132,378 flow against a wall 160 group dispersion 357, 362 flow around a plane plate 165 group index 357, 361, 362 flow around the circular cylinder 164 group velocity 357, 359, 361 flow without sources and sinks 155 fluid dynamics 82, 147 habitat 239 fluid kinematics 152, 259 Hamilton mechanics 92 flux 414 Hamiltonian 90, 152 focus 193 Hamiltonian flow 156 forbidden band 401, 410 Hamiltonian function 236 forbidden frequency range 50, 51, 59 HamiItonian system 91, 155, 158, 169 forbidden zone 51 harmonie 118, 134, 451 forced oscillation 111, 114 harmonie electromagnetic wave 349 forced standing waves 474 harmonie excitation 111, 290 Forsyth and lacobsthal formula 72 harmonie input 282, 288 Forsyth oscillator 40 harmonie modulation 62 Fourier analysis 282 harmonie oscillation 9, 10 Fourier integral 284 harmonie oscillator 6, 32, 470 Fourier series 120,283,455,479 harmonie wave 342 Fourier trans form 20, 44 harmonically modulated capacitance 62 Fourier transformation 20, 121,484 heat capacity 332 517 heat conduction 377 inherent degeneracy 247 heat pole 378 initial condition 105, 369 heat wave 377 initial state 17 Heaviside step function 42, 43, 124, initial value problem 33 280,488 instability theorem of Tchetayev 236 Heaviside unit step 492 instable 189 Heimholtz equation 457, 461 instable limit cyc1e 223, 228 Heimholtz resonator 462 instable oscillation 137 Helmholtz's vortex theorem 272 instationary flow 147 Hermite polynomial 39 integral transformation 127 Hertz equation 338, 369,453 integrating feedback 295, 298 Hertz equation, reduced 367, 370,416 intensity 350 Hessian matrix 178 intermittent route to chaos 322 heteropolar crystal 394 invariant 30 Hilbert transformation 389, 494 inverse Fourier transform 20, 44 HilI differential equation 47 inverse propagator 294 Hirota equation 421 inverted pendulum 64 homoeopolar crystal 391 irregular singular 77 homogeneous isotropie media 368 irregular singular point of time 69 homogeneously broadened spectralline irrotational flow 157, 273 21 irrotational system 154 Hooke's law 337 iteration 326 Hopf bifurcation 302, 314, 324 hump 432 Jacobi matrix 151,178 Hurwitz stability 253 Jacobi's elliptic function 93, 95, 99, 434 hybrid modulation 400 Jacobian 303 hyperbolic partial differential equation Jacobian elliptic eosine 93, 94 338, 429 Jacobian elliptic function 93, 95, 99, hyperbolic point 194 434 hysteresis 78, 88 Jacobian elliptic sine 100,449 Jacobian matrix 301, 303, 316 ideal flows 156 Johnson noise 286 ideal gas 332 Jones matrix 356 ideal telegraph line 375 Jones vector 355 impulse 121 Joukowski transformation 164 impulse excitation 112 jump phenomenon 136 impulse response 122 impulse response function 277, 280, KAM theorem 318 287, 294 KdV equation 379, 420 incompressib1e fluid 152,262 Kepler vortex 176 index modulation 399,405 kernell27 index of refraction 357 Kerr medium 436 index theorem of Poincare 232 Kerr-Iens modelocking 228 infinite sequence of bifurcations 321 kinetic energy 23 infrared 338 kink soliton 431 518 Kirchhoffs law 6, 25 linear media 382 Klein-Gordon equation 371,427 linear plane waves 368 Klein-Gordon wave 468 linear superposition 9 KLM laser 228 Iinearity of wave equation 341 Korteweg-de Vries equation 379, 415, linearly dependent 32 420 linearly polarized 351 Korteweg-de Vries soliton 423 Liouville-Neumann series 132 Korteweg-de Vries wave 425 LO wave 396 Kramers-Kronig relations 388 loeal rotation 152, 263, 265 loeal time derivative 262 LA wave 394, 396 logistie map 326 Lagrange function 244 logistie model, eontinuous 238, 239, Lagrange mechanics 244 326 Lagrange stability 133, 142 logistie parabola 327 Laguerre differential equation 