Abstract

We investigate the properties of the families of spatial walking solitons propagating in planar waveguides made of quadratic nonlinear media under conditions for second-harmonic generation in the presence of Poynting vector walk-off. We study in detail the shape and various features of the solitons, their stability on propagation, and their excitation.

© 1998 Optical Society of America

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  37. A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
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    [CrossRef]
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    [CrossRef]

1997

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 78, 2341 (1997).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Phys. Rev. E 56, 6294 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, J. Opt. Soc. Am. B 14, 3134 (1997).
[CrossRef]

A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997).
[CrossRef] [PubMed]

1996

S. Trillo, Opt. Lett. 21, 1111 (1996).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

B. A. Malomed, D. Anderson, and M. Lysak, Opt. Commun. 126, 251 (1996).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1996).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

1995

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[CrossRef] [PubMed]

L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, L. Torner, and G. I. Stegeman, Opt. Lett. 20, 1949 (1995).
[CrossRef] [PubMed]

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, Opt. Lett. 20, 1952 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

1994

1993

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

1986

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

1981

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

1976

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Agrawal, G. P.

Akhmediev, N.

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[CrossRef] [PubMed]

Akhmediev, N. N.

Anderson, D.

B. A. Malomed, D. Anderson, and M. Lysak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Assanto, G.

G. Leo, G. Assanto, and W. E. Torruellas, J. Opt. Soc. Am. B 14, 3134 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 78, 2341 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1996).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Buryak, A. V.

A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Cao, X. D.

Capobianco, A. D.

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

Conti, C.

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 78, 2341 (1997).
[CrossRef]

Costantini, B.

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Phys. Rev. E 56, 6294 (1997).
[CrossRef]

De Angelis, C.

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

De Rossi, A.

Drummond, P. D.

Etrich, C.

U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Fuerts, R. A.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Hayata, K.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[CrossRef] [PubMed]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Karlsson, M.

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[CrossRef] [PubMed]

Karpierz, M. A.

M. A. Karpierz and M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

Kivshar, Y. S.

A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Koshiba, M.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[CrossRef] [PubMed]

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

Laureti-Palma, A.

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Lawrence, B. L.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Lederer, F.

U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997).
[CrossRef]

Leo, G.

G. Leo, G. Assanto, and W. E. Torruellas, J. Opt. Soc. Am. B 14, 3134 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1996).
[CrossRef]

Lysak, M.

B. A. Malomed, D. Anderson, and M. Lysak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Malomed, B. A.

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997).
[CrossRef]

U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997).
[CrossRef]

B. A. Malomed, D. Anderson, and M. Lysak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Phys. Rev. E 56, 6294 (1997).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

Menyuk, C. R.

Meyerhofer, D. D.

Mihalache, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Phys. Rev. E 56, 6294 (1997).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

Nalesso, G. F.

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997).
[CrossRef]

A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, J. Opt. Soc. Am. B 14, 2602 (1997).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Peschel, U.

U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997).
[CrossRef]

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Schiek, R.

Stegeman, G. I.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, Opt. Lett. 20, 1952 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, L. Torner, and G. I. Stegeman, Opt. Lett. 20, 1949 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Sypek, M.

M. A. Karpierz and M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

Torner, L.

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Phys. Rev. E 56, 6294 (1997).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, Opt. Lett. 20, 1952 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, L. Torner, and G. I. Stegeman, Opt. Lett. 20, 1949 (1995).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, J. Opt. Soc. Am. B 14, 3134 (1997).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerts, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
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Figures (16)

Fig. 1
Fig. 1

Nonlinear wave-number shift versus energy flow for the families of walking solitons with different soliton velocities. (a) Solitons at positive wave-vector mismatch (β=3), (b) solitons at negative wave-vector mismatch (β=-3). In all cases δ=1. Dashed curves, unstable solutions; dotted curves, nonwalking solitons that exist in the absence of Poynting vector walk-off (δ=0).

Fig. 2
Fig. 2

Amplitude and local wave-front tilt, defined as the transverse derivative of the wave front, of solitons walking with different velocities at negative wave-vector mismatch, as a function of the transverse coordinate. Velocities: (a) v=-1, (b) v=-0.5, (c) v=0, (d) v=0.5. In all cases β=-3 and δ=1 and the nonlinear wave number shift is κ1=3.

