Abstract

Coupled nonlinear Schrödinger equations model several interesting physical phenomena. We used a trigonometric function transform method based on a homogeneous balance to solve the coupled higher-order nonlinear Schrödinger equations. We obtained four pairs of exact solitary-wave solutions including a dark and a bright-soliton pair, a bright- and a dark-soliton pair, a bright- and a bright-soliton pair, and the last pair, a combined bright–dark-soliton pair.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Mahalingam and K. Porsezian, “Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations,” J. Phys. A Math. Nucl. Gen. 35, 3099–3110 (2002).
    [CrossRef]
  2. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  3. R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
    [CrossRef]
  4. R. Radhakrishnan and M. Lakshmanan, “Bright and dark soliton solutions to coupled nonlinear Schrödinger equations,” J. Phys. A 28, 2683–2692 (1995).
    [CrossRef]
  5. K. Porsezian and K. Nakkeeran, “Optical solitons in birefringent fibre—Bäcklund transformation approach,” Pure Appl. Opt. 6, L7–L11 (1997).
    [CrossRef]
  6. K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
    [CrossRef]
  7. C. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A 224, 77–84 (1996).
    [CrossRef]
  8. N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
    [CrossRef]
  9. R. Radhakrishnan and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
    [CrossRef]
  10. D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
    [CrossRef]
  11. Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
    [CrossRef] [PubMed]

2002

A. Mahalingam and K. Porsezian, “Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations,” J. Phys. A Math. Nucl. Gen. 35, 3099–3110 (2002).
[CrossRef]

D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
[CrossRef]

2000

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

1999

K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
[CrossRef]

1997

K. Porsezian and K. Nakkeeran, “Optical solitons in birefringent fibre—Bäcklund transformation approach,” Pure Appl. Opt. 6, L7–L11 (1997).
[CrossRef]

1996

R. Radhakrishnan and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

C. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A 224, 77–84 (1996).
[CrossRef]

1995

R. Radhakrishnan and M. Lakshmanan, “Bright and dark soliton solutions to coupled nonlinear Schrödinger equations,” J. Phys. A 28, 2683–2692 (1995).
[CrossRef]

1991

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

1986

R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
[CrossRef]

1974

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Cao, D.

D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
[CrossRef]

Lakshmanan, M.

R. Radhakrishnan and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

R. Radhakrishnan and M. Lakshmanan, “Bright and dark soliton solutions to coupled nonlinear Schrödinger equations,” J. Phys. A 28, 2683–2692 (1995).
[CrossRef]

R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
[CrossRef]

Li, L.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Li, Z. H.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Mahalingam, A.

A. Mahalingam and K. Porsezian, “Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations,” J. Phys. A Math. Nucl. Gen. 35, 3099–3110 (2002).
[CrossRef]

K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
[CrossRef]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Nakkeeran, K.

K. Porsezian and K. Nakkeeran, “Optical solitons in birefringent fibre—Bäcklund transformation approach,” Pure Appl. Opt. 6, L7–L11 (1997).
[CrossRef]

Porsezian, K.

A. Mahalingam and K. Porsezian, “Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations,” J. Phys. A Math. Nucl. Gen. 35, 3099–3110 (2002).
[CrossRef]

K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
[CrossRef]

K. Porsezian and K. Nakkeeran, “Optical solitons in birefringent fibre—Bäcklund transformation approach,” Pure Appl. Opt. 6, L7–L11 (1997).
[CrossRef]

Radhakrishnan, R.

R. Radhakrishnan and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

R. Radhakrishnan and M. Lakshmanan, “Bright and dark soliton solutions to coupled nonlinear Schrödinger equations,” J. Phys. A 28, 2683–2692 (1995).
[CrossRef]

Sahadevan, R.

R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
[CrossRef]

Sasa, N.

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Satsuma, J.

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Shanmugha Sundaram, P.

K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
[CrossRef]

Tamizhmani, K. M.

R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
[CrossRef]

Tian, H. P.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Yan, C.

C. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A 224, 77–84 (1996).
[CrossRef]

Yan, J.

D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
[CrossRef]

Zhang, Y.

