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Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method

Year 2015, Volume: 3 Issue: 2, 126 - 132, 19.01.2015

M. Naci Ozer Murat Koparan

Abstract

We perform a multiple scales analysis on the modified nonlinear Schr¨odinger (MNLS) equation in the Hamiltonian form.We derive, as amplitude equations, Korteweg-de Vries (KdV) flow equations in the bi-Hamiltonian form

Keywords

Manakov Equation Multiple Scale method; Flow Equation

References

  • S. A. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic wawes, Sov. Phys. JETP 38 (1974) 248-2
  • B. Crosignani, A. Cutolo, P. di Porto, J. Opt. Soc. Am. 72 (1982) 1136-1141.
  • C. R. Menyuk, IEEE JI Quant. Electron. 23 (1987) 174-176.
  • J. U. Kang, G. I. Stegeman, J. S. Aitchison, N. Akhmediev, Phys. Rev. Lett. 76 (1996) 3699-3702.
  • V. Kutuzov, V. M. Petnikova, V. V. Shuvalov, V. A. Vysloukh, Phys. Rev. E 57 (1998) 6056-6065.
  • A. Hasegawa, Y. Kodama, Oxford: Clarendon 1995.
  • Y. Kodama, Mathematical theory of NZR. Preprint, solv-int 1997.
  • Y. Kodama, A. Maruta, S. Wabnitz, Opt. Lett. 21 (1996) 1815-1817.
  • L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Tevhnol. 9 (1991) 362-367.
  • M. R. Adams, J. Harnad, J. Hurtubise, Commun. Math. Phys. 155 (1993) 385-415.
  • E. Alfinito, M. Leo, G. Soliani, L. Solombrino, Phys. Rev. E 53 (1995) 3159-3165.
  • C. Polymilis, K. Hizanidis, D. J. Frantzeskakis, Phys. Rev. E 58 (1998) 1112-1124.
  • A. V. Porubov, D. F. Parker, Wave Motion. 29 (1999) 97-109.
  • V. I. Pulov, I. M. Uzunov, E. J. Chakarov, Phys. Rev. E 57 (1998) 3468-3477.
  • P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, N. A. Kostov, Proc. R. Soc. Lond. A 456 (2000) 2263-2281.
  • R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56 (1997) 2213.
  • V.E. Zakharov and E.A. Kuznetsov. Multiscale Expansions in The Theory of Systems Integrable by The Inverse Scattering Transform. Physica D, 18 : 455–463, 1986.
  • A.P. Fordy. Soliton Theory: A survey of Results, MUP, Manchester, 1990.
  • A.P. Fordy and J. Gibbons. Factorisation of operator I. Miura transformations. J. Math. Phys, 21, 2508–2510, 1980.
  • A.H. Nayfeh. ”Perturbation Methods” , Wiley , New York , (1973).
  • A. R. Osborne and G. Boffetta. A Summable ultiscale Expansion For The KdV Equation. Nonlinear Evolution Equations: Integrability and Spectral Methods., eds. A. Degasperis, A.P.Fordy, M. Lakshmanan, MUP, Manchester and New York, 559–571, 19 M. Koparan, Derivation of Integrable Equations from Nonlinear Partial Equations by Multiple Scales Methods. Ph. D. Eskis¸ehir Osmangazi University, 2008.

