Abstract

It is shown that the problem of dispersive self-modulation of waves including one transverse dimension and the three-dimensional self-focusing and self-defocusing theories has exact nontrivial analytic solutions. These are solutions compatible with cylindrical geometry and self-similar solutions that describe intermediate asymptotic, or long-time, behavior. These solutions are all obtained from symmetry considerations only.

© 1990 Optical Society of America

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References

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  1. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  2. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  3. P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  4. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).
  5. E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser-beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971); “…III. Filament formation,” Phys. Rev. A 6, 1162–1171 (1972).
    [CrossRef]
  6. V. E. Zakharov and V. S. Synakh, “The nature of the self-focusing singularity,” Sov. Phys. JETP 41, 465–468 (1976).
  7. M. V. Goldman and D. R. Nicholson, “Virial theory of direct Langmuir collapse,” Phys. Rev. Lett. 41, 406–410 (1978).
    [CrossRef]
  8. P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
    [CrossRef]
  9. D. N. Christodoulides and R. I. Joseph, “Exact radial dependence of the field in a nonlinear dispersive dielectric fiber: bright pulse solutions,” Opt. Lett. 9, 229–231 (1984).
    [CrossRef] [PubMed]
  10. D. N. Christodoulides and R. I. Joseph, “Dark solitary waves in optical fibers,” Opt. Lett. 9, 408–410 (1984).
    [CrossRef] [PubMed]
  11. J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schroedinger equations. I. A general review,” Phys. Scr. 33, 481–497 (1986); K. Rypdal and J. J. Rasmussen, “…II. Similarity structure of the blow-up singularity,” Phys. Scr. 33, 498–504 (1986).
    [CrossRef]
  12. A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
    [CrossRef]
  13. B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
    [CrossRef]
  14. J. T. Manassah, P. L. Baldek, and R. R. Alfano, “Self-focusing and self-phase modulation in a parabolic graded-index optical fiber,” Opt. Lett. 13, 589–591 (1988).
    [CrossRef] [PubMed]
  15. H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
    [CrossRef]
  16. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  17. G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).
  18. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).
  19. L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schrödinger equation. I. The symmetry group and its subgroups,” J. Phys. A. 21, 1493–1511 (1988).
    [CrossRef]
  20. L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. II. Exact solutions,” J. Phys. A 22, 369–497 (1989).
  21. L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
    [CrossRef]
  22. L. Gagnon and P. Winternitz, “Exact solutions of the cubic and quintic nonlinear Schroedinger equation for a cylindrical geometry,” Phys. Rev. A 39, 296–306 (1989).
    [CrossRef] [PubMed]
  23. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
  24. A. M. Grundland and J. A. Tsuzynski, “Multivalued solutions of the ϕ4-field equations, transition to ergodicity and bifurcations,” Phys. Lett. 133A, 298–304 (1988).

1989 (3)

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. II. Exact solutions,” J. Phys. A 22, 369–497 (1989).

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

L. Gagnon and P. Winternitz, “Exact solutions of the cubic and quintic nonlinear Schroedinger equation for a cylindrical geometry,” Phys. Rev. A 39, 296–306 (1989).
[CrossRef] [PubMed]

1988 (5)

A. M. Grundland and J. A. Tsuzynski, “Multivalued solutions of the ϕ4-field equations, transition to ergodicity and bifurcations,” Phys. Lett. 133A, 298–304 (1988).

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schrödinger equation. I. The symmetry group and its subgroups,” J. Phys. A. 21, 1493–1511 (1988).
[CrossRef]

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

J. T. Manassah, P. L. Baldek, and R. R. Alfano, “Self-focusing and self-phase modulation in a parabolic graded-index optical fiber,” Opt. Lett. 13, 589–591 (1988).
[CrossRef] [PubMed]

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

1986 (2)

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schroedinger equations. I. A general review,” Phys. Scr. 33, 481–497 (1986); K. Rypdal and J. J. Rasmussen, “…II. Similarity structure of the blow-up singularity,” Phys. Scr. 33, 498–504 (1986).
[CrossRef]

A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
[CrossRef]

1984 (2)

1983 (1)

P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
[CrossRef]

1978 (2)

M. V. Goldman and D. R. Nicholson, “Virial theory of direct Langmuir collapse,” Phys. Rev. Lett. 41, 406–410 (1978).
[CrossRef]

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

1976 (1)

V. E. Zakharov and V. S. Synakh, “The nature of the self-focusing singularity,” Sov. Phys. JETP 41, 465–468 (1976).

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1971 (1)

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser-beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971); “…III. Filament formation,” Phys. Rev. A 6, 1162–1171 (1972).
[CrossRef]

1970 (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

1967 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

1965 (1)

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Alfano, R. R.