D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
[CrossRef]

Zhou, G. S.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

J. Phys. A

R. Sahadevan, K. M. Tamizhmani, and M. Lakshmanan, “Painleve analysis and integrability of coupled non-linear Schrödinger equations,” J. Phys. A 19, 1783–1792 (1986).
[CrossRef]

R. Radhakrishnan and M. Lakshmanan, “Bright and dark soliton solutions to coupled nonlinear Schrödinger equations,” J. Phys. A 28, 2683–2692 (1995).
[CrossRef]

J. Phys. A Math. Nucl. Gen.

A. Mahalingam and K. Porsezian, “Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations,” J. Phys. A Math. Nucl. Gen. 35, 3099–3110 (2002).
[CrossRef]

K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Complete integrability of N-coupled higher-order nonlinear Schrödinger equations in nonlinear optics,” J. Phys. A Math. Nucl. Gen. 32, 8731–8738 (1999).
[CrossRef]

J. Phys. Soc. Jpn.

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Phys. Lett. A

C. Yan, “A simple transformation for nonlinear waves,” Phys. Lett. A 224, 77–84 (1996).
[CrossRef]

D. Cao, J. Yan, and Y. Zhang, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Phys. Lett. A 297, 68–71 (2002).
[CrossRef]

Phys. Rev. E

R. Radhakrishnan and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

Phys. Rev. Lett.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Pure Appl. Opt.

K. Porsezian and K. Nakkeeran, “Optical solitons in birefringent fibre—Bäcklund transformation approach,” Pure Appl. Opt. 6, L7–L11 (1997).
[CrossRef]

Sov. Phys. JETP

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(a) Relations S1N2 when N1=1. (b) Relations S2N2 when N1=1.

Fig. 2
Fig. 2

(a) Pulse shapes of q1 for five values of N2 and N1=1. (b) Pulse shapes of q2 for five values of N2 and N1=1.

Fig. 3
Fig. 3

(a) Initial profile, numerical result, and exact solution and (b) pulse evolution when z=40, =0.0099, and Nj=1 (j=1,2).

Fig. 4
Fig. 4

(a) Initial profile, numerical result, and exact solution and (b) pulse evolution, where z=40, =0.0099, N1=1, and N2=0.8.

Fig. 5
Fig. 5

(a) Initial profile, numerical result, and exact solution and (b) pulse evolution when z=40, =0.01, and Nj=1 (j=1,2).

Equations (28)

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

dφ(ξ)dξ=sin[φ(ξ)]
cos[φ(ξ)]=-tanh(ξ+ξ0),
sin[φ(ξ)]=sech(ξ+ξ0),
iqjz+12qjtt+(|qj|2+|q3-j|2)qj+i[qjttt+6(|qj|2+|q3-j|2)qjt+3qj(|qj|2+|q3-j|2)t]=0,(j=1,2),
qj(z, t)=Aj(z, t)exp[iψj(z, t)](j=1,2).
idAjdz-Ajkj+12d2Ajdt2-idAjdthj-12Ajhj2+Aj3+AjA3-j2+id3Ajdt3+3d2Ajdt2hj-3idAjdthj2-Ajhj3+12idAjdtAj2+6idAjdtA3-j2+6Aj3hj+6AjA3-j2hj+6iAjA3-jdA3-jdt=0,
Aj(z, t)=Mj+l=1m(Njl cosl φ+Sjl sinl φ)(j=1,2),
φ=φ(ξ),dφ(ξ)dξ=sin φ(ξ),
Aj(ξ)=Mj+Nj cos φ+Sj sin φ.
M1=M2=0,S1=0,N2=0,
β=722S22-242+112,
N1=±2S22-121/2,2S22-1>0,
q1(z, t)=-N1 tanh(t-βz+ξ0)×expi-11082z+16t,
q2(z, t)=S2 sech(t-βz+ξ0)×expi-11082z+16t.
M1=M2=0,S2=0,N1=0,
β=722N22+122+112,
S1=±1+2N2221/2,
q1(z, t)=S1 sech(t-βz+ξ0)×expi-11082z+16t,
q2(z, t)=-N2 tanh(t-βz+ξ0)×expi-11082z+16t.
Mj=0,Nj=0,(j=1,2),
β=122+112,Sj=±1-2S3-j221/2,
1-2S3-j2>0,
qj(z, t)=Sj sech(t-βz+ξ0)×expi-11082z+16t.
Mj=0,
β=722Nj2+722N3-j2+122+112,(j=1,2),
Sj=±(-1)j2Nj2+2N3-j2+12Nj2+2N3-j21/2,
N3-j=±(-1)jQN3-j.
qj(z, t)=[-Nj tanh(t-βz+ξ0)+Sj sech(t-βz+ξ0)]*×expi-11082z+16t(j=1,2).

Metrics