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ANALYTICAL AND NUMERICAL ASPECTS OF THE DISSIPATIVE NONLINEAR SCHRODINGER EQUATION

Year 2016, Volume: 6 Issue: 1, 135 - 142, 01.06.2016

Abstract

In this paper various analytical and numerical aspects of the dissipative nonlinear Schr¨odinger equation d-NLS equation are discussed. Decaying solitary wave type solutions derived by Demiray is reviewed and a new approximate solitary wave type solution of the d-NLS equation is introduced in order to make comparisons. Also a split-step Fourier scheme is proposed for numerical solution of the d-NLS equation and the analytical solutions are compared with the numerical results.

References

  • Bayındır,C., (2009), Implementation of a Computational Model for Random Directional Seas and Underwater Acoustics, MS Thesis, University of Delaware.
  • Bayındır,C., (2015), Compressive split-step Fourier method, TWMS J. Appl. and Engr. Math., vol. 5, no.2, pp. 298-306.
  • Bayındır,C., (2016), Early detection of rogue waves by the wavelet transforms, Physics Letters A , vol. 380, pp. 156-161.
  • Bayındır,C., (2016), Compressive spectral method for the simulation of the water waves, TWMS J. Appl. and Engr. Math., vol. 6, no.1, to appear.
  • Bayındır,C., (2015), Hesaplamalı akı¸skanlar mekani˘gi ¸calı¸smaları i¸cin sıkı¸stırılabilir Fourier tayfı y¨ontemi, XIX. T¨urk Mekanik Kongresi, Trabzon, (In Turkish)
  • Bayındır,C., (2015), Okyanus dalgalarının sıkı¸stırılabilir Fourier tayfı y¨ontemiyle hızlı modellenmesi, XIX. T¨urk Mekanik Kongresi, Trabzon. (In Turkish)
  • Bayındır,C., (2015), S¨on¨uml¨u de˘gi¸stirilmi¸s Korteweg de-Vries (KdV) denkleminin analitik ve hesapla- malı ¸c¨oz¨um kar¸sıla¸stırması, T¨urk Mekanik Kongresi, Trabzon. (In Turkish)
  • Bogomolov,Y.L. and Yunakovsky,A.D. , (2006), Split-step Fourier method for nonlinear Schrodinger equation, Proceedings of the International Conference Day on Diffraction, pp. 34-42.
  • Canuto,C., Hussaini,M.Y., Quarteroni,A. and Zang,T.A. , (2006), Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin.
  • Demiray,H., (2003), An analytical solution to the dissipative nonlinear Schrodinger equation, Applied Mathematics and Computation, 145, pp. 179-184.
  • Demiray,H. and Bayındır,C., (2015), A note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distribution, Physics of Plasmas, 22, 092105.
  • Hardin,R.H. and Tappert,F.D., (1973), Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation, SIAM Review Chronicles, 15, pp. 423-423. [13] Karjadi,E.A., Badiey,M. and Kirby,J.T., (2010), Impact of surface gravity waves on high-frequency acoustic propagation in shallow water, The Journal of the Acoustical Society of America, 127, pp. 1787-1787.
  • Karjadi,E.A., Badiey,M., Kirby,J.T. and Bayindir,C., (2012), The effects of surface gravity waves on high-frequency acoustic propagation in shallow water, IEEE Journal of Oceanic Engineering, 37, pp. 112-121.
  • Lamb,H., (1945), Hydrodynamics, Dover Publications, New York.
  • Sanz-Serna,J.M. and Calvo,M.P., (1994), Numerical Hamiltonian Problems, Chapman and Hall, Lon- don.
  • Taha,T.R. and Ablowitz,M.J., (1984), Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Nonlinear Schrodinger Equation, Journal of Computational Physics, 22, pp. 203-230.
  • Trefethen,L.N., (2000), Spectral Methods in MATLAB, SIAM, Philadelphia.
  • Zakharov,V.E., (1968), Stability of periodic waves of finite amplitude on the surface of a deep fluid, Soviet Physics JETP, 2, pp. 190-194.
  • C. Bayındır for the photography and short autobiography, see TWMS J. App. Eng. Math., V.5, N.2.
Year 2016, Volume: 6 Issue: 1, 135 - 142, 01.06.2016

