Short intense wave packets in smoothly inhomogeneous media

28 June 1999

Physics Letters A 257 Ž1999. 182–188

Short intense wave packets in smoothly inhomogeneous media E.M. Gromov 1, V.V. Tyutin, D.E. Vorontzov Institute of Applied Physics, Russian Academy of Science, 46 UljanoÕ Str., 603600 Nizhny NoÕgorod, Russia Received 2 April 1999; accepted 27 April 1999 Communicated by V.M. Agranovich

Abstract The propagation of short Žof the order of a few wavelengths. intense wave packets in smoothly inhomogeneous media with an arbitrary profile of the potential is considered within the framework of the third-order nonlinear Schrodinger ¨ equation with inhomogeneous potential. An equation for trajectories of the center of mass of a packet is studied under the Hirota conditions. Parabolic and periodic profiles of inhomogeneity are considered in detail. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Short wave packets; Trajectories of motion; Center of mass; Nonlinearity; Dispersion; Nonlinear dispersion; Inhomogeneity

The propagation of intense high-frequency wave packets F s c Ž x,t .exp Ž i v 0 t y ik 0 x . with small temporal and spatial widths in inhomogeneous dispersive media can be described to the third-order approximation of the nonlinear wave dispersion theory w1,2x. In this approximation, the envelope c Ž x,t . is described by the third-order nonlinear Schrodinger ¨ equation ŽNSE-3. with inhomogeneous potential, 2i

ž

Ec Et

Ec q ÕgL

E 2c qq

E x2

Ex

qb c 2

2

Ec Ex

q 2 a c c q ig

E c q mc

Ex

2

/

E 3c E x3

q U Ž x . c s 0,

Ž 1. ÕgL s

where Ž EvrE k . is the linear group velocity of the wave with the wave number k and fre-

quency v satisfying the nonlinear dispersion law 2 v s v Ž k , c ,U Ž x . . , q s y Ž E 2vrE k 2 . is the parameter of the second-order linear dispersion, a s Ž EvrE c 2 . 1is the parameter of cubic nonlinearity, and g s y 3 Ž E 3vrE k 3 . is the parameter of the third-order linear dispersion Žlinear aberration.. The last two terms in brackets in Eq. Ž1. Žwith parameters b and m . correspond to the dependence of the local 2 group velocity on the wave intensity c Žnonlinear dispersion.. The term with m is sometimes referred to as the Raman term: it corresponds to the nonlinear induced scattering effect. The dynamics of nonstationary wave packets within the framework of NSE-3 Ž1. was analyzed in Ref. w3x using the method of moments: the evolution of the first moment Žcenter of mass. of a packet

xŽ t . s 1

E-mail: [email protected]

1 N0

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 9 2 - 3

q`

Hy` x c

2

q`

dx ,

N0 s

Hy`

2

c dx

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

was considered. In particular, the following expressions for velocity and acceleration of center of mass of a packet were found: .

q

Õ Ž t . s x Ž t . sÕgL q

bq2m q 2 N0 3g y 2 N0

Ec

Hy`

..

1 2 N0

q`

Hy` q`

aŽ t . s xŽ t . s

q

N0

q`

Hy`

Ex

2

d

c dx

Ex

xŽ t.



dt

Ž E UrE x . x Ž t .

dx ,

E 2w

q`

Hy`

4

c dx

E x2

q y 3g

Ex

Ew

ž

/

2

c dx ,

2

c

Et Ex

Ew

ž

E

/

2

ž / ž Ex

xŽ t .

qy

3g N0

q`

Hy`

Ew Ex

2

s

E

Ew

ž

c

Et Ex

/

E

2

/

dx.

Ew

ž

c

4

c

4

Ew E Ex

Ž Ex

2

Ec

ž /

. yq E x

Ex Ex

ym

Ex

/ 1 EU

.q 2

Ex

2

c .

Ž 5. The substitution of Ž5. into Ž4. and integration of the equation obtained under the condition m s 0 and zero condition at the infinity, c Ž x ™ "`,t . ™ 0, give an equation for packet trajectories ..

d



dt

xŽ t.

Ž E UrE x . x Ž t .

0

3g E U

ž /

sy

Ex

4

.

Ž 6.

xŽt.

The initial conditions for Eq. Ž6. are the following: x 0 s x Ž 0. , .

Õ 0 s x Ž 0 . sÕgL q

b q 2 N0

q`

q N0

q`

Hy`

Ew

q`

ž

Hy` 4

Hy` Ž c .

3g 2 N0

c dx .

