Abstract

We consider a stochastic nonlinear Schrödinger equation related to signal propagation in waveguides and optical fibers. We first describe the modeling of the problem and the desired objectives concerning the transmission. We then present a new multilevel numerical method for its solution, which is based on a separation between low and high frequencies. We show that this method gives results of the same quality with significantly shorter CPU time compared with those of the other numerical methods commonly presented in the literature.

© 2000 Optical Society of America

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References

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  1. B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
    [CrossRef]
  2. B. M. Herbst, J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
  3. T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
    [CrossRef]
  4. M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
    [CrossRef]
  5. C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).
  6. C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).
  7. C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
    [CrossRef]
  8. T. Dubois, F. Jauberteau, R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).
  9. C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).
  10. C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
    [CrossRef]
  11. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Series, Vol. 68 of 2nd ed. (Springer-Verlag, New York, 1997).
  12. G. Moebs, “An efficient parallelization of a multilevel split-step Fourier method for a weakly damped nonlinear Schrödinger equation,” (available from the author).
  13. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  14. A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).
  15. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  16. J.-M. Ghidaglia, “Finite dimension behavior for the weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincaré 5, 365–405 (1988).
  17. O. Goubet, “Regularity of the attractor for the weakly damped driven Schrödinger equation,” Applic. Anal. 60, 99–119 (1996).
    [CrossRef]
  18. I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
    [CrossRef]
  19. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).
  20. D. Gottlieb, S. A. Orszag, “Numerical analysis of spectral methods: theory and applications,” Vol. 26 of CBMS–NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, Pa., 1997).
  21. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  22. A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]

1998 (1)

I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
[CrossRef]

1996 (1)

O. Goubet, “Regularity of the attractor for the weakly damped driven Schrödinger equation,” Applic. Anal. 60, 99–119 (1996).
[CrossRef]

1988 (4)

C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
[CrossRef]

J.-M. Ghidaglia, “Finite dimension behavior for the weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincaré 5, 365–405 (1988).

1987 (1)

C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).

1986 (1)

B. M. Herbst, J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).

1985 (2)

B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
[CrossRef]

C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).

1984 (1)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

1981 (1)

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

1976 (1)

A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Ablowitz, M. J.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Agrawal, G. P.

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

Batteh, J. H.

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Dubois, T.

T. Dubois, F. Jauberteau, R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).

Feit, M. D.

A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, A.

A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Foias, C.

C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).

C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
[CrossRef]

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).

C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).

Ghidaglia, J.-M.

J.-M. Ghidaglia, “Finite dimension behavior for the weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincaré 5, 365–405 (1988).

Gottlieb, D.

D. Gottlieb, S. A. Orszag, “Numerical analysis of spectral methods: theory and applications,” Vol. 26 of CBMS–NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, Pa., 1997).

Goubet, O.

O. Goubet, “Regularity of the attractor for the weakly damped driven Schrödinger equation,” Applic. Anal. 60, 99–119 (1996).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).

Herbst, B. M.

B. M. Herbst, J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).

B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
[CrossRef]

Hussaini, M. Y.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).

Jauberteau, F.

T. Dubois, F. Jauberteau, R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).

Jolly, M. S.

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

Kevrekidis, I. G.

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

Kodama, Y.

A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).

Lax, M.

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

Manley, O.

C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).

C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).

Mitchell, A. R.

B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
[CrossRef]

Moebs, G.

G. Moebs, “An efficient parallelization of a multilevel split-step Fourier method for a weakly damped nonlinear Schrödinger equation,” (available from the author).

Moise, I.

I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
[CrossRef]

Morris, J. Ll.

B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
[CrossRef]

Morris, J. R.

A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Orszag, S. A.

D. Gottlieb, S. A. Orszag, “Numerical analysis of spectral methods: theory and applications,” Vol. 26 of CBMS–NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, Pa., 1997).

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).

Rosa, R.

I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
[CrossRef]

Sell, G.

C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
[CrossRef]

C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).

Sell, G. R.

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

Taha, T. R.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Temam, R.

C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).

C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
[CrossRef]

C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).

C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Series, Vol. 68 of 2nd ed. (Springer-Verlag, New York, 1997).

T. Dubois, F. Jauberteau, R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).

Titi, E. S.

