Numerical computation of solitons for optical systems

Mathematical Modelling and Numerical Analysis

Volumn 43, Issue 1, 2009, Pages 173-208

  Menza, Laurent Di ^a  

^a UNIVERSITÉ DE REIMS CHAMPAGNE ARDENNE   (France)

Author keywords

Continuation method.; Elliptic problems; Nonlinear optics; Shooting method; Stationary states

Indexed keywords

EID: 59049093849     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2008044     Document Type: Article

References (19)

1. M. Balabane, J, Dolbeault and H. Ounaies, Nodal solutions for a sublines!*elliptic equation. Nonlinear Anal. 52 (2003) 219-237.
2. A.V. Buryak, V.V. Steblina and Y. Kivshar, Self-trapping of light beams and parametric solitons in diffractive quadratic media. Phys. Rev. A 52 (1995) 1670-1674.
3. A.V. Buryak, P. Di Trapani, D.V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlineaiities: from basic physics to futuristic applications. Phys. Rep. 370 (2002) 62-235.
4. L. Di Mcnza, Transparent and absorbing conditions for the Schrodingcr equation in a bounded domain. Numer. Fund. Anal. Optim,. 18 (1997) 759-775.
5. G. Fibich, N. Gavish and X.-P. Wang, Singular ring solutions of critical and supercritical nonlinear Schrödinger equations. Physica. D 18 (2007) 55-86.
6. W.J. Firth and D.V. Skryabin, Optical solitons carrying orbital angular momentum. Phys. Rev. Lett. 79 (1997) 2450-2453.
7. H. He, M.J. Werner and P.D. Drummond, Simultaneous solitary-wave solutions in a nonlinear parametric waveguide. Phys. Rev. E 54 (1996) 896-911.
8. J. Iaia and H. Warchall, Nonradial solutions of a semilinear elliptic equation in two dimensions. J. Diff. Equ. 119 (1995) 533-558.
9. R. Kajikiya. Norm estimates for radially symmetric solutions of semilinear elliptic equations. Trans. Amer. Math. Soc. 347 (1995) 1163-1199.
10. n. Arch. Rat. Mech. Anal. 105 (1989) 243-266.
11. D.J.B. Lloyd and A.R. Champneys, Efficient numerical continuation and stability analysis of spatiotcmporal quadratic optical solitons. SIAM J. Sci. Comput. 27 (2005) 759-773.
12. B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson and M. Lisak. Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity. Phys. Rev. E 56 (1997) 4725-4735.
13. K. McLeod, W.C. Troy and F.B. Weissler, Radial solutions of Δu + f(u) = 0 with prescribed number of zeros. J. Diff. Equ. 83 (1990) 368-378.
14. T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation. Diff. Int. Equ. 18 (2005) 431-450.
15. I.M. Moroz, R. Penrose and P. Tod. Spherically-symmetric solutions of the Schrödinger-Newton equation. Class. Quant. Grav. 15 (1998) 2733-2742.
16. (2) medium: variational approach. Optics Comm. 118 (1995) 345-352.
17. P.L. Sulem and C. Sulem, The nonlinear Schrödinger equation, Self-focusing and wave collapse. AMS, Springer-Verlag (1999).
18. I.N. Towers, B.A. Malomed and F.W. Wise, Light bullets in quadratic media with normal dispersion at the second harmonic. Phys. Rev. Lett. 90 (2003) 123902.
19. M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983) 567-576.