G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).

[CrossRef]

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).

[CrossRef]

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).

[CrossRef]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).

[CrossRef]

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).

[CrossRef]
[PubMed]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1560 (1995).

[CrossRef]
[PubMed]

V. Magni, G. Cerullo, S. De Silvesti, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).

[CrossRef]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).

[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).

[CrossRef]
[PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).

[CrossRef]

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).

[CrossRef]
[PubMed]

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).

[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).

[CrossRef]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).

[CrossRef]

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).

[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).

[CrossRef]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).

[CrossRef]

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).

[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).

[CrossRef]

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].

[CrossRef]

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

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).

[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).

[CrossRef]
[PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).

[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).

[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).

[CrossRef]

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).

[CrossRef]

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).

[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).

[CrossRef]

G. Cerullo, A. Dienes, and V. Magni, “Space–time coupling and collapse threshold for femtosecond pulses in dispersive nonlinear media,” Opt. Lett. 21, 65–67 (1996).

[CrossRef]
[PubMed]

V. Magni, G. Cerullo, S. De Silvesti, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).

[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).

[CrossRef]

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).

[CrossRef]

G. Fibich and A. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).

[CrossRef]

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. 76, 4356–4359 (1996).

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).

[CrossRef]
[PubMed]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).

[CrossRef]

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).

[CrossRef]

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).

[CrossRef]
[PubMed]

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).

[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).

[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).

[CrossRef]

G. Cerullo, A. Dienes, and V. Magni, “Space–time coupling and collapse threshold for femtosecond pulses in dispersive nonlinear media,” Opt. Lett. 21, 65–67 (1996).

[CrossRef]
[PubMed]

V. Magni, G. Cerullo, S. De Silvesti, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).

[CrossRef]

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).

[CrossRef]
[PubMed]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).

[CrossRef]

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).

[CrossRef]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).

[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).

[CrossRef]

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).

[CrossRef]

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).

[CrossRef]
[PubMed]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].

[CrossRef]

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).

[CrossRef]
[PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).

[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].

[CrossRef]

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

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].

[CrossRef]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).

[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).

[CrossRef]

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).

[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).

[CrossRef]

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).

[CrossRef]

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).

[CrossRef]

S. Konar and A. Sengupta, “Propagation of an elliptic Gaussian laser beam in a medium with saturable nonlinearity,” J. Opt. Soc. Am. B 11, 1644–1646 (1994).

[CrossRef]

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).

[CrossRef]

V. Magni, G. Cerullo, S. De Silvesti, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).

[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

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

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).

[CrossRef]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).

[CrossRef]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1560 (1995).

[CrossRef]
[PubMed]

G. G. Luther, A. C. Newell, J. V. Moloney, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994).

[CrossRef]
[PubMed]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).

[CrossRef]

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).

[CrossRef]

S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998).

[CrossRef]

G. Cerullo, A. Dienes, and V. Magni, “Space–time coupling and collapse threshold for femtosecond pulses in dispersive nonlinear media,” Opt. Lett. 21, 65–67 (1996).

[CrossRef]
[PubMed]

G. Fibich and A. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).

[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).

[CrossRef]

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).

[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).

[CrossRef]

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).

[CrossRef]

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. 76, 4356–4359 (1996).

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).

[CrossRef]
[PubMed]

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).

[CrossRef]
[PubMed]

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).

[CrossRef]
[PubMed]

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).

[CrossRef]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).

[CrossRef]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).

[CrossRef]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).

[CrossRef]

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].

[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).

[CrossRef]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999).

The values of e can be recovered by use of the equation e=1±h^{2}−1.

The relative error of relation (9) is less than 1.5% in the range 1/2.5≤e≤2.5.

The value of z_{c} here is twice that given in Eq. (13) below, because there ψ_{0}=c exp(−r^{2}).

I.e., if ∫_{x2+y2≤ri(z)}|ψ|^{2}dxdy=N_{i}, then V_{i}(z)=∫_{x2+y2≤ri(z)}r^{2}|ψ|^{2}dxdy.

I.e., the location of blowup in the unperturbed NLS [Eq. (4)].

Clearly, the critical power for collapse of ultrashort elliptic pulses should be higher than for circular pulses.