Nonlinear wave propagation in discrete and continuous systems
In this review we try to capture some of the recent excitement induced by a large volume of theoretical and computational studies addressing nonlinear Schrödinger models (discrete and continuous) and the localized structures that they support. We focus on some prototypical structures, namely the breather solutions and solitary waves. In particular, we investigate the bifurcation of travelling wave solution in Discrete NLS system applying dynamical systems methods. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. We also offer an outlook on interesting possibilities for future work on this theme.
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- 5.E.J. Doedel, A.R. Champneys, T.R. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.J. Wang. Auto97 continuation and bifurcation software for ordinary differential equations. (1997) ftp://ftp.es.concordia.ca/directory/doedel/auto
- 12.T.R.O. Melvin, A.R. Champneys, D.E. Pelinovsky (preprint) (University of Bristol, 2008)Google Scholar
- 18.M.J.D. Powell, A hybrid method for nonlinear algebraic equations, Numerical Methods for Nonlinear Algebraic Equations. (Gordon and Breach, 1970)Google Scholar
- 19.M. Remoissenet, M. Peyrard (eds.) Nonlinear Coherent Structures in Physics and Biology (Springer-Verlag Berlin, 1991)Google Scholar
- 25.C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)Google Scholar
- 26.L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003)Google Scholar
- 27.J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 2008)Google Scholar
- 28.Yu.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)Google Scholar