Baldek, P. L.

Banerjee, P. P.

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
[CrossRef]

P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
[CrossRef]

Burdet, G.

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

Chatterjee, M. J.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Christodoulides, D. N.

Gagnon, L.

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. II. Exact solutions,” J. Phys. A 22, 369–497 (1989).

L. Gagnon and P. Winternitz, “Exact solutions of the cubic and quintic nonlinear Schroedinger equation for a cylindrical geometry,” Phys. Rev. A 39, 296–306 (1989).
[CrossRef] [PubMed]

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schrödinger equation. I. The symmetry group and its subgroups,” J. Phys. A. 21, 1493–1511 (1988).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Goldman, M. V.

M. V. Goldman and D. R. Nicholson, “Virial theory of direct Langmuir collapse,” Phys. Rev. Lett. 41, 406–410 (1978).
[CrossRef]

Grammaticos, B.

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

Grundland, A. M.

A. M. Grundland and J. A. Tsuzynski, “Multivalued solutions of the ϕ4-field equations, transition to ergodicity and bifurcations,” Phys. Lett. 133A, 298–304 (1988).

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

Joseph, R. I.

Kelly, P. L.

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kerr, E. L.

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser-beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971); “…III. Filament formation,” Phys. Rev. A 6, 1162–1171 (1972).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Korpel, A.

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
[CrossRef]

P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
[CrossRef]

Lemesurier, B. J.

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Lonngren, K. E.

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
[CrossRef]

P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
[CrossRef]

Manassah, J. T.

Nicholson, D. R.

M. V. Goldman and D. R. Nicholson, “Virial theory of direct Langmuir collapse,” Phys. Rev. Lett. 41, 406–410 (1978).
[CrossRef]

Papanicolaou, G.

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Patera, J.

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

Perrin, M.

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

Ramani, A.

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

Rasmussen, J. J.

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schroedinger equations. I. A general review,” Phys. Scr. 33, 481–497 (1986); K. Rypdal and J. J. Rasmussen, “…II. Similarity structure of the blow-up singularity,” Phys. Scr. 33, 498–504 (1986).
[CrossRef]

Rypdal, K.

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schroedinger equations. I. A general review,” Phys. Scr. 33, 481–497 (1986); K. Rypdal and J. J. Rasmussen, “…II. Similarity structure of the blow-up singularity,” Phys. Scr. 33, 498–504 (1986).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Sim, H. K.

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

A. Korpel, K. E. Lonngren, P. P. Banerjee, H. K. Sim, and M. J. Chatterjee, “Split-step-type angular plane wave spectrum method for the study of self-refractive effects in nonlinear wave propagation,” J. Opt. Soc. Am. B 3, 885–890 (1986).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Sulem, C.

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Sulem, P. L.

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Synakh, V. S.

V. E. Zakharov and V. S. Synakh, “The nature of the self-focusing singularity,” Sov. Phys. JETP 41, 465–468 (1976).

Talanov, V. I.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Tsuzynski, J. A.

A. M. Grundland and J. A. Tsuzynski, “Multivalued solutions of the ϕ4-field equations, transition to ergodicity and bifurcations,” Phys. Lett. 133A, 298–304 (1988).

Winternitz, P.

L. Gagnon and P. Winternitz, “Exact solutions of the cubic and quintic nonlinear Schroedinger equation for a cylindrical geometry,” Phys. Rev. A 39, 296–306 (1989).
[CrossRef] [PubMed]

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. II. Exact solutions,” J. Phys. A 22, 369–497 (1989).

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schrödinger equation. I. The symmetry group and its subgroups,” J. Phys. A. 21, 1493–1511 (1988).
[CrossRef]

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

Zakharov, V. E.

V. E. Zakharov and V. S. Synakh, “The nature of the self-focusing singularity,” Sov. Phys. JETP 41, 465–468 (1976).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Ann. Sci. Math. Québec (1)

G. Burdet, J. Patera, M. Perrin, and P. Winternitz, “Sous-algèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Québec 2, 81–108 (1978).

J. Opt. Soc. Am. (1)

H. K. Sim, A. Korpel, K. E. Lonngren, and P. P. Banerjee, “Simulation of two-dimensional nonlinear envelope pulse dynamics by a two-step spatiotemporal angular spectrum method,” J. Opt. Soc. Am. 5, 1900–1909 (1988).
[CrossRef]

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

J. Phys. A (2)

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. II. Exact solutions,” J. Phys. A 22, 369–497 (1989).