4

c

Ž

4 Ex yb

y xŽ t . s

2 N0

a E

Ž 2.

In order to find the trajectories of wave packets we will analyze the relation Ž2. under the condition 3 ag y q b s 0 and consider smoothly inhomogeneous media, for which the scale of potential inhomogeneity L ; Ur Ž E UrE x . is much larger than the packet’s width D: L 4 D. By factoring Ž E UrE x . outside the integral sign at the point of the center of mass x Ž t . , we obtain the relation Ž2. in the form 1 EU

E

q`

Hy`

Multiplying Ž1. by c ) , where c ) is the complex conjugate of the field c , and adding the resulting equation to the complex conjugate yield the following relation:

where P' drdt and w is the phase of packet envelope c s c exp Ž i w . . Packet trajectories, however, cannot be found from these relations. They were found only for the linear profile U s px and under the Hirota conditions Ž 3 ag y q b . s 0, m s 0 in Ref. w4x. In this Letter we will derive an equation for the center of mass of a wave packet x Ž t . within the framework of NSE-3 Ž1. with an arbitrary profile of potential U Ž x . and under the Hirota conditions. Then, detailed analysis of trajectories in the cases of parabolic U s px 2 and periodic profiles U s psin Ž l x . will be made.

..

0

3g sy

Ž 4.

2

2 N0

Hy`

..

Ew

4

Ex

EU

nonlinear Schrodinger equation w5x. By dividing Ž3. ¨ by Ž E UrE x . x Ž t . and differentiating the obtained equation we have

c dx

3 ag y q b

q`

183

Ec

ts0

c

Ex

2

/

dx ts0

dx

2

ž / Ex

dx , ts0

..

Ž 3. In a particular case, for g s 0, Eq. Ž3. reduces to an equation for packet trajectories in the frame of the

x Ž 0. 1 EU s 2

ž / Ex

qy x Ž0.

3g N0

Ew

q`

Hy`

ž

Ex

c

2

/

dx . ts0

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

184

For a linear profile of potential U s px, Eq. Ž6. has ...

the form x Ž t . s y Ž 34 . g p 2 obtained in Ref. w4x. By multiplying Ž6. by a Ž t . r Ž E UrE x . x Ž t . and integrating the obtained relation we get the second-order equation 2

..



xŽ t.

Ž E UrE x . x Ž t .

3g

0

q 2

.

xŽ t . s d 2 q

3g 2

Õ0 s H ,

˙

For g x Ž t . < H and d s 0, from Ž9. we have .

x Ž t . , Õ 0 y 32 g 1r3  12 U Ž x Ž t . . y U Ž x 0 .

4

2r3

.

For an arbitrary value of parameter g we will consider two cases of inhomogeneity in the form of parabolic U s px 2 and periodic profiles U s psin Ž l x . .

Ž 7. Parabolic inhomogeneity. For a parabolic profile U s px 2 , Eq. Ž8. has the form

where ..

ds

x Ž 0.

.

ž

d xŽ t.

Ž E UrE x . x Ž 0 . 3g

q s

y 2

Ew

q`

2 N0

ž

Hy`

c

Ex

2

/

(

dx.

.

ž

/

s"

.

(

H y 3g x Ž t . r2

ž / Ex

dt.

Ž 8.

3n

ž

2

2H y

n2

ž

2

3r2

.

Hy

3g 2

1r2

.

xŽ t.

/

Ž 9.

/ , Õ q 2 d U Ž x Ž t . . y UŽ x . q 3g

0

½

Õ 03 2

d

d q 2

. 2

ž y/

g

ž

y 2 y Õ 0 q pd y y

2

p 2 y 2 s x 02 .

/

g

2 2 0

Ž 11 .

By multiplying Ž11. by y and integrating the equation obtained under the initial condition x Ž 0 . s x 0 we find the first integral of Eq. Ž11.

ž

In a particular case, for g x Ž t . < H and d / 0, from Ž9. we have

ž

p2 y2 .

W Ž y . s y2 y Õ 0 q pd y y ˙

.

2

yd

s " U Ž x Ž t . . y UŽ x0 . .

xŽ t.

3g

Ž 12 .

Eq. Ž11. is the equation of motion of a particle with ‘mass’ m s 2 and total energy x 02 in the effective potential

yd 3

/

xŽ t.

y s Õ 0 q 2 d py y

.

˙

3g

˙

x Ž t . s y˙ and x Ž t . s y˙ taken into account, Eq. Ž10. takes the form

xŽ t .

the sign of parameter d . By multiplying Ž8. by x Ž t . and integrating the resulting equation we obtain the first-order equation for packet trajectories Hy

Ž 10 .