256 longitudinal 333, 334 Laguerre oscillator 41 longitudinal aeoustie (LA) 394, 396 Laguerre polynomial 41 longitudinal optieal (LO) 396 laminar flow 24 longitudinal wave 335, 340, 391 Landau-Hopfmodel320 Lorentz dispersion 21, 387 Laplace equation 155, 157, 158 Lorentz funetion 43 Laplace integral 123 Lorentz line shape 21, 387 Laplace operator 152 Lorenz attraetor 325 Laplace transformation 44, 123, 124, Lorenz equation 323 282, 291, 293, 295, 487 Lorenz model 322 laser 318 Lotka-Volterra model 238, 240 laser photoacousties 462 Lotka-Volterra system 242 Laurent matrix series 251 L TI system 277 Laurent series 69 Lyapunov exponent 319 LC eireuit 6, 62 Lyapunov funetion 133, 233, 236 LCR eireuit 25 Legendre differential equation 256 magnetie energy 26 Legendre function 464, 465 magnetic permeability 338, 383 Legendre polynomial 465, 486 Malthus model 238, 239 Legendre's elliptie integral 93 mathematical pendulum 7, 98 Levinson and Smith theorem 90, 104 Mathieu differential equation 62 Lienard equation 91, 103, 133 Mathieu function 62 Lienard oseillator 89, 134 Matthieu differential equation 406 light 338 Maxwell equations 338, 383, 398 light veetor 351 Maxwell-Bloch equations 442 limit eyc1e 90,105,106,180,223,315, Maxwell-Bloch model 318, 323 317 mean square 284 limit eyc1es, existenee of 224, 231 medium without dispersion 390 linear ehirp 36 medium without loss or gain 345 linear dependenee 31 membrane 456 linear independenee 9 merry-go-round 171, 263 519 method by averaging 105 pendulum inverted 64 method of characteristics 413 period 9 microwaves 338 period doubling 130,329 mode splitting 320 period tripling 130, 134, 139, 146 modulated linear oscillator 30 periodic distortion 55, 400 modulation 47, 63 periodic excitation 120 modulus of elasticity 332 periodic iterative sequence 327 molecular vibration 246 periodic media 391,398 momentary intensity 350 periodic perturbation 55, 400 momentary rotation 264 periodic potential 408 momentum-energy relation 380 periodic pulse modulation 52, 53, 56 monochromatic 20 periodic step modulation 46 periodically modulated oscillator 130 Newton's second law 7, 24 periodicity 14 NLS equation 436 permeability 338, 386 nodallines 452 permittivity 338, 386 node 178, 180, 193,452 perturbation vector 149 non-autonomous 247 phase 10 non-causal system 123 phase diagram 12 non-degenerate 199 phase dispersion 357, 360 non linear delay circuit 298 phase lag 114, 115, 116 nonlinear optical refractive index 436 phase plane 334 nonlinear phenomena 117 phase shift 61 nonlinear spring 447 phase space. 12 nonlinear transformation 28 phase velocity 356, 358 normal dispersion 357 phase-frequency relation 135 Nyquist diagram 117 phasor 282 Nyquist noise 286 physical-optics approximation 77 piston engine 228 order in chaos 330 pitchfork bifurcation 308, 311, 315, orthogonal system 469 324, 329 oscillating pulse 433 planar flow 152 oscillating resonant circuit 291 Planck's constant 76, 349 oscillation, absence of 35 Planck's law 75 oscillation condition 292 Planck's relations 349 oscillation mode 462 plane flow 274 oscillations of continuous media 347 plane waves 334 oscillatory behavior 34 Poincare and Bendixon theorem 231 overtone 451 Poincare sphere 353 Poincare-Lindstedt-Lighthill parametric oscillator 47, 400 approximation 96, 101, 105, 138 parametric resonance 47, 52, 60, 64 point of time 65 particle velocity 381 point source or sink 268 Pauli spin matrices 355 Poisson equation 170, 177, 182, 267 pendulum 7 Poisson integral 268, 271 520 polar coordinates 153 real dispersion relation 345 polarization 351, 386, 443 real representation 10, 342 polarization vector 336 rectangle function 