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­ .., .., .., .. () «» (). . - , . COMPUTER SIMULATION OF ASYMPTOTIC NLSHSOLITON IN EXTERNAL FIELDS OF SPECIAL FORM Borisov A.V., Kistenev Yu.V., Trifonov A.Yu., Shapovalov A.V. (Tomsk) Soliton dynamics in a "strong" external fields of special forms is considered for the nonlinear Schrodinger equation (NLSE) with external field. The initial condition is differ from the standard NLSE one-soliton form and is determined by the external field. Both computer simulation and asymptotic WKB-type method developed by authors is applied. ()
ih,t + h2 , 2
xx

+ 2g = 0 ,
176

2

(1)


.. . -- -10, 2002, .176-177

. ( x, t ) ­ , , x ; g > 0 . h . (1) () (., , [1]). ( + i ) , (1), [1]
s ( x, t ) = 1 2 i exp 2( 2 - 2 )t + 2x + 0 . (2) h 2 g ch[ 1 (2 ( x - x ) - 4t )] 0 h

[

]

(1) , v ( x, t ) , :
ih, t + h2 2 , xx + 2 g - v ( x, t ) = 0 . 2

(

)

(3)

(3) , - . v ( x, t ) . , , (. [2]). «» . [3-4] h , h 0 , . 177


(2) v ( x ) . (3). , [2] , (2) (1) t = 0. v ( x, t ) , , .. v ( x, t ) 0 (3) , , , . , , (1), t = 0 (2) ( t = 0 ). (3) (2) (1). (3) , v 0 (2) (1). (3) , v ( x, t ) . v-. v- , , [3], v- (3), h , h 0 , , x, v ( x, t ) = v ( x ) . , (3) , . , , «» v( x ) v- (3). v- .
178


.. . -- -10, 2002, .176-179

. h ( h 0 ) v - (3), [3], O ( h 2 ) h ( h 0 )
(x, t , h ) = (
( 0) ,x

)

2

2g

i exp S h

(

( 0)

(x, t )

+ hS

(1)

(x, t ))

cosh

.

(4)

1 = ( 0) (x, t ) + (1) (x, t ) ­ « »; h S (k ) ( x, t ) , (k ) ( x, t ) k = 0,1 ,


S

( 0)

( x, t ) =4

-



x

1 1 V ( x ) - 2 dx - t + 2 ( x - x0 ), 0

( 0)

( x, t ) = 2( 2 - 2 ) t +
,x

-

(V0

x

( x ) - 2 )dx + 2 x + 0 ,


S,

(1)

=-


2 q( x )

(V0 )

, xx

(48 2 2 V0- 3 + V0 ) ,

x

(1)

= 4


q( x )

(V0 )

-1 , xx V0

.

(5)

, 0 ( (4) , 0 ) :
q ( x ) = 16 2 2V0-2 + V02 , = sign .

(6)

V0 ( x ) v ( x )
8 2 2 1 2 - V0 ( x ) + 2( 2 - 2) . V02 ( x ) 2 v( x ) =
V02 ( x ) = -v ( x ) - 2 ( 2 - 2) +

[v

( x ) + 2( 2 - 2 ) + 16 2

]

2

2

.

(7)

179


v ( x ) 0 S (k ) ( x, t ) , (k ) ( x, t ) (4) (2), (1). , v (4)
(x, t , h ) = 1 4 2 g V0 ( x ) i exp S (x h cosh , t ) .

S (1) = (1) = 0

S (0 ) = 2 2 - 2 t + 2x + 0 , (0 ) = -4 t + 2 (x - x0 ) ,

(

)

(8)

« » ­ ­

( x, t ) =

4 h



-



x

2 1 1 dx - t + ( x - x0 ) + - V ( x ) 2 h 0
x

(1)

( x ),

(9)

v-
S ( x, t ) = 2( 2 - 2 ) t +

-

(V0

( x ) - 2 )dx + 2 x + 0 + hS (1) ( x ) .
-1

(10)

h (1) V ( x ) = V0 ( x ) 1 + 4 V0 ( x ) ( x ) = - ,t , x v- ( ( x, t ) = const , x = x (t ) ­ , V ( x (t )) ­ ). V0 ( x ) ­ h ( h 0 ) . V0 ( x ) 2 v ( x ) 0 . , v ( x ) v- ­ (2) 2 . v- (3). , (3) ( x,0) , (4-(10). -



180


.. . -- -10, 2002, .176-181

(4-(10) v- t > 0 , (9, (10) . . x 0.05, t 0.0000025. g = 1 , h = 1 , (1) , S (1) . ( x,0) (9)-(10) v( x ) . ( = (3).
2

-



( x, t ) dx ), -

2

. 1. : v( x ) = 10 exp( - x 2 ) , = 3, = 10-7

. 1 v- (3) v ( x ) = 10 exp( - x 2 ) . . 1 . , . , . ( t ) , (
181


t ). v-, .

. 2. : v ( x ) = -10 exp( - x 2 ) , = 4, = 10-7

. 2,3 v ( x ) = -10 exp( - x 2 ) . . 2,3 . . 2 . . (. 2) , ( ).

. 3. : v ( x ) = -10 exp( - x 2 ) , = 2, = 0,1

. 3 , ( "" («» ) ). .
182


.. . -- -10, 2002, .176-183





. 4. : v ( x ) = 1 - cos(10 x ) , = 1,5, = 3

.4 v (3) v ( x ) = 1 - cos (10 x ) . . 4 . , t . , . . 1. .., .., .., .. : . -- .: , 1980. 2. Kivshar Yu.S., Malomed B.A. Dynamics of solitons in nearly integrable systems.// Rev. Mod. Phys. 1989. V.61. No. 4. P.763-915. 3. Shapovalov A.V. and Trifonov A.Yu. Semiclassical Solutions of the Nonlinear Schr\"odinger Equation// J. Nonlin. Math. Phys. 1999, V.6, No.2. P.1-12. 4. Belov V.V., Trifonov A.Yu., and Shapovalov A.V. The Trajectory-Coherent Approximation and the System of Moments for the Hartree Type Equation// Int. J. Math. Math. Sci. (USA). 2002. V.32. No.6. P.325-370.

183