With a new variable y Ž t . s H0t x Ž t . dt and with

..

EU

The sign in the right-hand side of Ž8. corresponds to

2

s "2 px Ž t . dt.

H y 3g x Ž t . r2

ts0

Further, we will consider media with g ) 0. Packet trajectories in media with g - 0 can be found from Ž7. using the transformations g ™ yg and t ™ yt. Eq. Ž7. can be rewritten in the form d xŽ t.

/

.

U Ž x Ž t . . y UŽ x0 .

5

.

2

p2 y2 .

/

Ž 13 .

Let us analyze the phase portrait of Eq. Ž12.. The phase trajectories are localized in the region W F x 02 and originate at the point Ž y s 0, y˙ s x 0 . . The topology of the phase portrait depends on the number and signs of extreme points of the effective potential well X W defined by Wy s 0: y2 y

4 d 3 pg

yy

2 Õ0 3 g p2

s 0.

Ž 14 .

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

185

The roots of Ž14. are the following: y 1,2 s

2 d 3 pg

ž ( 1"

1q

3 2

/

S ,

Ss

Õ0g

d2

.

For S ) 0, roots y 1 and y 2 have different signs, y 1 P y 2 - 0. The potential well and the phase portrait for this case are shown in Fig. 1. Curve 1 corresponds to x 02 s W Ž ymax . ) 0, 2 - x 02 s 0. For x 02 - W Ž ymax . , the motion on phase plane Ž y, y˙ . is periodic. It corresponds, taking into account x Ž t . s y, ˙ to the periodic motion of center of mass Ž x Ž t q T . s x Ž t . . . For x 02 ) W Ž ymax . , the motion of the wave packets is infinite. For y2r3F S - 0, both the roots y 1,2 have the same sign defined by the sign of parameter pd . For pd - 0, both the roots y 1,2 are negative. For this case, the potential well and the phase portrait are shown in Fig. 2. Curve 1 corresponds to x 02 s W Ž ymax . ) 0. The phase trajectory is in this case part of the separatrix Žcontinuous part of curve 1.. Curve 2 corresponds to x 02 s 0. In a particular case, Fig. 2. The effective potential well W Ž y . Ž13. and the phase trajectories of Eq. Ž12. for y1r2 F S - 0 and pq - 0.

for x 02 s W Ž ymax . s 0, curves 1 and 2 coincide. In this case the motion along one half of the separatrix from the point Ž y s 0, y˙ s 0 . is described by the relation

d yŽ t . s

pg

tan2

ž

t 2

'y pd

/

.

It corresponds to the packet trajectory

d xŽ t . s

Fig. 1. The effective potential well W Ž y . Ž13. and the phase trajectories of Eq. Ž12. for S ) 0.

(

y

g

d sinh Ž t'y pd r2 . p cosh3 Ž t y pd r2 .

'

.

For pd ) 0, both the roots y 1,2 are positive. The corresponding potential well and phase portrait are shown in Fig. 3. For S - y2r3, the effective potential well W does not have extreme points. It gives nonlocalized packet trajectories: wave packets go to the infinity both at the ‘barrier’ Ž pq ) 0 . and at the ‘hole’ Ž pq - 0 . .

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

186

into Ž16. and integration of the resulting relation under the initial condition x Ž 0 . s 0, in variable t s l t, yield . 2

žz/

2

q z 2 Ž Õ 0 q 12 d z y 18 g z 2 . s p 2 .

Ž 17 .

Eq. Ž17. is an equation of motion of a particle with effective ‘mass’ m s 2 and total energy p 2 in the effective potential well 2

W Ž z . s z 2 Ž Õ 0 q 12 d z y 18 g z 2 . .

Ž 18 .

Let us analyze the phase plane of Eq. Ž17.. The phase trajectories are localized in the region W Ž z . F p 2 and originate at the point Ž z s 0, z˙ s pl . . The topology of the phase portrait depends on the num-

Fig. 3. The effective potential well W Ž y . Ž13. and the phase trajectories of Eq. Ž12. for y1r2 F S - 0 and pq ) 0.

Periodic inhomogeneity. For a periodic profile U s psin Ž l x . , Eq. Ž8. has the form .

ž

d xŽ t.

(

/

s "plcos Ž l x Ž t . . dt.

.

Ž 15 .

H y n xŽ t .

Introducing a new variable z Ž t . s plH0t cos Ž l x Ž t . . dt and integrating Eq. Ž15. we have .

x Ž t . s Õ 0 q d z y 38 g z 2 .