485, 493 polarizer 356 rectilinear flow 159 pole 43 rectilinear source or sink 160, 182 population dynamics 238 rectilinear vortex 162 population inversion 444 regular point of time 66 potential 90, 152, 302 regular singular point of time 67 potential energy 23, 92 regularly singular 251 potential flow 158, 267 relative permeability 383 power spectral density 285 relative permittivity 383 power spectrum 285, 319 relativistic particle 372 Poynting relation 350 relaxation 281 Poynting vector 350 relaxation function 281, 294 Prandtl's boundary layer 173 relaxation oscillation 108 predator and prey 240 relaxation time 444 press ure wave 332 relay 77 principle of unperturbed propagation 341 resolvent 248 probability density 382, 408 resonance 111, 115, 451 projection matrix 356 resonance width 115 propagation constant 346, 401 resonant mode 455 propagator 16, 18, 33,45, 58, 195 retarded time 437 propagator matrix 11,48,53 Riccati differential equation 28, 29, 30 propagator of pulse modulation 47 Riccati transformation 29, 31 propagator representation 45 rigid body 264 propagator solution 248 ripples 364 proper rotation 263 Rogowski profile 169 Prüfer substitution 256 rotating pendulum 31 1 pseudo-soliton 441 rotating system 255 pulse modulation 46 rotation 152, 246 pulse-area theorem 445 Routh-Hurwitz conditions 275 pure rotation 171 Ruelle-Takens-Newhouse model 320 Q - factor 14 saddle 194, 254 quadratic system 198 saddle point 170, 178 quality factor 14,22,25,27, 115 saddle-node bifurcation 303 quasi-circular frequency 15 sampling function 491 quasi-period 15 sawtooth 480 scalar wave 331, 340 Rabi circular frequency 444 Scherrer's Hamiltonian system 212 radial flow 181 Schrödinger envelope equation 367 radial gradient 181 Schrödinger equation 349, 367, 380, radial velocity 153 382,408,435,451,469 radiowaves 338 secular equation 196,245,274,324 Rayleigh equation 103 self similarity 68, 71 521 self trapping 436, 439 spiral 193 self-adjoint differential equations 256 SPM 438 self-adjoint system 255, 467 spring-mass system 7, 23 self-induced transpareney 443 square wave 480 self-phase modulation 438 square-wave generator 108 semistable limit eycle 223, 229 square-wave modulation 57, 60, 403 shah funetion 410, 491 stability 188, 252, 301, 328 shah potential 410 stability eriteria 232, 238, 275 shear flow 173 stability eriteria of Hurwitz 275 shear modulus 337 stability eriteria of Lyapunov 232 shear point 194 stability diagram 189 shoek exeitation 112, 121 stability theorem of Lyapunov 234 shoek response 122 stabilization 27 shoek wave 418,419 stable 189 Shohat approximation 106 stable limit eycle 223, 225 Shohat's method 106 stable oseillation 137 short-cireuit eonditions 475 standing wave 346,347,451 sign funetion 42 standing-wave pattern 452 silent zone 141, 143, 145 star 194, 254 similarity faetor 69, 71 static friction 80 similarity transformation 186, 198 stationary flow 149 simple pendulum 90, 98 stationary motion 127 Sine-Gordon equation 429, 471 statistical measures 284 Sine-Gordon soliton 430 step exeitation 112 singular point 149, 178,301,302 step modulation 44 singularity 65 step response 280 sink 152 Stokes veetor 354 skew-symmetric matrix 264 Stokes' law 24, 163, 170, 174 Smith and Levinson theorem 101 Stokes' theorem 266 Smith oseillator 90, 101 strange attraetor 320, 325 solid top 264 stream funetion 156, 159, 170 solitary solution 428 streamline 155, 159, 259, 260 solitary wave 413, 414, 422 strietly stable 252 soliton 413, 416, 422 string 451 sound 340 strueturally instable equilibrium 313 souree 152 Sturm's eomparison theorem 34 souree or sink density 262 Sturm's separation theorem 34 souree or sink