Abstract

We investigate stability of moving solitons in a nonlinear optical fiber with short segments of a relatively strong Bragg grating (BG) periodically inserted into it. The model is related to a class of composite and artificial optical media that have recently attracted much interest. The analysis is focused on moving solitons, as this is relevant to the experiment. By means of systematic simulations, we find that, in accordance with a qualitative consideration, the stability region for moving solitons first increases with velocity c but then decreases (in contrast to the uniform fiber BG, where the stability of the solitons depends very weakly on c). As well as in the uniform system, solitons do not exist for c exceeding the group velocity of light in the fiber, nor may they be stable with negative intrinsic frequencies. Collisions between solitons moving in opposite directions are also studied in a systematic way. A difference from the situation in the uniform fiber BG is strong shrinkage and eventual disappearance of a region where the collision results in a merger of the two solitons into a single one, which is explained too. Fiber waveguides with BG superstructures being currently available, moving solitons can be created in them by means of experimental techniques similar to those that have helped to generate solitons in uniform fiber gratings.

© 2008 Optical Society of America

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2006 (6)

J. T. Mok, C. M. de Sterke, I. C. M. Litte, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775-780 (2006).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416-421 (2006).
[CrossRef]

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

N. C. Panoiu, R. M. Osgood, and B. A. Malomed, “Semi-discrete composite solitons in arrays of quadratically nonlinear waveguides,” Opt. Lett. 31, 1097-1099 (2006).
[CrossRef] [PubMed]

O. F. Oxtoby, D. E. Pelinovsky, and I. V. Barashenkov, “Travelling kinks in discrete phi(4) models,” Nonlinearity 19, 217-235 (2006).
[CrossRef]

B. A. Malomed, J. Fujioka, A. Espinosa-Ceron, R. F. Rodriguez, and S. González, “Moving embedded lattice solitons,” Chaos 16, 013112 (2006).
[CrossRef] [PubMed]

2005 (5)

D. E. Pelinovsky and V. M. Rothos, “Bifurcations of travelling wave solutions in the discrete NLS equations,” Physica D 202, 16-36 (2005).
[CrossRef]

I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, “Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity,” Phys. Rev. E 71, 016613 (2005).
[CrossRef]

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, “Trapping Bragg solitons by a pair of defects,” Phys. Rev. E 71, 066601 (2005).
[CrossRef]

F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express 13, 4314-4324 (2005).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

2004 (7)

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

W. Li and A. Smerzi, Phys. Rev. E 70, 016605 (2004).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and splitting of gap solitons in apodized Bragg Gratings,” J. Mod. Opt. 51, 2141-2158 (2004).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Dynamics of positive- and negative-mass solitons in optical lattices and inverted traps,” J. Phys. B 37, 1443-1459 (2004).
[CrossRef]

B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, “Bright Bose-Einstein gap solitons of atoms with repulsive interaction,” Phys. Rev. Lett. 92, 230401 (2004).
[CrossRef] [PubMed]

2003 (10)

E. A. Ostrovskaya and Yu. S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
[CrossRef] [PubMed]

P. G. Kevrekidis, “On a class of discretizations of Hamiltonian nonlinear partial differential equations,” Physica D 183, 68-86 (2003).
[CrossRef]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Interaction of a soliton with a localized gain in a fiber Bragg grating,” Phys. Rev. E 67, 026608 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Interaction of a soliton with a local defect in a fiber Bragg grating,” J. Opt. Soc. Am. B 20, 725-735 (2003).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, “Discrete gap solitons in a diffraction-managed waveguide array,” Eur. Phys. J. D 23, 421-236 (2003).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

2002 (6)

A. A. Sukhorukov and Yu. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112-2114 (2002).
[CrossRef]

R. H. Goodman, R. E. Slusher, and M. I. Weinstein, “Stopping light on a defect,” J. Opt. Soc. Am. B 19, 1635-1652 (2002).
[CrossRef]

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

M. J. Ablowitz, Z. H. Musslimani, and G. Biondini, “Methods for discrete solitons in nonlinear lattices,” Phys. Rev. E 65, 026602 (2002).
[CrossRef]

I. Carrusotto, D. Embriaco, and G. C. La Rocca, “Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices,” Phys. Rev. A 65, 053611 (2002).
[CrossRef]

B. B. Baizakov, V. V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105-5119 (2002).
[CrossRef]

2001 (1)

R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481-489 (2001).
[CrossRef]

2000 (2)

1999 (4)

1998 (2)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

1997 (1)

A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309-4341 (1997).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1995 (1)

F. Ouelette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broad-band and WDM dispersion compensationusing chirped sampled fiber Bragg gratings,” Electron. Lett. 31, 899-901 (1995).
[CrossRef]

1994 (1)

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

1993 (1)

D. B. Duncan, J. C. Eilbeck, H. Feddersen, and J. A. D. Wattis, “Solitons on lattices,” Physica D 68, 1-11 (1993).
[CrossRef]

1989 (2)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structure,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear-optical response of superlattices,” Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

1986 (1)

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344-3355 (1986).
[CrossRef]

1981 (1)

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with large group retardation,” Zh. Tekh. Fiz. 51, 902 (1981) Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin,[Sov. Phys. Tech. Phys. 26, 541 (1981)].