L. Gagnon, B. Grammaticos, A. Ramani, and P. Winternitz, “Lie symmetries of a generalised non-linear Schroedinger equation. III. Reduction to third-order ordinary differential equations,” J. Phys. A 22, 499–509 (1989).
[CrossRef]

J. Phys. A. (1)

L. Gagnon and P. Winternitz, “Lie symmetries of a generalised non-linear Schrödinger equation. I. The symmetry group and its subgroups,” J. Phys. A. 21, 1493–1511 (1988).
[CrossRef]

JETP Lett. (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Opt. Lett. (3)

Phys. D (1)

B. J. Lemesurier, G. Papanicolaou, C. Sulem, and P. L. Sulem, “Focusing and multi-focusing solutions of the nonlinear Schrödinger equation,” Phys. D 31, 78–102 (1988); “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Phys. Fluids (1)

P. P. Banerjee, A. Korpel, and K. E. Lonngren, “Self-refraction of nonlinear capillary-gravity waves,” Phys. Fluids 26, 2393–2398 (1983).
[CrossRef]

Phys. Lett. (1)

A. M. Grundland and J. A. Tsuzynski, “Multivalued solutions of the ϕ4-field equations, transition to ergodicity and bifurcations,” Phys. Lett. 133A, 298–304 (1988).

Phys. Rev. A (2)

L. Gagnon and P. Winternitz, “Exact solutions of the cubic and quintic nonlinear Schroedinger equation for a cylindrical geometry,” Phys. Rev. A 39, 296–306 (1989).
[CrossRef] [PubMed]

E. L. Kerr, “Filamentary tracks formed in transparent optical glass by laser-beam self-focusing. II. Theoretical analysis,” Phys. Rev. A 4, 1195–1218 (1971); “…III. Filament formation,” Phys. Rev. A 6, 1162–1171 (1972).
[CrossRef]

Phys. Rev. Lett. (3)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

M. V. Goldman and D. R. Nicholson, “Virial theory of direct Langmuir collapse,” Phys. Rev. Lett. 41, 406–410 (1978).
[CrossRef]

Phys. Scr. (1)

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schroedinger equations. I. A general review,” Phys. Scr. 33, 481–497 (1986); K. Rypdal and J. J. Rasmussen, “…II. Similarity structure of the blow-up singularity,” Phys. Scr. 33, 498–504 (1986).
[CrossRef]

Sov. Phys. JETP (3)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Development of an optical waveguide in the propagation of light in a nonlinear medium,” Sov. Phys. JETP 24, 198–210 (1967); “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

V. E. Zakharov and V. S. Synakh, “The nature of the self-focusing singularity,” Sov. Phys. JETP 41, 465–468 (1976).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (2)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

Cited By

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Figures (2)

Fig. 1
Fig. 1

Comparison between the Bessel function J1/3(ρ) (dashed curve) and the field distribution of solution (3.1) with ρ 0 = K ( 1 / 2 ) 1.8541, α = 1.4, and a1 = −1. Such parameter values lead to the same behavior as ρ → 0.

Fig. 2
Fig. 2

Graphic solution of Eq. (3.4). The intersections between the dashed curve and curves 1, 2, 3, and 4 (n = 2, 3, 4, 5, respectively) provide the first four values of k that make Eq. (3.3) periodic.

Equations (42)