strength 152, 263 Sturm-Liouville eonditions 453, 456 spatial period 398 Sturm-Liouville equation 466, 473 speetral analysis 20 Sturm-Liouville system 467,469 speetral half-width 22 Sturm-Liouville wave pattern 466 spherieal Bessel funetion 463, 464 suberitieal 15, 113, 122, 375 spherieal harmonie 465 subharmonic 118, 134, 138, 146, 329 spherieal wave 460 substitution, sueeessive 132 spin matrices 355 supereritieal 18, 113, 122,375 522 superposition law 28, 29 transient waves 389 superposition of incoherent polarized translation 246, 265 light beams 355 translation operator 392, 395 superposition of two coherent polarized transmission line 374 light beams 355 transversal 334 superposition principle 27, 31 transversal electromagnetic wave 338 superposition principle of Huygens 341 trans verse 333 surface waves of liquids 363 trans verse velocity 153 swing 59 transverse wave 335 switch 77 traveling wave 346, 421, 429 switching 78, 123 tri angle function 485, 493 switching functions 78 triangle wave packet 417 system of differential equations 147 two-Ievellaser 323 system operator 276 two-Ievel system 443 system parameter 300, 311 two-soliton solution 424 system propagator 195,248 ultraharmonic 118, 134 tandem connection 278 uItraharmonic osciIIations 137 tandem equations 291 uItraviolet 338 Taylor matrix series 250 unimodular 12, 33, 45 Taylor profile 419 uniqueness theorem 33 Taylor series 66, 73, 158, 302 unit impulse 43, 112, 121 telegraph equation 374, 475 unit shock 121 telegraph li ne 375, 475 unit step 42 TEM wave 339, 349 unit-step excitation 125 temperature wave 377 unit-step response 280, 294 Thomson's theorem on the permanence units 4 of circulation 272 up-chirp 35 time contraction 108 time dilatation 109 vacuum wavelength 357 time-dependent flows 261 van der Pol equation 103, 141, 142 time-dependent wave velocity 371 van der Poloscillator 90, 103, 141,316 time-invariant system 276 variation ofparameters 128 Toda chain 447 variation of the constant 257 Toda oscillator 318 vector equation 147 Toda soliton 448 vector potential 270 torsional wave 337 vectorial 333 total energy 23, 24 vectorial wave 331 trace 48 velocity field 149 trajectory 12, 13, 259 velocity field 259, 273 transcritical bifurcation 306, 328 velocity of light in vacuum 338 transfer differential equation 287 vibration 246 transferfunction 117, 120 vibration of astring 453 transfer system 276 viscosity 24 transient 111, 113, 124, 291, 476 vital space 239 523 Volterra integral equation 132 wave without dispersion 345, 416 Volterra kernel 277 waveguide 372 Volterra series 289 Weber function 40 Volterra system 276 Weber oscillator 39 vortex 155, 170, 176, 194, 255, 263 weight function 453 vortex filament 162, 175,271 Wentzel-Kramers-Brillouin vortex motion 272 approximation 75 vortex strength 162, 163,266 white noise 286 vorticity 266 Whittaker function 42 Whittaker oscillator 42 Watt's centrifugal governor 311 Wiener-Khintchine theorem 285 wave damping or gain 346 WKB approximation 75 wave form 452 work 61 wave function 381 Wronski determinant 31,32, 127,258 wave impedance of vacuum 339 Wronski matrix 249, 257 wave in aspring 333 wave in a string 333, 474 X-rays 338 wave in a thin solid rod 337 wave in an isotropie solid 336 Young's modulus 337 wave mechanics 348, 408, 468 wave on a membrane 456 zero rotation 157 wave packet 381 zeros 34 wave with dispersion 357, 361, 363, 379,420 Springer and the environment At Springer we firmly believe that an international science publisher has a special obligation to the environment, and our corporate policies consistently reflect this conviction. 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