Abstract

References

  • Bayındır,C., (2009), Implementation of a Computational Model for Random Directional Seas and Underwater Acoustics, MS Thesis, University of Delaware.
  • Bayındır,C., (2015), Compressive split-step Fourier method, TWMS J. Appl. and Engr. Math., vol. 5, no.2, pp. 298-306.
  • Bayındır,C., (2016), Early detection of rogue waves by the wavelet transforms, Physics Letters A , vol. 380, pp. 156-161.
  • Bayındır,C., (2016), Compressive spectral method for the simulation of the water waves, TWMS J. Appl. and Engr. Math., vol. 6, no.1, to appear.
  • Bayındır,C., (2015), Hesaplamalı akı¸skanlar mekani˘gi ¸calı¸smaları i¸cin sıkı¸stırılabilir Fourier tayfı y¨ontemi, XIX. T¨urk Mekanik Kongresi, Trabzon, (In Turkish)
  • Bayındır,C., (2015), Okyanus dalgalarının sıkı¸stırılabilir Fourier tayfı y¨ontemiyle hızlı modellenmesi, XIX. T¨urk Mekanik Kongresi, Trabzon. (In Turkish)
  • Bayındır,C., (2015), S¨on¨uml¨u de˘gi¸stirilmi¸s Korteweg de-Vries (KdV) denkleminin analitik ve hesapla- malı ¸c¨oz¨um kar¸sıla¸stırması, T¨urk Mekanik Kongresi, Trabzon. (In Turkish)
  • Bogomolov,Y.L. and Yunakovsky,A.D. , (2006), Split-step Fourier method for nonlinear Schrodinger equation, Proceedings of the International Conference Day on Diffraction, pp. 34-42.
  • Canuto,C., Hussaini,M.Y., Quarteroni,A. and Zang,T.A. , (2006), Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin.
  • Demiray,H., (2003), An analytical solution to the dissipative nonlinear Schrodinger equation, Applied Mathematics and Computation, 145, pp. 179-184.
  • Demiray,H. and Bayındır,C., (2015), A note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distribution, Physics of Plasmas, 22, 092105.
  • Hardin,R.H. and Tappert,F.D., (1973), Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equation, SIAM Review Chronicles, 15, pp. 423-423. [13] Karjadi,E.A., Badiey,M. and Kirby,J.T., (2010), Impact of surface gravity waves on high-frequency acoustic propagation in shallow water, The Journal of the Acoustical Society of America, 127, pp. 1787-1787.
  • Karjadi,E.A., Badiey,M., Kirby,J.T. and Bayindir,C., (2012), The effects of surface gravity waves on high-frequency acoustic propagation in shallow water, IEEE Journal of Oceanic Engineering, 37, pp. 112-121.
  • Lamb,H., (1945), Hydrodynamics, Dover Publications, New York.
  • Sanz-Serna,J.M. and Calvo,M.P., (1994), Numerical Hamiltonian Problems, Chapman and Hall, Lon- don.
  • Taha,T.R. and Ablowitz,M.J., (1984), Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Nonlinear Schrodinger Equation, Journal of Computational Physics, 22, pp. 203-230.
  • Trefethen,L.N., (2000), Spectral Methods in MATLAB, SIAM, Philadelphia.
  • Zakharov,V.E., (1968), Stability of periodic waves of finite amplitude on the surface of a deep fluid, Soviet Physics JETP, 2, pp. 190-194.
  • C. Bayındır for the photography and short autobiography, see TWMS J. App. Eng. Math., V.5, N.2.
There are 19 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

C. Bayındır This is me

Publication Date June 1, 2016
Published in Issue Year 2016 Volume: 6 Issue: 1

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