ANALYTICAL AND NUMERICAL ASPECTS OF THE DISSIPATIVE NONLINEAR SCHRODINGER EQUATION

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.

Keywords:

Dissipative nonlinear Schr¨odinger equation, solitary waves, spectral methods split-step Fourier method.,

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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.