Abstract

We consider the propagation of periodic waves with initially narrow spatial spectra in a Kerr medium. The set of constants of motion closely related to the order amplitudes is introduced. It is shown that the spatial spectrum remains uniformly narrow with propagation for both self-focusing and self-defocusing nonlinearity. In addition, for sufficiently weak nonlinearity, initially strong orders always remain strong. Thus the problem is shown to be essentially finite dimensional and well approximated by a proper set of coupled ordinary differential equations.

© 2004 Optical Society of America

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References

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J. Math. Phys. (1)

M. J. Ablowitz and J. F. Ladik, �??Nonlinear differential-difference equations and Fourier analysis,�?? J. Math. Phys. 17, 1011-1018 (1976).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Fluids (2)

A. Thyagaraja �??Recurrent motions in certain continuum dynamical systems,�?? Phys. Fluids 22, 2093-2096 (1979).
[CrossRef]

D. U. Martin and H. C. Yuen �??Spreading of energy in solutions of the nonlinear Schrödinger equation,�?? Phys. Fluids 23, 1269-1271 (1980).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, �??Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media,�?? Sov. Phys. JETP 61, 62-69 (1972).

Stud. Appl. Math. (1)

Y. C. Ma and M. J. Ablowitz, �??The periodic cubic Schrödinger equation,�?? Stud. Appl. Math. 65, 113-158 (1981).

Other (1)

L. A. Takhtajan and L. D. Faddeev Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).

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Figures (2)

Fig. 1.
Fig. 1.

Open squares, maximal intensity in kth diffraction order along propagation distances 1, 10, 100, 1000, and 10000 (lower to upper traces). The zeroth and the first diffraction orders with equal amplitudes (1/2)1/2 are taken initially. Solid squares, initially there are three orders, the zeroth and the first with equal amplitudes (0.8)(1/2)1/2; the second order 0.6i, propagation length 1000. For both cases K=1,ϰ=-0.3,ν=-1.π 2.

Fig. 2.
Fig. 2.

Evolution of the zeroth order in the phase plane for propagation length 500. Points are taken every 0.1 along the propagation distance. The initial amplitude of orders are the zeroth and the first with equal amplitudes (0.8)(1/2)1/2; the second order 0.6i, K=1,ϰ=-0.3,ν=-1.2π 2.

Equations (35)

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i ψ t = 2 ψ x 2 + 2 ϰ ψ 2 ψ .
F ( x , t , λ ) x = U ( x , t , λ ) F ( x , t , λ ) ,
U ( x , t , λ ) = ( λ 2 i ϰ ψ ¯ ( x , t ) ϰ ψ ( x , t ) ψ 2 i ) .
T ( x , y , λ ) = E ( x y , λ ) + y x E ( x z , λ ) U 0 ( z ) T ( z , y , λ ) dz ,
E ( x y , λ ) = ( exp ( λ 2 i ( x y ) ) 0 0 exp ( λ 2 i ( x y ) ) ) ,
U 0 ( z ) = ϰ ( 0 ψ ¯ ( z ) ψ ( z ) 0 ) .
tr ( T L ( λ ) ) 2 cos ( λ L + ϰ n = 1 I n λ n + O ( λ ) ) .
I n = L L P n ( x ) dx ,
P 1 = ψ ( x ) ψ ¯ ( x ) ,
P n + 1 = i ψ ¯ x ( P n ψ ¯ ) + ϰ k = 1 n 1 P k P n k .
I 1 = L L ψ 2 dx = 2 LI ,
I 3 = L L ( ψ x 2 + ϰ ψ 4 ) dx .
I 5 = L L ( ψ xx 2 + ϰ ( x ψ 2 ) 2 + 6 ϰ ψ x 2 ψ 2 + 2 ϰ 2 ψ 6 ) dx .
ψ ( x , t ) = k = exp ( ikKx ) ψ k ( t ) ,
L L ψ x 2 dx = 2 LK 2 k = k 2 ψ k ( t ) 2 I 3 ;
ψ k ( t ) 2 I 3 2 LK 2 k 2 .
Φ ( x ) = x 0 x ψ ( z ) 2 dz .
x 0 x 0 + 2 L ψ 4 dx = Φ ψ 2 x 0 x 0 + 2 L x 0 x 0 + 2 L Φ ψ x ψ ¯ dx x 0 x 0 + 2 L Φ ψ ψ ¯ x dx .
L L f ( x ) g ( x ) dx 2 L L f ( x ) 2 dx L L g ( x ) 2 dx .
x 0 x 0 + 2 L Φ ψ x ψ ¯ dx x 0 x 0 + 2 L Φ 2 ψ 2 dx x 0 x 0 + 2 L ψ x 2 dx ( 2 LI ) 3 2 L L ψ x 2 dx .
L L ψ x 2 dx ( I 3 + ϰ 2 LI 2 + ϰ 2 ( 2 LI ) 3 + ϰ ( 2 LI ) 3 2 ) 2 = A 1 2 .
k = ψ k 2 I k 2 < ν 2 π 2 ,
L L ψ xx 2 dx I 5 + 10 ϰ ( A 1 2 I + 2 A 1 3 I 1 2 ) + 2 ϰ 2 ( Q 1 I + 2 A 1 Q 1 I 1 2 ) .
μ k ( z ) = exp ( i λ k z ) ψ ( z ) ,
tr ( T L ( λ k ) ) = J k = ( 1 ) k 2 Re [ 1 + G 1 + G 2 + ]
G 1 = ϰ L L dz μ k ¯ ( z ) L z dz μ k ( z ) ,
G 2 = ϰ 2 L L dz μ k ¯ ( z ) L z dz μ k ( z ) L z dz″ μ k ¯ ( z ) L z dz ″′ μ k ( z ″′ ) ,
Re ( G 1 ) = 2 ϰ L 2 ψ k 2 .
G n ν n ( 2 n ) ! .
ψ k 2 I ν 6
ψ k ( t ) = ψ k ( t ) exp ( ik 2 K 2 t ) .
x 0 x 0 + 2 L ψ 6 dx = Φ 1 ψ 2 x 0 x 0 + 2 L x 0 x 0 + 2 L Φ 1 ψ x ψ ¯ dx x 0 x 0 + 2 L Φ 1 ψ ψ ¯ x dx .
φ 1 ( i 1 ) , , φ p ( ip ) ,
L z dz ″′ μ ( z ″′ ) 2 L z dz ″′ μ ( z ″′ ) 2 L z dz ″′ 1 2 LI ( z + L ) .
L z dz μ ¯ ( z ) L z dz ″′ μ ( z ″′ ) 2 L z dz μ ( z ) 2 L z dz L z dz ″′ μ ¯ ( z ″′ ) 2 ( 2 LI ) 2 ( z + L ) 2 2 ! ,

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