Nonlinear_Schodinger_Equation_

Nonlinear_Schodinger_Equation_ - P r,t ˆ FE r t(1.1 E P ˆ...

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Unformatted text preview: P r ,t ˆ FE r , t (1.1) E P ˆ F E Pi r , t 1 ij Ej 2 ijk E j Ek 3 ijkl E j E k El Ei Pi J 0 0 E D 1B ct 0 B H 1D ct B D 0 D M 0 E 4P B H 4M M H 2 E E 1 c2 2 DL t2 4 c2 2 P NL t2 0 P L (1) E D L ˆE E 4P L ˆ E E ˆ ˆ D 0 E E 0 2 E 1 c2 2 DL t2 4 c2 2 P NL t2 t 0 E x, y , z , t A x, y, z, t exp ik 0 z i k0 0 A 0 0 D j x, y , z , t t t ' E j x, y, z, t ' dt ' t t' t t' exp i t t' d A x, y , z , t A x, y , z , t A x, y , z exp i td D D x, y , z , t exp(ik 0 z ) dt ' d exp i t t' d ' A x, y, z ' exp i ' 0 t' exp i 1t ' dt ' 1 ' 0 D x, y , z , t exp(ik 0 z i 0 t) 0 A x, y , z exp i td 1 c2 2 D 2 t exp(ik 0 z i 0 t) k 2 0 A x, y , z exp i td 2 k2 0 0 c2 0 ! 0 k2 0 1 c2 2 D 2 t exp(ik 0 z i 0 ' t ) k 02 A 2ik 0 k 0 A t ' k0 ' k0 2 ' k 0 k 0' 2 A 2 t2 k 0'' k 2 0 k0 k( 0 ), k 0 v g1 " vg 2ik 0 A z 2 A 2 2 A 2 2 A 2 x y z ' 2ik 0 k 0 A t k 0' 2 ' k 0 k 0' 2 A 2 4 2 0 t c 2 xxxx AA 2 0 P NL xxxx AA 2 k0 z t k 0' z A, 1 k0 A z ' k0 A t 2 A 2 z k 0' 2 2 A 2 t z 1 2A 2 y2 ' k0 t ' k 0' 2k 0 z 2 ' k0 t A A i A 1 2A 2 x2 2 2 0 22 A 2 AA 2 0 . k0 c xxxx A Ax 2 k0 x y k0 y ! ! k" # ! k" k" GVD " (group-velocity dispersion) GVD " k 0 (zero-dispersion wavelength) ! ZD GVD ! " ! ZD x,y A# . A! " k0 # " k0 k0 Bright soliton i A 1 2 2 A 2 AA 2 0 . i A 1 2 2 A 2 AA 2 0 $$ " i A 1 2 2 A 2 AA 2 0 (2.1.1) A, eik i' &% % V (2.1.2) &% " k , ', V (2.1.1) (2.1.2) 1 d 2& 2 d% 2 ik V d& d% ' k2 & &3 2 0 (2.1.3) kV 0 k2 2 ' (2.1.3) d 2& d% 2 2& 2& 3 .$ % (2.1.4) & % (2.1.5) " 0 A=$ d& d% 2 A 2 &2 &4 & % d& &2 d% &2 2 "& 2 (2.1.6) (2.1.7) (7) &2 & 2 1 2 arccosh 2 & % (2.1.8) (2.1.8) &% 2 cosh 2 % cosh V (2.1.9) (2.1.1) A, 2 cosh 2 V exp i V V2 2 V=0 (2.1.10) A, 2 cosh 2 exp i (2.1.11) (2.1.10) 0 0.5 V 10 (2.1.10) ...
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This note was uploaded on 10/13/2010 for the course PHYSICS 2238 taught by Professor Asdf during the Spring '07 term at Bar-Ilan University.

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