Abstract

In the framework of second-harmonic generation with type I phase matching, we investigated the role played by temporal or spatial walk-off. Using a variational technique, we found that a two-parameter class of solutions always exists, even when nonnegligible spatial or temporal walk-off is present. We checked solitonlike behavior for the approximate stationary solutions by numerical integration of the governing equations; expressions of phase condition for stationary solutions and velocity of the locked waves were derived. To outline the possibilities offered by our approach in predicting the behavior of χ(2) devices, several examples for beam and pulse propagation in KTiOPO4 and LiNbO3 are presented.

© 1997 Optical Society of America

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1997

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

1996

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

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

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

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 21, 940 (1996).
[CrossRef] [PubMed]

M. Picciau, G. Leo, and G. Assanto, J. Opt. Soc. Am. B 13, 661 (1996).
[CrossRef]

R. Mollame, S. Trillo, and G. Assanto, Opt. Lett. 21, 1969 (1996).
[CrossRef] [PubMed]

1995

Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

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

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

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

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

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

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

A. V. Buryak and Y. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

P. Ferro and S. Trillo, Phys. Rev. E 51, 4994 (1995).
[CrossRef]

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

1994

1993

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

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

R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993).
[CrossRef]

M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993).
[CrossRef]

1992

1991

1983

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

1982

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, IEEE J. Quantum Electron. 18, 2048 (1982).
[CrossRef]

1980

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[CrossRef]

1974

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

V. E. Zakharov and A. M. Rubenchick, Sov. Phys. JETP 38, 494 (1974).

1973

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Aitchison, J. S.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

Akhmediev, N.

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

Anderson, D.

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

Arnold, J. M.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

Assanto, G.

M. Picciau, G. Leo, and G. Assanto, J. Opt. Soc. Am. B 13, 661 (1996).
[CrossRef]

R. Mollame, S. Trillo, and G. Assanto, Opt. Lett. 21, 1969 (1996).
[CrossRef] [PubMed]

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

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

G. Assanto, A. Laureti-Palma, C. Sibilia, and M. Bertolotti, Opt. Commun. 110, 599 (1994).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

Baek, Y.

Baumann, I.

Bertolotti, M.

G. Assanto, A. Laureti-Palma, C. Sibilia, and M. Bertolotti, Opt. Commun. 110, 599 (1994).
[CrossRef]

Buryak, A. V.

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Y. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

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

A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994).
[CrossRef] [PubMed]

De Angelis, C.

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

DeSalvo, R.

Drummond, P. D.

Fedosejevs, R.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, IEEE J. Quantum Electron. 18, 2048 (1982).
[CrossRef]

Ferro, P.

P. Ferro and S. Trillo, Phys. Rev. E 51, 4994 (1995).
[CrossRef]

Fuerst, R. A.

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

Goldman, M. V.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[CrossRef]

Hadley, G. R.

Hafizi, B.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[CrossRef]

Hagan, D. I.

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

Hagan, D. J.

Hayata, K.

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

Ironside, C. N.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Karlsson, M.

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

Kivshar, Y. S.

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

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Y. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994).
[CrossRef] [PubMed]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Koshiba, M.

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

Krijnen, G.

Laureti-Palma, A.

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

G. Assanto, A. Laureti-Palma, C. Sibilia, and M. Bertolotti, Opt. Commun. 110, 599 (1994).
[CrossRef]

Lawrence, B. L.

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

Leo, G.

Lisak, M.

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Malomed, B. A.

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Mazilu, D.

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

Menyuk, C. R.

Mihalache, D.

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

Mollame, R.

Nalesso, G. F.

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

Offenberger, A. A.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, IEEE J. Quantum Electron. 18, 2048 (1982).
[CrossRef]

Pelinovsky, D. E.

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

Picciau, M.

Rubenchick, A. M.

V. E. Zakharov and A. M. Rubenchick, Sov. Phys. JETP 38, 494 (1974).

Rypdal, K.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[CrossRef]

Schiek, R.

Sheik-Bahae, M.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

Shiek-Bahae, M.

Sibilia, C.

G. Assanto, A. Laureti-Palma, C. Sibilia, and M. Bertolotti, Opt. Commun. 110, 599 (1994).
[CrossRef]

Sohler, W.

Someda, C. G.

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

Steblina, V. V.

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Stegeman, G. I.