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Wang, X.

I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
[CrossRef]

Weideman, J. A. C.

B. M. Herbst, J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).

Zang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).

Ann. Inst. Henri Poincaré (1)

J.-M. Ghidaglia, “Finite dimension behavior for the weakly damped driven Schrödinger equation,” Ann. Inst. Henri Poincaré 5, 365–405 (1988).

Appl. Phys. (1)

A. Fleck, J. R. Morris, M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Applic. Anal. (1)

O. Goubet, “Regularity of the attractor for the weakly damped driven Schrödinger equation,” Applic. Anal. 60, 99–119 (1996).
[CrossRef]

C. R. Acad. Sci. Ser. I: Math. (2)

C. Foias, G. Sell, R. Temam, “Variétés inertielles des équations différentielles dissipatives,” C. R. Acad. Sci. Ser. I: Math. 301, 139–142 (1985).

C. Foias, O. Manley, R. Temam, “Sur l’interaction des petits and grands tourbillons dans les écoulements turbulents,” C. R. Acad. Sci. Ser. I: Math. 307, 497–500 (1987).

J. Appl. Phys. (1)

M. Lax, J. H. Batteh, G. P. Agrawal, “Channeling of intense electromagnetic beams,” J. Appl. Phys. 52, 109–125 (1981).
[CrossRef]

J. Comput. Phys. (2)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical, nonlinear Schrödinger, equations,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

B. M. Herbst, J. Ll. Morris, A. R. Mitchell, “Numerical experience with the nonlinear Schrödinger equation,” J. Comput. Phys. 60, 282–305 (1985).
[CrossRef]

J. Diff. Eqns. (1)

C. Foias, G. Sell, R. Temam, “Inertial manifolds for nonlinear evolutionary equations,” J. Diff. Eqns. 77, 309–353 (1988).
[CrossRef]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Math. Modell. Numer. Anal. (1)

C. Foias, O. Manley, R. Temam, “On the interaction of small and large eddies in two-dimensional turbulent flows,” Math. Modell. Numer. Anal. 22, 93–114 (1988).

Nonlinearity (1)

I. Moise, R. Rosa, X. Wang, “Attractors for non-compact semigroups via energy equations,” Nonlinearity 11, 1369–1393 (1998).
[CrossRef]

Phys. Lett. A (1)

C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, E. S. Titi, “On the computation of inertial manifolds,” Phys. Lett. A 131, 433–436 (1988).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

B. M. Herbst, J. A. C. Weideman, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).

Other (8)

T. Dubois, F. Jauberteau, R. Temam, Dynamical Multilevel Methods and the Numerical Solution of Turbulence (Cambridge U. Press, Cambridge, UK, 1998).

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1988).

D. Gottlieb, S. A. Orszag, “Numerical analysis of spectral methods: theory and applications,” Vol. 26 of CBMS–NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, Pa., 1997).

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Series, Vol. 68 of 2nd ed. (Springer-Verlag, New York, 1997).

G. Moebs, “An efficient parallelization of a multilevel split-step Fourier method for a weakly damped nonlinear Schrödinger equation,” (available from the author).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, UK, 1995).

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

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Figures (7)

Fig. 1
Fig. 1

Nodal values of the initial condition u(t, z=0) with a zoom between 400 and 600.

Fig. 2
Fig. 2

Nodal values of the solution u(t, z=13).

Fig. 3
Fig. 3

Energy spectrum of u(t, z) at z=0.400 (left) and at z=0.413 (right).

Fig. 4
Fig. 4

Energy spectrum of u(t, z) at z=0.426 (left) and at z=0.439 (right).

Fig. 5
Fig. 5

Ratio of the averages of the nonlinear terms NLT2j/NLT1j.

Fig. 6
Fig. 6

Ratio of the averages of the nonlinear terms NLT2j/NLT1j.

Fig. 7
Fig. 7

Description of a V-cycle.

