V. Tikhonenko, J. Christour, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).

[CrossRef]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 52.

Solitons are stable to arbitrary perturbations when dI0/dP is positive. Mighty morphing solitons have I0=γP, where γ is a positive constant and P is the beam power. See A. W. Snyder, D. J. Mitchell, and A. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).

[CrossRef]

A. Buryak, Optical Sciences Centre, Australian National University, Canberra, Australia (personal communication, August5, 1996).

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.

[CrossRef]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

Solitons are stable to arbitrary perturbations when dI0/dP is positive. Mighty morphing solitons have I0=γP, where γ is a positive constant and P is the beam power. See A. W. Snyder, D. J. Mitchell, and A. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.

[CrossRef]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

[CrossRef]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

Solitons are stable to arbitrary perturbations when dI0/dP is positive. Mighty morphing solitons have I0=γP, where γ is a positive constant and P is the beam power. See A. W. Snyder, D. J. Mitchell, and A. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).

[CrossRef]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.

[CrossRef]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.

[CrossRef]

Y. Silberberg, Opt. Lett. 15, 1282 (1990). Here the usual ∇t2 operator is generalized to ∂2/∂x2+∂2/∂y2+∂2/∂t¯2, where t¯=d2ω/dk2t, with t the retarded time.

[CrossRef]
[PubMed]

Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

[CrossRef]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).

[CrossRef]
[PubMed]

The inverse scattering technique does not give a closed-form expression for the evolution of sech profile beam of arbitrary amplitude and width, only for special values. It provides an implicit description at asymptotic distances for 1+1-D beams of a cubic (Kerr) nonlinearity.

This is seen from Eq. (4) with the identification b=n0 Imq˙/q/kΔ.

This is in homage to the creators of “Mighty Morphin Power Ranges.” Morphing is used because the beams can change dramatically, and mighty because they are impervious to changes in intensity and can be propagated numerically, even with asymmetric perturbations and even in an ln1+I/It medium.

The solution of the Schrödinger equation for a (linear) parabolic index medium n2=n12z-η2zx2, which changes in the axial z direction, is generalized from Arnaud12 to have the Gaussian form ψ of Eq. (3) but with q obeying q¨+η2zq˙=0. By inverting this expression for Ix=ψ2 to find x2I and substituting this into x2 in the expression for n2x,z above, we find that the Gaussian must be a soliton of a In I nonlinearity as in Eq. (1). Substituting Ix into Eq. (1) shows that η2z=Δkn0 Imq˙/q for self-consistency. This leads to Eq. (4) and hence to our general solution.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 52.

A. Buryak, Optical Sciences Centre, Australian National University, Canberra, Australia (personal communication, August5, 1996).