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

1977 (1)

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634-3646 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Chaos (1)

B. A. Malomed, J. Fujioka, A. Espinosa-Ceron, R. F. Rodriguez, and S. González, “Moving embedded lattice solitons,” Chaos 16, 013112 (2006).
[CrossRef] [PubMed]

Electron. Lett. (1)

F. Ouelette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broad-band and WDM dispersion compensationusing chirped sampled fiber Bragg gratings,” Electron. Lett. 31, 899-901 (1995).
[CrossRef]

Eur. Phys. J. D (1)

P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, “Discrete gap solitons in a diffraction-managed waveguide array,” Eur. Phys. J. D 23, 421-236 (2003).
[CrossRef]

Europhys. Lett. (1)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006-1012 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268-1270 (1999).
[CrossRef]

J. Appl. Phys. (1)

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344-3355 (1986).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and splitting of gap solitons in apodized Bragg Gratings,” J. Mod. Opt. 51, 2141-2158 (2004).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. B (2)

H. Sakaguchi and B. A. Malomed, “Dynamics of positive- and negative-mass solitons in optical lattices and inverted traps,” J. Phys. B 37, 1443-1459 (2004).
[CrossRef]

B. B. Baizakov, V. V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105-5119 (2002).
[CrossRef]

Nat. Phys. (1)

J. T. Mok, C. M. de Sterke, I. C. M. Litte, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775-780 (2006).
[CrossRef]

Nature (1)

W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147-150 (2003).
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Nonlinearity (1)

O. F. Oxtoby, D. E. Pelinovsky, and I. V. Barashenkov, “Travelling kinks in discrete phi(4) models,” Nonlinearity 19, 217-235 (2006).
[CrossRef]

Opt. Commun. (2)

R. Driben and B. A. Malomed, “Split-step solitons in long fiber links,” Opt. Commun. 185, 439-456 (2000).
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R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481-489 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (3)

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416-421 (2006).
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J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

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[CrossRef]

Phys. Rev. A (1)

I. Carrusotto, D. Embriaco, and G. C. La Rocca, “Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices,” Phys. Rev. A 65, 053611 (2002).
[CrossRef]

Phys. Rev. E (10)

M. J. Ablowitz, Z. H. Musslimani, and G. Biondini, “Methods for discrete solitons in nonlinear lattices,” Phys. Rev. E 65, 026602 (2002).
[CrossRef]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

S. Flach, Y. Zolotaryuk, and K. Kladko, “Moving lattice kinks and pulses: An inverse method,” Phys. Rev. E 59, 6105-6115 (1999).
[CrossRef]

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

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[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

W. Li and A. Smerzi, Phys. Rev. E 70, 016605 (2004).
[CrossRef]

I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, “Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity,” Phys. Rev. E 71, 016613 (2005).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Interaction of a soliton with a localized gain in a fiber Bragg grating,” Phys. Rev. E 67, 026608 (2003).
[CrossRef]

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, “Trapping Bragg solitons by a pair of defects,” Phys. Rev. E 71, 066601 (2005).
[CrossRef]

Phys. Rev. Lett. (11)

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D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 (2004).
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Physica D (3)

P. G. Kevrekidis, “On a class of discretizations of Hamiltonian nonlinear partial differential equations,” Physica D 183, 68-86 (2003).
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D. E. Pelinovsky and V. M. Rothos, “Bifurcations of travelling wave solutions in the discrete NLS equations,” Physica D 202, 16-36 (2005).
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Rev. Sci. Instrum. (1)

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Zh. Tekh. Fiz. (1)

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Other (6)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1994), Vol. XXXIII, Chap. III, pp. 203-260.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Typical example of the motion of a robust soliton in the model with spatial period a = 0.5 . ABS(U) and ABS(V) stand for u ( x , t ) and v ( x , t ) in (a) and (b), respectively. The soliton was created from initial configuration (3, 4) with θ = π 3 and c = 0.3 . The soliton remained stable during the integration time, keeping constant values of its amplitude and velocity much longer than shown in the figure, and cycling in the system with periodic boundary conditions. In this and all other figures, time is shown in units of T a c (in the present figure, T = 5 3 ).

