Abstract

We solve the nonlinear Schrodinger equation with an unsupervised neural network with the optical axis position z and time t as inputs. The network outputs the real and imaginary components of the solution. Unsupervised training aims to minimize a non-negative energy function derived from the equation and the boundary conditions. The trained network is generalizing - a solution value is determined at any (z, t)-combination including those not considered during training. Solutions with normalized mean-squared errors of order 10-2, are obtained when the average energy is reduced to 10-2 from order 104. The NN method is universal and applies to other complex differential equations.

©2001 Optical Society of America

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References

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  1. G. Joos and I. Freeman, Theoretical Physics (Dover Publications, New York1986).
  2. G. Arfken and H. Weber, Mathematical Methods for Physicists4th Ed. (Academic Press, New York1995).
  3. W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).
  4. K. Kunz and R. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton1993).
  5. C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
    [Crossref]
  6. M. Lim and C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
    [Crossref]
  7. C. Monterola and C. Saloma, “Characterizing the dynamics of constrained physical systems with unsupervised neural network,” Phys. Rev. E 57, 1247R–1250R (1998).
    [Crossref]
  8. M. Quito, C. Monterola, and C. Saloma, “Solving N-body problems with neural networks,” Phys. Rev. Lett. 86, 4741–4744 (2001).
    [Crossref] [PubMed]
  9. B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
    [Crossref] [PubMed]
  10. G. Agrawal and R. Boyd, Contemporary Nonlinear Optics, (Academic Press, New York1992).
  11. D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
    [Crossref] [PubMed]
  12. M. Saffman, “Self-Induced Dipole Force and Filamentation Instability of a Matter Wave,” Phys. Rev. Lett. 81, 65–68 (1998).
    [Crossref]
  13. S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
    [Crossref]
  14. J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
    [Crossref]
  15. D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
    [Crossref]
  16. W. Fushchych and A. Nikitin, “Higher symmetries and exact solutions of linear and nonlinear Schrdinger equation,” J. Math. Phys. 38, 5944–5959 (1997).
    [Crossref]
  17. S. Haykin, Neural Networks: A Comprehensive Foundation2nd Ed. (Prentice-Hall, New York1999).
  18. D. Saad and S. Solla, “Exact solution for on-line learning in multilayer neural networks,” Phys. Rev. Lett. 74, 4337–4340 (1995).
    [Crossref] [PubMed]
  19. J. Yam and T. Chow, “A weight initialization method for improving training speed in feedforward neural networks,” Neurocomputing 30, 219–232 (2000).
    [Crossref]
  20. Z. Luo, “On the convergence of the LMS algorithm with adaptive learning rate for linear feedforward neural networks,” Neural Computation 3, pp. 226–245 (1991).
    [Crossref]
  21. J. Satsuma and N. Yajima, “Initial Value Problems of One-dimensional Self-Modulation of Nonlinear Waves in Dispersive Media,” Suppl. Prog. Theo. Phys. 55, 284–295 (1974).
    [Crossref]
  22. M. Soriano and C. Saloma, “Improved classification robustness for noisy cell images represented as principal-component projections in a hybrid recognition system,” Appl. Opt. 37, 3628–3639 (1998).
    [Crossref]
  23. M. Soriano, M.L. Garcia, and C. Saloma, “Fluorescent image classification by major color histograms and a neural network,” Opt. Express 8, pp. 271–277 (2001), http://www.opticsexpress.org/oearchive/source/30248.htm
    [Crossref] [PubMed]
  24. S. Saarinen, R. Bramley, and G. Cybenko, The numerical solution of the neural network training problems, CRSD Report 1089 (Center for Supercomputing Research and Development, University of Illinois, Urbana1991).
  25. T. Poggio and F. Girosi, “A Sparse Representation for Function Approximation,” Neural computation 10, 1445–1454 (1998).
    [Crossref] [PubMed]

2001 (2)

2000 (1)

J. Yam and T. Chow, “A weight initialization method for improving training speed in feedforward neural networks,” Neurocomputing 30, 219–232 (2000).
[Crossref]

1998 (8)

T. Poggio and F. Girosi, “A Sparse Representation for Function Approximation,” Neural computation 10, 1445–1454 (1998).
[Crossref] [PubMed]

M. Soriano and C. Saloma, “Improved classification robustness for noisy cell images represented as principal-component projections in a hybrid recognition system,” Appl. Opt. 37, 3628–3639 (1998).
[Crossref]

M. Lim and C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[Crossref]

C. Monterola and C. Saloma, “Characterizing the dynamics of constrained physical systems with unsupervised neural network,” Phys. Rev. E 57, 1247R–1250R (1998).
[Crossref]

M. Saffman, “Self-Induced Dipole Force and Filamentation Instability of a Matter Wave,” Phys. Rev. Lett. 81, 65–68 (1998).
[Crossref]