Fig. 3
Fig. 3

Same as in Figs. 2(b) and 2(d) but for the walking solitons existing at positive wave-vector mismatch (β=3). Velocities: (a) v=-0.5, (b) v=0.5. In both cases δ=1.

Fig. 4
Fig. 4

Same as in Figs.2(b) and 2(d) but for two walking solitons carrying the same energy flow. Conditions: β=-3, I=30. Velocities: (a) v=-0.5, (b) v=0.5. In both cases δ=1.

Fig. 5
Fig. 5

Fraction of power carried by the second-harmonic beam (i.e., I2/I) as a function of the nonlinear wave-number shift κ1 at several wave-vector mismatches and soliton velocities. Solid curves, walking solitons with v=0; dashed curves, solitons walking with the velocity v=-0.5. The short portions of the curves near I2/I=1 plotted as dotted curves stand for the corresponding unstable solutions. In all cases δ=1.

Fig. 6
Fig. 6

Fraction of power carried by the second-harmonic beam as a function of the soliton velocity at several wave-vector mismatches. Conditions: in (a) the nonlinear wave-number shift is fixed to κ1=3; in (b) the total energy flow is fixed to I=40. The short portion of the curve corresponding to β=-3 near I2/I=1 plotted as a dotted curve stands for unstable solutions. In all cases δ=1.

Fig. 7
Fig. 7

Widths (FWHM) of the fundamental and the second-harmonic beams of representative walking solitons that exist at several wave-vector mismatches as a function of the soliton velocity. In (a) the nonlinear wave-number shift is fixed to κ1=3; in (b) the total energy flow is fixed to I=40. In all cases δ=1.

Fig. 8
Fig. 8

Threshold (minimum) energy flow for the existence of solitons walking with a given velocity as a function of the walk-off parameter δ for various values of the wave-vector mismatch. Solid curves, threshold energy for solitons walking with a velocity v=-δ/4; dashed curve, threshold energy for solitons walking with a velocity v=-δ/2. The plot contains only one curve for β=0 and none for β=3, because in the missing cases there is no threshold energy for the existence of solitons in the range of walk-off parameters and soliton velocities displayed.

Fig. 9
Fig. 9

Energy Hamiltonian curves and energy wave-number curves to determine geometrically the stability of the nonwalking solitons that exist in the absence of Poynting vector walk-off (δ=0). Solid curves, stable solitons; dashed curves, unstable solutions. (From Ref. 30).

Fig. 10
Fig. 10

Hamiltonian versus energy flow for some of the families of walking solitons. Notice that the curves correspond to fixed soliton velocities; therefore the momentum is not constant along the curves. For the sake of clarity, only the curves for a few velocities are shown. Solid curves, stable walking solitons; dashed curves, unstable solutions; dotted curves, nonwalking solitons that exist in the absence of Poynting vector walk-off (δ=0). A zoom of the curve corresponding to β=-3, v=-1 near the point of marginal stability is shown in Fig. 12(b) below.

Fig. 11
Fig. 11

Momentum versus energy flow for the families of walking solitons. Notice that the Hamiltonian is not constant along the curves. Solid curves, stable walking solitons; dashed curves, unstable solutions.

Fig. 12
Fig. 12

Zooms of two representative curves of Figs. 1(b) and Fig. 10(b) that show the small differences that exist between the numerical values for the condition of marginal stability given by the exact criterion [Eq. (28)] and by the Vakhitov–Kolokolov (V-K) criterion [Eq. (30)]. The curves correspond to β=-3, v=-1.

Fig. 13
Fig. 13

Summary of the stability analysis for the family of solitons that exist at β=-3, δ=1, with velocities in the range [-2, 2]. Values of the nonlinear wave-number shift κ1 are shown at the cutoff for the soliton existence (dashed line) and at the condition of marginal stability (solid curve). The dotted curve that almost coincides with the solid curve corresponds to the condition of marginal stability given by criterion (30). Above the solid curve, all solutions are stable. The existing unstable solutions occur in the narrow band that appears between the cutoff and the marginal stability lines.