Fig. 2
Fig. 2

Typical example of the decay of an unstable soliton in the model with spatial period a = 0.75 is displayed by means of contour plots (top view) of v ( x , t ) (the picture for u ( x , t ) is quite similar). The soliton was created from initial configuration (3, 4) with θ = π 2 and c = 0.9 .

Fig. 3
Fig. 3

Stability regions for moving solitons in the plane of the velocity and superstructure period, c and a, for four different values of the soliton’s intrinsic parameter, θ. Maximum values of a at which stable solitons could be found are a max = 2.5 , 1.3, 1.0, and 0.75, for θ = π 6 , π 3 , 5 π 12 , and π 2 , respectively. Note that the expansion of the stability region to entire interval 0 c < 1 at small a complies with the known stability properties of moving solitons in the standard model of the uniform fiber Bragg grating [13].

Fig. 4
Fig. 4

Typical examples of elastic (a) and destructive (b) collisions between two solitons with, respectively, θ = 0.1 π (i.e., small θ, corresponding to the quasi-NLS limit), and θ = π 2 (i.e., close to the stability limit). In either case, the solitons collided with velocities of c = ± 0.4 . The spatial period is a = 0.5 (a), and a = 0.75 (b). In this and following figures, the collision is displayed by mean of contour plots of u ( x , t ) (top view), supplemented, if relevant, with the gray-shade bar. The picture is quite similar for v ( x , t ) .

Fig. 5
Fig. 5

Typical example of the merger of two slowly colliding solitons into a single standing pulse. Parameters are a = 0.5 , θ = 0.3 π , and c = ± 0.05 .

Fig. 6
Fig. 6

Diagrams showing outcomes of collisions between identical solitons with intrinsic parameter θ [see Eqs. (3, 4)] moving at velocities ± c , for different periods a of the Bragg-grating superstructure. The collision center coincides with the central point of one of the Bragg-grating segments. Capital letters mark the following outcomes of the collision: E, elastic collision; M, merger of the solitons into a single pulse; S, separation of the solitons after the collision with reduced (slower) velocities; R, the velocities are not affected by the collision, but conspicuous radiation loss is observed; F, large radiation loss takes place, while the velocities increase after the collision (the solitons move faster); D, the collision leads to strong deformation of the solitons.

Fig. 7
Fig. 7

Examples of collisions at values of the parameters that would correspond to the merger in the uniform counterpart of the model, cf. Fig. 6a: θ = 0.2 π , a = 0.5 , and c = ± 0.10 (a) or c = ± 0.15 (b). Note that the example of the merger displayed in Fig. 5 pertains to the same parameters, except for c = ± 0.05 . In both cases (a) and (b), the separation of the solitons after the collision, with some reduction of the velocities, is observed, which places the outcomes in region S of the chart shown in Fig. 6c (that pertains to the same period, a = 0.5 ). In panel (a), the solitons are stronger perturbed by the collision, which is explained by the smaller collision velocity in this case [in other words, by proximity to the border of the merger region in Fig. 6c]. We stress that, despite the local increase of the field amplitude in panel (a), the total energy is conserved in this case [actually, the total nonconservation (loss)] of energy in the entire numerical simulation displayed in panel (a) is 0.29 % ; in other simulations, the accuracy of the conservation of the total energy was higher than this.

Equations (11)

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i u t + i u x + ( 1 2 u 2 + υ 2 ) u + κ υ = 0 ,
i υ t i υ x + ( 1 2 υ 2 + u 2 ) υ + κ u = 0 .
i u t + i u x + ( 1 2 u 2 + υ 2 ) u + [ n = + δ ( x a n ) ] υ = 0 ,
i υ t i υ x + ( 1 2 υ 2 + u 2 ) υ + [ n = + δ ( x a n ) ] u = 0 .
u sol = 2 κ ( 1 + c ) 3 c 2 ( 1 c 2 ) 1 4 W ( X ) exp [ i ϕ ( X ) i T cos θ ] ,
v sol = 2 κ ( 1 c ) 3 c 2 ( 1 c 2 ) 1 4 W * ( X ) exp [ i ϕ ( X ) i T cos θ ] ,
X = κ x c t 1 c 2 , T = κ t c x 1 c 2 ,
W ( X ) = ( sin θ ) sech ( X sin θ i θ 2 ) ,
ϕ ( X ) = 4 c 3 c 2 tan 1 { tan ( θ 2 ) tanh ( X sin θ ) X } .
E + ( u 2 + v 2 ) d x = 8 θ ( 1 c 2 ) 3 c 2 ,
P = i + ( u x * u + v x * v ) d x = 8 κ c 1 c 2 [ 7 c 2 ( 3 c 2 ) 2 ( sin θ θ cos θ ) + θ cos θ 3 c 2 ] .

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