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
[Crossref]

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

1997 (1)

W. Fushchych and A. Nikitin, “Higher symmetries and exact solutions of linear and nonlinear Schrdinger equation,” J. Math. Phys. 38, 5944–5959 (1997).
[Crossref]

1996 (1)

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

1995 (2)

D. Saad and S. Solla, “Exact solution for on-line learning in multilayer neural networks,” Phys. Rev. Lett. 74, 4337–4340 (1995).
[Crossref] [PubMed]

B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
[Crossref] [PubMed]

1993 (1)

C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[Crossref]

1991 (1)

Z. Luo, “On the convergence of the LMS algorithm with adaptive learning rate for linear feedforward neural networks,” Neural Computation 3, pp. 226–245 (1991).
[Crossref]

1974 (1)

J. Satsuma and N. Yajima, “Initial Value Problems of One-dimensional Self-Modulation of Nonlinear Waves in Dispersive Media,” Suppl. Prog. Theo. Phys. 55, 284–295 (1974).
[Crossref]

Agrawal, G.

G. Agrawal and R. Boyd, Contemporary Nonlinear Optics, (Academic Press, New York1992).

Andrews, M.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Arfken, G.

G. Arfken and H. Weber, Mathematical Methods for Physicists4th Ed. (Academic Press, New York1995).

Bohn, J.

J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
[Crossref]

Boyd, R.

G. Agrawal and R. Boyd, Contemporary Nonlinear Optics, (Academic Press, New York1992).

Bramley, R.

S. Saarinen, R. Bramley, and G. Cybenko, The numerical solution of the neural network training problems, CRSD Report 1089 (Center for Supercomputing Research and Development, University of Illinois, Urbana1991).

Chen, S.

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

Chow, T.

J. Yam and T. Chow, “A weight initialization method for improving training speed in feedforward neural networks,” Neurocomputing 30, 219–232 (2000).
[Crossref]

Cybenko, G.

S. Saarinen, R. Bramley, and G. Cybenko, The numerical solution of the neural network training problems, CRSD Report 1089 (Center for Supercomputing Research and Development, University of Illinois, Urbana1991).

D.,

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

Esry, B.

J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
[Crossref]

Flannery, S.

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).

Freeman, I.

G. Joos and I. Freeman, Theoretical Physics (Dover Publications, New York1986).

Fushchych, W.

W. Fushchych and A. Nikitin, “Higher symmetries and exact solutions of linear and nonlinear Schrdinger equation,” J. Math. Phys. 38, 5944–5959 (1997).
[Crossref]

Garcia, M.L.

Girosi, F.

T. Poggio and F. Girosi, “A Sparse Representation for Function Approximation,” Neural computation 10, 1445–1454 (1998).
[Crossref] [PubMed]

Greene, C.

J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
[Crossref]

Haykin, S.

S. Haykin, Neural Networks: A Comprehensive Foundation2nd Ed. (Prentice-Hall, New York1999).

Inouye, S.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Jimenez, J.

B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
[Crossref] [PubMed]

Joos, G.

G. Joos and I. Freeman, Theoretical Physics (Dover Publications, New York1986).

Ketterle, W.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Kunz, K.

K. Kunz and R. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton1993).

Lim, M.

M. Lim and C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[Crossref]

Luebbers, R.

K. Kunz and R. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton1993).

Luo, Z.

Z. Luo, “On the convergence of the LMS algorithm with adaptive learning rate for linear feedforward neural networks,” Neural Computation 3, pp. 226–245 (1991).
[Crossref]

Maksimchuk, A.

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

Mazilu, D.

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

Meisner, H.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

Monterola, C.

M. Quito, C. Monterola, and C. Saloma, “Solving N-body problems with neural networks,” Phys. Rev. Lett. 86, 4741–4744 (2001).
[Crossref] [PubMed]

C. Monterola and C. Saloma, “Characterizing the dynamics of constrained physical systems with unsupervised neural network,” Phys. Rev. E 57, 1247R–1250R (1998).
[Crossref]

Mourou, G.

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

Nikitin, A.

W. Fushchych and A. Nikitin, “Higher symmetries and exact solutions of linear and nonlinear Schrdinger equation,” J. Math. Phys. 38, 5944–5959 (1997).
[Crossref]

Poggio, T.

T. Poggio and F. Girosi, “A Sparse Representation for Function Approximation,” Neural computation 10, 1445–1454 (1998).
[Crossref] [PubMed]

Press, W.

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).

Quito, M.

M. Quito, C. Monterola, and C. Saloma, “Solving N-body problems with neural networks,” Phys. Rev. Lett. 86, 4741–4744 (2001).
[Crossref] [PubMed]

Saad, D.

