Abstract

We study defect solitons (DSs) in a parity-time (PT) symmetric superlattice with focusing Kerr nonlinearity. The properties of the DSs with a PT symmetrical potential are obviously different from those in a superlattice with a real refractive index. Unusual features stemming from PT symmetry can be found. Research results show that the solitons with a zero defect or a positive defect can exist and stably propagate in the semi-infinite gap, but they cannot exist in the first gap. For the case of a negative defect, the soliton can stably exist in both the semi-infinite gap and the first gap.

© 2011 OSA

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References

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  1. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef] [PubMed]
  3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  4. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef] [PubMed]
  5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  6. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
    [CrossRef]
  7. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
    [CrossRef]
  8. Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007).
    [CrossRef] [PubMed]
  9. X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010).
    [CrossRef] [PubMed]
  10. X. Zhu, H. Wang, T. W. Wu, and L. X. Zheng, “Defect solitons in triangular optical lattices,” J. Opt. Soc. Am. B 28(3), 521 (2011).
    [CrossRef]
  11. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
    [CrossRef]
  12. Z. Lu and Z. Zhang, “Surface line defect solitons in square optical lattice,” Opt. Express 19(3), 2410 (2011).
    [CrossRef] [PubMed]
  13. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
    [CrossRef] [PubMed]
  14. A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. 30(14), 1849–1851 (2005).
    [CrossRef] [PubMed]
  15. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
    [CrossRef]
  16. A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
    [CrossRef]
  17. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
    [CrossRef] [PubMed]
  18. H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express 19(5), 4030 (2011).
    [CrossRef] [PubMed]
  19. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
    [CrossRef]
  20. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
    [CrossRef]

2011 (3)

2010 (4)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010).
[CrossRef] [PubMed]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
[CrossRef] [PubMed]

2009 (3)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[CrossRef] [PubMed]

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

2008 (5)

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[CrossRef]

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
[CrossRef]

2007 (2)

Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007).
[CrossRef] [PubMed]

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

2006 (1)

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
[CrossRef]

2005 (1)

1998 (1)

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Assanto, G.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Bender, C. M.

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

Böttcher, S.

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Chen, W. H.

Chen, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Chen, Z.

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Christodoulides, D. N.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Dreisow, F.

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

Guo, Z.

He, Y. J.

Heinrich, M.

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

Kottos, T.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

Lederer, F.

Li, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Liu, S.

Lu, Z.

Lucchetti, L.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Malomed, B. A.

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

Nolte, S.

Pang, W.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Pertsch, T.

Piccardi, A.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

Simoni, F.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Sukhorukov, A. A.

Szameit, A.

Torner, L.

Tünnermann, A.

Vysloukh, V. A.

Wang, H.

Wang, H. Z.

Wang, J.

Wu, T. W.

Yang, J.

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Ye, F.

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[CrossRef]

Yu, Z.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhang, H.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhang, Z.

Zheng, L. X.

Zhou, J.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhou, K.

Zhu, X.

Appl. Phys. Lett. (1)

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

J. Comput. Phys. (1)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. A (3)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[CrossRef]

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[CrossRef]

Phys. Rev. E (1)

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

Stud. Appl. Math. (1)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(Color online) (a) Band structure of the superlattice with V 0 = 6 and W 0 = 0.4. Lattices intensity profile of the PT superlattice with (b) ε = 0, (c) ε = 0.5, and (d) ε = −0.5. Blue line: real part, red line: imaginary part.

Fig. 2
Fig. 2

(Color online) (ε = −0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −6.0 (point A in Fig. 2(a)). (d) Stable soliton with μ = −2.0 (point B in Fig. 2(a)). (e) Unstable soliton with μ = −1.74 (point C in Fig. 2(a)). In (c)–(e): black lines plot the real part of f and red lines plot the imaginary part of f. (f) Soliton propagation for (c). (g) Soliton propagation for (d). (h) Soliton propagation for (e).

Fig. 3
Fig. 3

(Color online) (ε = 0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Stable soliton with μ = −6.5 (point A in Fig. 3(a)). (c) Stable soliton with μ = −5.0 (point B in Fig. 3(a)). In (b) and (c): black lines plot the real part of f and red lines plot the imaginary part of f. (d) Soliton propagation for (b). (e) Soliton propagation for (c).

Fig. 4
Fig. 4

(Color online) (ε = 0) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −5.0 (point A in Fig. 4(a)). (d) Unstable soliton with μ = −3.3 (point B in Fig. 4(a)). In (c) and (d): black lines plot the real part of f and red lines plot the imaginary part of f. (e) Soliton propagation for (c). (f) Soliton propagation for (d).

Fig. 5
Fig. 5

(Color online) Stable (green) and unstable domains (μ, ε) of PT solitons in the semi-infinite gap (a) and (b), and the first gap (c).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

i U z + U xx + [ V ( x ) + i W ( x ) ] U + | U | 2 U = 0 ,
V ( x ) = { V 0 { ε 1 sin 2 [ ( x + π 2 ) ] + ( 1 ε 1 ) sin 2 [ 2 ( x + π 2 ) ] } , ( x > π 2 or x < π 2 ) , 0.78125 V 0 sin 2 [ ( x + π 2 ) ] [ 1 + ε exp ( x 8 128 ) ] , ( π 2 x π 2 ) ,
W ( x ) = W 0 sin ( 2 x ) ,
f xx + [ V ( x ) + iW ( x ) ] f + | f | 2 f + μ f = 0.
U = exp ( i μ z ) { f ( x ) + [ v ( x ) w ( x ) ] exp ( δ z ) + [ v ( x ) + w ( x ) ] * exp ( δ * z ) }
{ δ v = i [ d 2 w / d x 2 + μ w + Vw iWv + 2 | f | 2 w 1 2 ( f 2 f * 2 ) v 1 2 ( f 2 + f * 2 ) w ] , δ w = i [ d 2 v / d x 2 + μ v + Vw iWw + 2 | f | 2 v + 1 2 ( f 2 f * 2 ) w + 1 2 ( f 2 + f * 2 ) v ] .

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