Fig. 14
Fig. 14

Threshold energy flow for stationary solitons to exist as a function of the soliton velocity, at two values of the wave-vector mismatch. Above the existence curve there is always a stable walking soliton solution.

Fig. 15
Fig. 15

Typical excitation of a slow and a fast walking soliton. Amplitudes of the fundamental and the second harmonic beams at ξ=10 of two selected walking solitons that have been numerically excited with tilted input beams and with different input conditions. Tilts: (a) μ=0, (b) μ=-1.5. In both cases A=3, B=3, β=-3, and δ=1.

Fig. 16
Fig. 16

Excitation of two rare walking solitons. (a) Soliton that walks against the Poynting vector walk-off. (b) Walking soliton that carries an energy much smaller than the minimum required for formation of a nonwalking soliton. Conditions: (a) A=4, B=4, μ=1; (b) A=1, B=1, μ=-1. Conditions and features are as in Fig. 15.

Equations (43)

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i a1ξ-r2 2a1s2+a1*a2 exp(-iβξ)=0,
i a2ξ-α2 2a2s2-iδ a2s+a12 exp(iβξ)=0,
aν(ξ, s)=Uν(η)exp[iϕν(ξ, η)],ν=1, 2,
½U¨1-(κ1-vf˙1+½ f˙12)U1+U1U2 cos(f2-2 f1)
=0,
½ f¨1U1+(f˙1-v)U˙1+U1U2 sin(f2-2 f1)=0,
½αU¨2+[2κ1+β-(v+δ)f˙2-½αf˙22]U2
-U12 cos(f2-2 f1)=0,
½αf¨2U2+(αf˙2+v+δ)U˙2+U12 sin(f2-2 f1)
=0,
H=-12 rA1s2+α2 A2s2-β|A2|2+i δ2 A2 A2*s-A2* A2s+(A1*2A2+A12A2*)ds,
I=I1+I2=[|A1|2+|A2|2]ds
J=J1+J2=14i 2A1* A1s-A1 A1*s+A2* A2s-A2 A2*sds.
δF(H+κ1I-vJ)sta=0,
H=-35 κ1I+15 βI2+45 vJ-15 δJ2.
J=½ (2U12f˙1+U22f˙2)ds.
J=Iv+δI2+(2α+1)J2.
J=Iv,
H=Hv=0+½ Iv2,
f˙1(η)=v.
Γ22=-2α2κ1+β+(v+δ)22α,
f˙2(η)=-1α(δ+v).
1α(2α+1)(2κ1-v2)+v2+β+(v+δ)22α>0.
δ(2v+δ)>β.
κ1,cut(2)=½[(δ+v)2-β].
κ1,cut(1)=½v2.
κ1μκ1,vμv,βμβ,δμδ,
U1μU1,U2μU2,
ss/μ,Iμ3/2I,
[f˙1(η)-v]U12(η)-[αf˙2(η)+v+δ]U22(η)=0.
U12ξ=-ηϕ1η-vU12-2U12U2 sin(ϕ2-2ϕ1-βξ),
U22ξ=ηαϕ2η+v+δ)U22+2U12U2 sin(ϕ2-2ϕ1-βξ).
U12(0)=2κ1+β2[U2(0)-κ1]U22(0).
Iκ1 Jv-Iv Jκ1=0.
n=(J,H)(κ1,v) xˆ+(H,I)(κ1,v) yˆ+(I,J)(κ1,v) zˆ,
Iκ1=0,
v=JI-δ I2I-(2α+1) J2I.
v=JI-δ2 I2I-δ3 I3I-(α2+1) J2I-(2α3+1) J3I.
i a1ξ-α12 2a1s2+a3A2* exp(-iβξ)=0,
i a2ξ-α22 2a2s2-iδ2 a2s+a3a1* exp(-iβξ)=0,
i a3ξ-α32 2a3s2-iδ3 a3s+a1a2 exp(iβξ)=0,
I=I1+I2+I3=(½|a1|2+½|a2|2+|a3|2)ds
J=J1+J2+J3=14i a1* a1s-a1 a1*s+A2* a2s-a2 A2*s+a3* a3s-a3 a3*sds.

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