D. Saad and S. Solla, “Exact solution for on-line learning in multilayer neural networks,” Phys. Rev. Lett. 74, 4337–4340 (1995).
[Crossref] [PubMed]

Saarinen, S.

S. Saarinen, R. Bramley, and G. Cybenko, The numerical solution of the neural network training problems, CRSD Report 1089 (Center for Supercomputing Research and Development, University of Illinois, Urbana1991).

Saffman, M.

M. Saffman, “Self-Induced Dipole Force and Filamentation Instability of a Matter Wave,” Phys. Rev. Lett. 81, 65–68 (1998).
[Crossref]

Saloma, C.

M. Quito, C. Monterola, and C. Saloma, “Solving N-body problems with neural networks,” Phys. Rev. Lett. 86, 4741–4744 (2001).
[Crossref] [PubMed]

M. Soriano, M.L. Garcia, and C. Saloma, “Fluorescent image classification by major color histograms and a neural network,” Opt. Express 8, pp. 271–277 (2001), http://www.opticsexpress.org/oearchive/source/30248.htm
[Crossref] [PubMed]

M. Soriano and C. Saloma, “Improved classification robustness for noisy cell images represented as principal-component projections in a hybrid recognition system,” Appl. Opt. 37, 3628–3639 (1998).
[Crossref]

C. Monterola and C. Saloma, “Characterizing the dynamics of constrained physical systems with unsupervised neural network,” Phys. Rev. E 57, 1247R–1250R (1998).
[Crossref]

M. Lim and C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[Crossref]

C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[Crossref]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial Value Problems of One-dimensional Self-Modulation of Nonlinear Waves in Dispersive Media,” Suppl. Prog. Theo. Phys. 55, 284–295 (1974).
[Crossref]

Solla, S.

D. Saad and S. Solla, “Exact solution for on-line learning in multilayer neural networks,” Phys. Rev. Lett. 74, 4337–4340 (1995).
[Crossref] [PubMed]

Soriano, M.

Stamper-Kurn, D.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Stenger, J.

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Teukolsky, S.

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).

Torner, L.

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

Tribaldos, V.

B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
[Crossref] [PubMed]

Umstadter, D.

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

van Milligen, B. Ph.

B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
[Crossref] [PubMed]

Vetterling, W.

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).

Wagner, R.

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

Weber, H.

G. Arfken and H. Weber, Mathematical Methods for Physicists4th Ed. (Academic Press, New York1995).

Yajima, N.

J. Satsuma and N. Yajima, “Initial Value Problems of One-dimensional Self-Modulation of Nonlinear Waves in Dispersive Media,” Suppl. Prog. Theo. Phys. 55, 284–295 (1974).
[Crossref]

Yam, J.

J. Yam and T. Chow, “A weight initialization method for improving training speed in feedforward neural networks,” Neurocomputing 30, 219–232 (2000).
[Crossref]

Appl. Opt. (1)

J. Appl. Phys. (1)

C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[Crossref]

J. Math. Phys. (1)

W. Fushchych and A. Nikitin, “Higher symmetries and exact solutions of linear and nonlinear Schrdinger equation,” J. Math. Phys. 38, 5944–5959 (1997).
[Crossref]

Nature (London) (1)

S. Inouye, M. Andrews, J. Stenger, H. Meisner, D. Stamper-Kurn, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature (London) 392, 151–154 (1998).
[Crossref]

Neural Computation (1)

Z. Luo, “On the convergence of the LMS algorithm with adaptive learning rate for linear feedforward neural networks,” Neural Computation 3, pp. 226–245 (1991).
[Crossref]

T. Poggio and F. Girosi, “A Sparse Representation for Function Approximation,” Neural computation 10, 1445–1454 (1998).
[Crossref] [PubMed]

Neurocomputing (1)

J. Yam and T. Chow, “A weight initialization method for improving training speed in feedforward neural networks,” Neurocomputing 30, 219–232 (2000).
[Crossref]

Opt. Express (1)

Phys. Rev. A (1)

J. Bohn, B. Esry, and C. Greene, “Effective potentials for dilute Bose-Einstein condensates,” Phys. Rev. A 58, 584–597 (1998).
[Crossref]

Phys. Rev. E (2)

M. Lim and C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[Crossref]

C. Monterola and C. Saloma, “Characterizing the dynamics of constrained physical systems with unsupervised neural network,” Phys. Rev. E 57, 1247R–1250R (1998).
[Crossref]

Phys. Rev. Lett. (5)

M. Quito, C. Monterola, and C. Saloma, “Solving N-body problems with neural networks,” Phys. Rev. Lett. 86, 4741–4744 (2001).
[Crossref] [PubMed]

B. Ph. van Milligen, V. Tribaldos, and J. Jimenez, “Neural network differential equation and plasma equilibrium solver,” Phys. Rev. Lett. 75, 3594–3597 (1995).
[Crossref] [PubMed]