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

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

R. Schiek, Y. Baek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 21, 940 (1996).
[CrossRef] [PubMed]

Y. Baek, R. Schiek, and G. I. Stegeman, Opt. Lett. 20, 2168 (1995).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

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

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

L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[CrossRef] [PubMed]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

R. DeSalvo, D. J. Hagan, M. Shiek-Bahae, G. I. Stegeman, E. W. VanStryland, and H. V. Vanherzeele, Opt. Lett. 17, 28 (1992).
[CrossRef] [PubMed]

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Tomov, I. V.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, IEEE J. Quantum Electron. 18, 2048 (1982).
[CrossRef]

Torner, L.

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

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

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

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

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

L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[CrossRef] [PubMed]

Torruellas, W. E.

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

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

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

Trillo, S.

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Vanherzeele, H. V.

VanStryland, E.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

VanStryland, E. W.

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

R. DeSalvo, D. J. Hagan, M. Shiek-Bahae, G. I. Stegeman, E. W. VanStryland, and H. V. Vanherzeele, Opt. Lett. 17, 28 (1992).
[CrossRef] [PubMed]

VanStryland, W.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

Wang, Z.

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

Werner, M. J.

Zakharov, V. E.

V. E. Zakharov and A. M. Rubenchick, Sov. Phys. JETP 38, 494 (1974).

Appl. Phys. Lett.

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and W. VanStryland, Appl. Phys. Lett. 62, 1323 (1993).
[CrossRef]

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

IEEE J. Quantum Electron.

C. N. Ironside, J. S. Aitchison, and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650 (1993).
[CrossRef]

G. Assanto, G. I. Stegeman, M. Sheik-Bahae, and E. VanStryland, IEEE J. Quantum Electron. 31, 673 (1995).
[CrossRef]

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, IEEE J. Quantum Electron. 18, 2048 (1982).
[CrossRef]

J. Opt. Soc. Am. B

JETP Lett.

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Opt. Commun.

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

G. Assanto, A. Laureti-Palma, C. Sibilia, and M. Bertolotti, Opt. Commun. 110, 599 (1994).
[CrossRef]

V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

R. Schiek, Opt. Quantum Electron. 26, 415 (1994).
[CrossRef]

C. De Angelis, A. Laureti-Palma, G. F. Nalesso, and C. G. Someda, Opt. Quantum Electron. 29, 217 (1997).
[CrossRef]

Phys. Fluids

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[CrossRef]

Phys. Lett. A

A. V. Buryak and Y. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Phys. Rev. A

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

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Y. Baek, R. Schiek, and G. I. Stegeman, in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA TechnicalDigest Series (Optical Society of America, Washington, D.C., 1995), p. 24.

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Figures (9)

Fig. 1
Fig. 1

Characteristic parameters of stationary solutions plotted versus the ratio of linear momentum to total mass, P0/M0: (a) normalized spatial widths α and β of the FF and the SH, respectively; (b) θ1 and velocity v0; (c) normalized peak amplitudes A of the FF and B of the SH. In all cases the linear normalized mismatch is γ=-10, the normalized spatial walk-off is δ=-2, s/r=0.5, and M0=750. Continuous curves, stable solutions; dashed curves, unstable solutions.

Fig. 2
Fig. 2

Region of existence (shaded area) of stable stationary solutions in the (M0, P0) plane for normalized linear mismatch γ=-10, normalized spatial walk-off δ=-2, and s/r=0.5. Dotted curve, solutions with constant velocity v0=0; dashed curve, solutions with v0=-5.

Fig. 3
Fig. 3

Phase constant ϕa of the FF versus total mass M0 for two values of the common velocity v0. The parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

ξ evolution of FF and SH envelopes from a numerical BPM code for linear normalized mismatch γ=-10, normalized spatial walk-off δ=-2, s/r=0.5, and initial conditions corresponding to P0/M0=-0.6 in Fig. 1.

Fig. 5
Fig. 5

Characteristic parameters of stationary solutions plotted versus the ratio of linear momentum to total mass P0/M0: (a) normalized temporal widths α and β of the FF and the SH, respectively; (b) θ1 and velocity v0; (c) normalized peak amplitude A of the FF and B of the SH. In all cases the normalized phase mismatch is γ=10, the normalized temporal walk-off is δ=2, s/r=0.25, and M0=750. Continuous curves, stable solutions; dashed curves, unstable solutions.

Fig. 6
Fig. 6

Region of existence (shaded area) of stable stationary solutions in the (M0, P0) plane for normalized linear mismatch γ=10, normalized temporal walk-off δ=2, and s/r=0.25. Dotted curve, solutions with constant velocity v0=0; continuous curves, loci of the symmetry points Pcr=δ/(r-2s)M0.