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i u z + u s 1 s 1 + u s 2 s 2 = a 1 u | u | 2 ,
s 1 = s 1 cos φ + s 2 sin φ , s 2 = s 1 sin φ + s 2 cos φ ,
z = a 2 z , s i = a s i , a u ( z , s 1 , s 2 ) = u ( z , s 1 , s 2 ) ,
s i = s i + υ z , u = u exp [ i u 4 ( u z + 2 s i ) ] ,
z = z 1 λ z , s i = s i 1 λ z , u = u ( 1 λ z ) exp ( i λ 4 s 1 2 + s 2 2 1 λ z ) .
u = ( 2 / a 1 ) 1 / 2 sech ( s 1 cos φ + s 2 sin φ ) exp ( i z ) .
u = f ( ξ ) exp [ i ( a θ b z ) ] , ξ = ρ ,
u = f ( ξ ) ρ 1 , ξ = θ + a ln ρ ,
u = f ( ξ ) z 1 / 2 exp [ i ( b θ a 2 ln z ) ] , ξ = z / ρ 2 ,
u = f ( ξ ) ( 1 + z 2 ) 1 / 2 exp ( i b arctan z ) exp ( i z 2 1 4 z ξ ) × exp ( i s 1 2 4 z ) , ξ = ( z s 2 s 1 ) 1 1 + z 2 ,
u = f ( ξ ) ( 1 + z 2 ) 1 / 2 exp ( i b arctan z ) exp ( i ξ z 4 ) exp ( i a θ ) ξ = ρ 2 1 + z 2 ,
M ρ ρ S 2 ρ 2 M 3 + 1 ρ M ρ a 2 ρ 2 M = b M + a 1 M 3 ,
χ ρ = S ρ 1 M 2 ,
( a 2 + 1 ) M ξ ξ S 2 ( a 2 + 1 ) M 3 exp [ 4 a ξ / ( a 2 + 1 ) ] 2 a M ξ + M = a 1 M 3 ,
χ ξ = S M 2 exp [ 2 a ξ / ( a 2 + 1 ) ] ,
4 ξ 3 M ξ ξ + 4 ξ 2 M ξ 4 ξ S 2 M 3 + ( a 2 b 2 ξ + 1 16 ξ ) M = a 1 M 3 ,
χ ξ = S ξ 1 M 2 1 8 ξ 2 ,
M ξ ξ S 2 M 3 + ( b ξ 2 ) M = a 1 M 3 ,
χ ξ = S M 2 ,
4 ξ M ξ ξ 4 S 2 ξ M 3 + 4 M ξ + ( b ξ 4 a 2 ξ ) M = a 1 M 3 ,
χ ξ = S ξ 1 M 2 .
M = [ 2 a 1 λ 0 ρ 2 / 3 W ( η ) ] 1 / 2 , η = 3 2 λ 0 1 / 2 ρ 2 / 3 ρ 0 ,
W η η = 1 2 W ( W η ) 2 + 4 W 2 4 3 b λ 0 3 / 2 η W + S 2 a 1 2 2 λ 0 3 W 1 .
( W η ) 2 = 4 W 3 + 4 K W S 2 a 1 2 λ 0 3 ,
M = [ λ 0 W ( η ) ] 1 / 2 , η = ( λ 0 a 1 2 ) 1 / 2 θ θ 0 ,
( W η ) 2 = 4 W 3 8 a 1 λ 0 W 2 + 4 K W 8 S 2 a 1 λ 0 3 .
u = 2 α 3 ( a 1 ) 1 / 2 ρ 1 / 3 cn ( α ρ 2 / 3 ρ 0 , 1 2 ) exp ( i θ / 3 ) , a 1 < 0 ,
u = ρ 1 ( 2 / a 1 ) 1 / 2 k ( 1 + k 2 ) 1 / 2 sn [ θ ( 1 + k 2 ) 1 / 2 , k ] , 0 < k 2 1 , a 1 > 0 ,
u = ρ 1 ( 2 / a 1 ) 1 / 2 k ( 1 2 k 2 ) 1 / 2 cn [ θ ( 1 2 k 2 ) 1 / 2 , k ] , 0 < k 2 < 1 / 2 , a 1 < 0 ,
K ( k ) = π 2 n ( 1 2 k 2 ) 1 / 2 .
| u | 2 = 2 α 2 9 a 1 ρ 2 / 3 ( 1 + λ z ) 4 / 3 × c n 2 [ α ( 1 + λ z ) 2 / 3 ρ 2 / 3 ρ 0 , 1 2 ] a 1 < 0 ,
Coordinate translations : Z = δ z , S i = δ s i , i = 1 , 2 , Constant change of phase : M = i 2 ( u δ u u * δ u * ) , Rotation : J = s 2 δ s 1 s 1 δ s 2 , Dilation : D = 2 z δ z + s 1 δ s 1 + s 2 δ s 2 u δ u u * δ u * , Galilean boosts : B j = z δ s j i 2 s j ( u δ u u * δ u * ) , j = 1 , 2 , Talanov's lens transformation : C = z 2 δ z + z ( s 1 δ s 1 + s 2 δ s 2 ) z ( u δ u + u * δ u * ) + i 4 ( s 1 2 + s 2 2 + z 2 ) ( u δ u u * δ u * ) .
{ J + 2 a M , Z + 2 b M } ,
{ D + a J , Z } ,
{ D + 2 a M , J + 2 b M } ,
{ C + Z + J + a M , B 1 S 2 } , = ± 1 ,
{ J + 2 a M , C + Z + 2 b M } .
X i F ( z , s 1 , s 2 , u , u * ) = 0 ,
d s 1 z = d s 2 = 2 d u i s 1 u = 2 d u * i s 1 u *
u = u ˜ ( z , ζ ) exp ( i s 1 2 / 4 z ) , ζ = s 2 z s 1 .
d z 1 + z 2 = z d ζ ζ ( z 2 1 ) = d u ˜ u ˜ ( b i z + i 4 ζ 2 ) = d u ˜ * u ˜ * ( b i z i 4 ζ 2 ) ,
u ˜ = f ( ξ ) ( 1 + z 2 ) 1 / 2 exp ( b i arctan z ) exp ( i z 2 1 4 z ξ 2 ) , ξ = ( z s 2 s 1 ) 1 1 + z 2 .

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