Ž 16 .

The substitution of the expression ..

.

xŽ t . s "

z

(

. 2

l p 2l2 y z

ž /

Fig. 4. The effective potential W Ž z . Ž18. and the phase trajectories of Eq. Ž17. in the case of five extreme points of W Ž y . .

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

187

effective potential well W Ž z . in these maxima are the following: W Ž z 3,4 . s W "s

32 d 6 27 g 4 p 2

f"Ž S . ,

f " Ž S . s S q 43 1 " 1 q 32 S =

ž ( / S q ž 1 " (1 q S / 1 3

3 2

2

.

The corresponding potential well and the phase portrait of Eq. Ž17. are shown in Fig. 4. There are three

˙˙ l x for the Fig. 5. Trajectories of wave packets on the plane x, case of five extreme points of W Ž z . Ž18.: Ž 1 . motion to the minus infinity Žspan packets. with four turn points in a period; Ž 2 . motion between two turn points Žcapture packets.; Ž 3 . motion to the plus infinity Žspan packets. without turn points.

Ž

.

ber of extreme points of the function W Ž z . defined X by the relation Wz s 0:

ž

d

z Õ0 q

2

g zy

8

z2

/ž

Õ0 q d z y

3g 8

z 2 s 0.

/

Ž 19 .

The expression Ž19. has the following roots: z 0 s 0, z 1,2 s z 3,4 s

4 d 3 g

2d

g

Ž 1 " '1 q 2 S . , 3 2

ž 1 " (1 q S / .

For S ) y1r2 and S / 0, the effective potential well W Ž z . has five extrema. Three of them, z 0,1,2 , are the minima at which the effective potential well is equal to zero Ž W Ž z 0,1,2 . s 0 . . Two other extremum, z 3,4 , are the maxima. The values of the

Fig. 6. The effective potential well W Ž z . Ž18. and the phase trajectories of Eq. Ž17. for the case of three extreme points of WŽ z ..

E.M. GromoÕ et al.r Physics Letters A 257 (1999) 182–188

188

centers and two saddle equilibrium states on the phase plane of Eq. Ž17.. Curve 1 corresponds to p 2 ) W ", curve 2 to Wy- p 2 - Wq, and curve 3 to p 2 - W ". For these three cases, the packet trajectories on the plane x˙ , l x are shown in Fig. 5. For p 2 ) W ", the packets move to the minus infinity Žspan packets. with four turn points in a period ŽFig. 5.1.; for Wy- p 2 - Wq, the packets move between two turn points Žcapture packets. ŽFig. 5.2.; for p 2 - W ", the packets move to the plus infinity Žspan packets. without turn points ŽFig. 5.3.. For S s 0, roots z 0,2,4 are equal to zero and the effective potential well W has two minima W Ž z 0,2,4 . s W Ž z 1 . s 0 and one maximum Wqs 2 32 d 3r Ž 27g 2 p . . For p 2 ) Wq, the packets move to the minus infinity Žspan packets. with four turn points in a period; for p 2 - Wq, the packets move between two turn points Žcapture packets.. For y2r3- S - y1r2, the effective potential W has three extreme points: z 0,3,4 . The potential well and the corresponding phase portrait are shown in Fig. 6. There are two centers and a saddle equilibrium state on the phase plane of Eq. Ž17.. Curve 1 corresponds to p 2 ) W Ž z max . , curve 2 to p 2 W Ž z max . . In both cases the wave packets move to the minus infinity: with four turn points in a period

ž

/

ŽFig. 5.2. in the first case, and without turn points ŽFig. 5.3. in the second case. For S - y2r3, the effective potential well W Ž z . has only one extreme point Žminimum. z 0 s 0. There is a center equilibrium state on the phase plane of Eq. Ž17.. In this case, the wave packets move to the minus infinity without turn points.

Acknowledgements This work was supported by the Russian Foundation for Basic Research Žproject No 96 y 15 y 96592. and the Russian Ministry of Higher Education Žgrant for basic research in mathematics..

References w1x G.P. Agraval, Nonlinear Fiber Optics, Academic, Orlando, Flo., 1989. w2x A. Hasegava, Optical Solitons in Fibers, Springer-Verlag, Berlin, 1989. w3x E.M. Gromov, V.I. Talanov, Zh. Eksp. Teor. Fiz. 110 Ž1996. 137 wJETP 83 Ž1996. 73x. w4x E.M. Gromov, Phys. Lett. A 227 Ž1997. 67. w5x H.H. Chen, C.S. Liu, Phys. Fluids 21 Ž1978. 377.