D. Mihalache, D. Mazilu, D., and L. Torner, “Stability of Walking Vector Solitons,” Phys. Rev. Lett. 81, 4353–4356(1998).
[Crossref]

M. Saffman, “Self-Induced Dipole Force and Filamentation Instability of a Matter Wave,” Phys. Rev. Lett. 81, 65–68 (1998).
[Crossref]

D. Saad and S. Solla, “Exact solution for on-line learning in multilayer neural networks,” Phys. Rev. Lett. 74, 4337–4340 (1995).
[Crossref] [PubMed]

Science (1)

D. Umstadter, S. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear Optics in Relativistic Plasmas and Laser Wake Field Acceleration of Electrons,” Science 273, 472–475 (1996).
[Crossref] [PubMed]

Suppl. Prog. Theo. Phys. (1)

J. Satsuma and N. Yajima, “Initial Value Problems of One-dimensional Self-Modulation of Nonlinear Waves in Dispersive Media,” Suppl. Prog. Theo. Phys. 55, 284–295 (1974).
[Crossref]

Other (7)

S. Saarinen, R. Bramley, and G. Cybenko, The numerical solution of the neural network training problems, CRSD Report 1089 (Center for Supercomputing Research and Development, University of Illinois, Urbana1991).

S. Haykin, Neural Networks: A Comprehensive Foundation2nd Ed. (Prentice-Hall, New York1999).

G. Agrawal and R. Boyd, Contemporary Nonlinear Optics, (Academic Press, New York1992).

G. Joos and I. Freeman, Theoretical Physics (Dover Publications, New York1986).

G. Arfken and H. Weber, Mathematical Methods for Physicists4th Ed. (Academic Press, New York1995).

W. Press, S. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, New York1986).

K. Kunz and R. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton1993).

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Figures (6)

Fig. 1.
Fig. 1. Gaussian input pulse. Average energy <Eq > curve for: a) β=1, γ=0 (diamond), b) β=0, γ=1 (circle), and c) β=1, γ=1 (cross-hair). At q=200, E(β=1, γ=1)=8.3×10-4, E(β=1, γ=0)=4.3×10-5, and E(β=0, γ=1)=1.3×10-5.
Fig. 2.
Fig. 2. Gaussian input pulse. Intensity profiles Ψ254Ψ254* for: a) β=0, γ=1, b) β=1, γ=0, and c) β=1, γ=1. Ranges considered: 0 ≤ z (in L 0 units)≤15 (Δz=0.1), and -10≤t (in τp units)≤10 (Δt=0.2),
Fig. 3.
Fig. 3. Sech input pulse. Intensity profiles Ψ q Ψ q* for: (q=1200): a) β=1, γ=1 b) β=1, γ=2. Range considered: 0≤z (in L 0 units) ≤ 4 (Δz=0.1), and -4≤t (in τp units) ≤4 (Δt=0.2),
Fig. 4.
Fig. 4. Sech input pulse. Modulus plot of the β=1, γ=3: a) ISM solution, and b)NN solution at q=10200. Range considered: 0 ≤z (in L 0 units) ≤3 (Δz=0.025), and -5≤t (in τp units) ≤5 (Δt=0.2),
Fig. 5.
Fig. 5. Sech input pulse. Modulus profile of the β=1, γ=4 : a) ISM solution, and b) NN at q=3200. Only selected z values (Δz=0.4) are considered: 0≤z (in L 0 units) ≤1.6, and -5≤t (in τp units) ≤5 (Δt=0.2),
Fig. 6.
Fig. 6. Sech input pulse β=1, γ=4. Behavior of <Eq > plots for different hidden unit number H of the three-layer 3-input/2-output feedforward NN.

Equations (7)

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

F ( x 1 , x 2 , , x M 1 ) = [ D R + j D I ] Ψ = 0
w m h q + 1 = w m h q η ( E q / w m h q )
E q r = E 1 q r + E 2 q r + + E K + 1 q r = F q r 2 + C 1 q r 2 + + C K q r 2
j ( Ψ / z ) + ( β / 2 ) ( 2 Ψ / 2 t ) γ 2 Ψ 2 Ψ = F ( z , t ) = 0
Ψ R 2 ( 0 , t ) + Ψ 1 2 ( 0 , t ) Ψ 0 Ψ 0 * = 0 = C 1
Ψ R 2 ( z , t ± ) + Ψ I 2 ( z , t ± ) + [ Ψ R 2 ( z , t ± ) / t ] 2 + [ Ψ I 2 ( z , t ± / t ) ] 2 = 0 = C 2 2
E q = E 1 q + E 2 q + E 3 q = F q ( z , t ) 2 + C 1 q 2 + C 2 q 2

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