Mehmet Naci Ozer1and Murat Koparan2

Year 2015, Volume: 3 Issue: 2, 126 - 132, 19.01.2015

M. Naci Ozer Murat Koparan

Abstract

References

  • S. A. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic wawes, Sov. Phys. JETP 38 (1974) 248-2
  • B. Crosignani, A. Cutolo, P. di Porto, J. Opt. Soc. Am. 72 (1982) 1136-1141.
  • C. R. Menyuk, IEEE JI Quant. Electron. 23 (1987) 174-176.
  • J. U. Kang, G. I. Stegeman, J. S. Aitchison, N. Akhmediev, Phys. Rev. Lett. 76 (1996) 3699-3702.
  • V. Kutuzov, V. M. Petnikova, V. V. Shuvalov, V. A. Vysloukh, Phys. Rev. E 57 (1998) 6056-6065.
  • A. Hasegawa, Y. Kodama, Oxford: Clarendon 1995.
  • Y. Kodama, Mathematical theory of NZR. Preprint, solv-int 1997.
  • Y. Kodama, A. Maruta, S. Wabnitz, Opt. Lett. 21 (1996) 1815-1817.
  • L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Tevhnol. 9 (1991) 362-367.
  • M. R. Adams, J. Harnad, J. Hurtubise, Commun. Math. Phys. 155 (1993) 385-415.
  • E. Alfinito, M. Leo, G. Soliani, L. Solombrino, Phys. Rev. E 53 (1995) 3159-3165.
  • C. Polymilis, K. Hizanidis, D. J. Frantzeskakis, Phys. Rev. E 58 (1998) 1112-1124.
  • A. V. Porubov, D. F. Parker, Wave Motion. 29 (1999) 97-109.
  • V. I. Pulov, I. M. Uzunov, E. J. Chakarov, Phys. Rev. E 57 (1998) 3468-3477.
  • P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, N. A. Kostov, Proc. R. Soc. Lond. A 456 (2000) 2263-2281.
  • R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56 (1997) 2213.
  • V.E. Zakharov and E.A. Kuznetsov. Multiscale Expansions in The Theory of Systems Integrable by The Inverse Scattering Transform. Physica D, 18 : 455–463, 1986.
  • A.P. Fordy. Soliton Theory: A survey of Results, MUP, Manchester, 1990.
  • A.P. Fordy and J. Gibbons. Factorisation of operator I. Miura transformations. J. Math. Phys, 21, 2508–2510, 1980.
  • A.H. Nayfeh. ”Perturbation Methods” , Wiley , New York , (1973).
  • A. R. Osborne and G. Boffetta. A Summable ultiscale Expansion For The KdV Equation. Nonlinear Evolution Equations: Integrability and Spectral Methods., eds. A. Degasperis, A.P.Fordy, M. Lakshmanan, MUP, Manchester and New York, 559–571, 19 M. Koparan, Derivation of Integrable Equations from Nonlinear Partial Equations by Multiple Scales Methods. Ph. D. Eskis¸ehir Osmangazi University, 2008.
There are 21 citations in total.

Details

Journal Section Articles
Authors

M. Naci Ozer This is me

Murat Koparan This is me

Publication Date January 19, 2015
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Ozer, M. N., & Koparan, M. (2015). Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method. New Trends in Mathematical Sciences, 3(2), 126-132.
AMA Ozer MN, Koparan M. Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method. New Trends in Mathematical Sciences. January 2015;3(2):126-132.
Chicago Ozer, M. Naci, and Murat Koparan. “Korteweg-De Vries Flow Equations from Manakov Equation by Multiple Scale Method”. New Trends in Mathematical Sciences 3, no. 2 (January 2015): 126-32.
EndNote Ozer MN, Koparan M (January 1, 2015) Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method. New Trends in Mathematical Sciences 3 2 126–132.
IEEE M. N. Ozer and M. Koparan, “Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method”, New Trends in Mathematical Sciences, vol. 3, no. 2, pp. 126–132, 2015.
ISNAD Ozer, M. Naci - Koparan, Murat. “Korteweg-De Vries Flow Equations from Manakov Equation by Multiple Scale Method”. New Trends in Mathematical Sciences 3/2 (January 2015), 126-132.
JAMA Ozer MN, Koparan M. Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method. New Trends in Mathematical Sciences. 2015;3:126–132.
MLA Ozer, M. Naci and Murat Koparan. “Korteweg-De Vries Flow Equations from Manakov Equation by Multiple Scale Method”. New Trends in Mathematical Sciences, vol. 3, no. 2, 2015, pp. 126-32.
Vancouver Ozer MN, Koparan M. Korteweg-de Vries Flow Equations from Manakov Equation by Multiple Scale method. New Trends in Mathematical Sciences. 2015;3(2):126-32.