Fig. 7
Fig. 7

Characteristic parameters θ1 and v0 plotted versus ratio of linear momentum to total mass, P0/M0, for normalized linear mismatch γ=-10, normalized temporal walk-off δ=-2, s/r =0.6, and M0=750. Continuous curve, stable solutions; dashed curve, unstable solutions.

Fig. 8
Fig. 8

Characteristic parameters of stationary solutions plotted versus the ratio of linear momentum to total mass, P0/M0, in the anomalous–normal regime: (a) normalized temporal widths α and β of the FF and the SH, respectively; (b) θ1 and velocity v0; (c) normalized peak amplitudes A of the FF and B of the SH. In all cases the normalized linear mismatch is γ=-10, the normalized temporal walk-off is δ=-2, s/r=-0.155, and M0 =750.

Fig. 9
Fig. 9

ξ evolution of FF (anomalous regime) and SH (normal regime) envelopes from a numerical BPM code for normalized linear mismatch γ=-10, a normalized temporal walk-off δ=-2, s/r=-0.155, and initial conditions corresponding to P0/M0=1.5 in Fig. 8.

Equations (28)

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iξu-(r/2)ττu+u*w=0,
iξw-iδτw-(s/2)ττw+γw+(1/2)u2=0,
u(τ, ξ)=2χ1χ2(r02/|k1|)A1,
w(τ, ξ)=χ1(r02/|k1|)A2 exp[iΔk(r02/|k1|)ξ],
χ1=ω02k1c2χ(2)(2ω0,-ω0),
χ2=2ω02k2c2χ(2)(ω0, ω0).
u(τ, ξ)=u0(τ-v0ξ)exp[iΦ0(τ-v0ξ)]exp(iϕaξ),
w(τ, ξ)=w0(τ-v0ξ)exp[iΨ0(τ-v0ξ)]exp(iϕbξ),
ϕb-2ϕa=0,
L=(i/2)(u*ξu-uξu*+w*ξw-wξw*)+(r/2)τuτu*+(s/2)τwτw*+i(δ/2)(wτw*-w*τw)+γww*+(1/2)(u2w*+u*2w).
M0=-+uu*dτ+-+2ww*dτ=M1+M2,
H=--+[(r/2)τuτu*+(s/2)τwτw*+γww*+i(δ/2)(wτw*-w*τw)+(1/2)(u2w*+u*2w)]dτ,
P=(i/2)-+(uτu*-u*τu)dτ+(i/2)-+(wτw*-w*τw)dτ=-P1-P2.
u(τ, ξ)=A exp-τ-τf2α2+i(a0+a1τ+a2τ2),
w(τ, ξ)=B exp-τ-τs2β2+i(b0+b1τ+b2τ2),
L=-+Ldτ=i=02Mμi dμidξ+Mθi dθidξ-H(A, B, α, β, τf, τs, μ1, μ2, θ0, θ1, θ2),
Mμi=-(1/4)-+τi(uu*+2ww*)dτ,
Mθi=-(1/4)-+τi(-uu*+2ww*)dτ,
L=i=01Mμi dμidξ+Mθi dθidξ+(1/8)2πrA2α+sB2β-(1/16)(μ12+θ12)[r(Mμ0-Mθ0)+s(Mμ0+Mθ0)]-(1/8)μ1θ1[-r(Mμ0-Mθ0)+s(Mμ0+Mθ0)]+2π αβA2B(2β2+α2)1/2×exp-αβθ12(2β2+α2)cos(θ0+τ0θ1)-(δ/2)(μ1+θ1)(Mμ0+Mθ0)-γ(Mμ0+Mθ0).
dMθ0dξ+2π αβA2B(2β2+α2)1/2exp-αβθ12(2β2+α2)
×sin(θ0+τ0θ1)=0.
θ0+τ0θ1=0, π.
dMμ1dξ=dτ0dξMμ0=Hμ1,
dMθ1dξ=dτ0dξMθ0=Hθ1,
Lηi=Mμ0ηidμ0dξ+Mθ0ηidθ0dξ-Hηi=0,
M(A, B, α, β)=2π(αA2+2βB2)=M0=M1+M2=const.,
P(A, B, α, β, μ1, θ1)=-(Mμ0μ1+Mθ0θ1)=-P0=-P1-P2=const,
dτ0dξ=-δM2M0-rP0M0-(2s-r)P2M0.

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