Tables (7)

Tables Icon

Table 1 Results with the Symmetrized Split-Step Fourier Method (key1)

Tables Icon

Table 2 Results with the Split-Step Fourier Method (key1)

Tables Icon

Table 3 Results with the Split-Step Fourier Method (key2)

Tables Icon

Table 4 Results with the MLSSF Method for N1=4096, 2048 and Δz=10-3 (key1)

Tables Icon

Table 5 Results with the MLSSF Method for N1=4096, 2048 and Δz=10-3 (key1)

Tables Icon

Table 6 Results with the (MLSSF) Method for N1=4096 (key1 and key2)

Tables Icon

Table 7 Results with the MLSSF Method for N1=2048 (key1 and key2)

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

uz-i22ut2+M2|u|2u+iαu=0.
uz=Lu+iN(u)u,
Lu=i22ut2-αu,N(u)=|u|2;
K2za0zaexp(-2αz)dz=1,i.e.,
K=2αza1-exp(-2αza)1/21.35.
u(t, z=0)=Kj=0127ajcosh(t-8j-4),
cj=8j8(j+1)(t-8j-4)v(t)dt8j8(j+1)v(t)dt.
σtg=(cj2-cj2)1/2,
Qt=0.7Tb2σtg.
Q=μ1-μ0σ1+σ0,
u(t, z=0)=Kj=0127ajcosh(t-8j-4),
u(t, z)=kZu^k(z)Φk(t),
Φk(t)=expik 2πT t.
B(t)=k=-N/2N/2-1b^kΦk(t).
b^k=r exp(iΨk),
uz=Lu+iN(u)u,
u(t, z+h)=exp(hL)exp[ihN(u(t, z))]u(t, z)
u(t, z+h)=exph2Lexpzz+hiN(u(z))dz×exph2Lu(t, z).
zz+hiN(u(z))dz=ih2 [N(u(z))+N(u(z+h))].
y˜(z+h)=y(z)+hf(y, z);
y(z+h)=y(z)+h2 [f(y, z)+f(y˜, z+h)].
UNn+1=F-1exph2LˆFexp(ih|UNn|2)F-1exph2LˆFUNn,
U˜Nn+1=F-1exph2LˆFexp(ih|UNn|2)×F-1exph2LˆFUNn;
UNn+1=F-1exph2LˆFexpih2 (|UNn|2+|U˜Nn+1|2)×F-1exph2LˆFUNn.
u=v+w,
v=|k|N1/2u^kΦk(t),w=N1/2<|k|<N/2u^kΦk(t),
Φk(t)=expik 2πT t.
i uz+122ut2+iαu=0,α=17.
|w^lin(k)| 0.9832|w^nlin(k)|forN1/2< |k| <N/2,
uz=Lu+iN(u)u,
Lu=i22ut2-αu,N(u)=|u|2.
v=PN1u=|k|N1/2u^kΦk(t);
w=QN1u=N1/2<|k|<N/2u^kΦk(t);
IN1(u)(tj)=u(tj).
PN1uz=PN1Lu+PN1{iN(u)u};
vz=Lv+iPN1{N(u)v}.
N(u)=N(v+w)=|v+w|2=|v|2+(|u|2-|v|2)=N(v)+[N(u)-N(v)].
PN1{[PN1N(v)+PN1{N(u)-N(v)}]v}.
IN1vz=IN1Lv+iIN1{PN1{PN1N(v)+PN1{N(u)-N(v)}}v}.
uz=Luwith Lu=i22ut2-αu.
QN1uz=QN1Lu,
wz=Lw.
IN1vz=IN1Lv+iIN1{PN1{PN1N(v)+PN1{N(u)-N(v)}}v},
wz=Lw.
NLT1=PN1N(u(t, z))
NLT2=PN1N(v(t, z))+exp(-2αδz)PN1N(vw(t, z0)),
N(vw(t, z0))=N(u(t, z0))-N(v(t, z0))
NLT1j=IN1{PN1N(u(t, z))}j=1zaz0z0+zaIN1{PN1N(u(tj, z))}dz,
NLT2j=IN1{PN1N(v(t, z))+exp(-2αδz)PN1N(vw(t, z0))}j=1zaz0z0+zaIN1{PN1N(v(tj, z))+exp[-2α(z-z0)]PN1N(vw(tj, z0))}dz.
PN1{N(u)-N(v)}.
nbiter1+(numberofV-cycles)(nbiter2+nbiter3)
=40.
nbiter1+(numberofV-cyles)(nbiter2+nbiter3)
=zaΔz.
nbiter1+nbiter2+nbiter3=zaΔz.

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