NONLINEAR SCHR ODINGER EQUATIONS ON R : GLOBAL BIFURCATION, ORBITAL STABILITY AND NONLINEAR WAVEGUIDES

[1] S.A. Akhmanov, A.P. Sukhorukov, and R.V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Sov. Phys. Usp. 10 (1968), 609–636.
[2] R.Y. Chiao, E. Garmire, and C.H. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 13 (1964), 479–482.
[3] T. Cazenave, Semilinear Schrodinger Equations, Courant Lecture Notes in Mathematics, 2003.
[4] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schr ̈odinger equations, Comm. Math. Phys. 85 (1982), 549–561.
[5] B. Crosignani, P. Di Porto, M. Segev, G. Salamo, and A. Yariv, Nonlinear optical beam propagation and solitons in photorefractive media, Riv. Nuovo Cimento 21 (1998), 1-37.
[6] F. Genoud, Existence and orbital stability of standing waves for some nonlinear Schrodinger equations, perturbation of a model case, J. Differential Equations 246 (2009), 1921–1943.
[7] F. Genoud, Bifurcation and stability of travelling waves in self-focusing planar waveguides, Adv. Nonlinear Stud. 10 (2010), 357–400.
[8] F. Genoud, A smooth global branch of solutions for a semilinear elliptic equation on R N , Calc. Var. Partial Differential Equations 38 (2010), 207–232.
[9] F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line, Nonlinear Anal. 74 (2011), 4533–4543.
[10] F. Genoud and C.A. Stuart, Schrodinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst. 21 (2008), 137–186.
[11] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987), 160–197.
[12] H. Jeanjean and C.A. Stuart, Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states, Adv. Differential Equations 4 (1999), 639–670.
[13] J.B. McLeod, C.A. Stuart, and W.C. Troy, Stability of standing waves for some nonlinear Schr ̈odinger equations, Differential Integral Equations 16 (2003), 1025–1038.
[14] P.H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973) 462–475.
[15] P. Raphael, Stability and Blow Up for the Nonlinear Schrodinger Equation, Lecture notes for the Clay summer school on evolution equations, ETH, Zurich (2008), available at http://www.math.univ-toulouse.fr/~raphael/Publications/cours-zurich.pdf
[16] B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, Wiley, New York, 1991.
[17] Y. Sivan, G. Fibich, B. Ilan, and M.I. Weinstein, Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons, Phys. Rev. E 78 (2008), 046602.
[18] G.I. Stegeman and M. Segev, Optical spatial solitons and their interactions: universality and diversity, Science 286 (1999), 1518–1523.
[19] C.A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Differential Equations 36 (1980), 391–407.
[20] C.A. Stuart, A global branch of solutions to a semilinear equation on an unbounded interval Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 273–282.
[21] C.A. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rational Mech. Anal. 125 (1993), 145–200.
[22] C.A. Stuart, Uniqueness and stability of ground states for some nonlinear Schr ̈odinger equations, J. Eur. Math. Soc. 8 (2006), 399–414.
[23] C.A. Stuart and H.-S. Zhou, Global branch of solutions for nonlinear Schr ̈odinger equations with deepening potential well, Proc. London Math. Soc. 92 (2006), 655–681.
[24] C.A. Stuart, Existence and stability of TE modes in a stratified non-linear dielectric, IMA J. Appl. Math. 72 (2007), 659–679.
[25] C.A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schr ̈odinger equation, Milan J. Math. 76 (2008), 329–399.
[26] O. Svelto, Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams, in: Prog. Opt. 12, North-Holland, Amsterdam, 1974, 1-51.
[27] T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc. (N.S.) 46 (2009), 1–33.
[28] J.F. Toland, Asymptotic nonlinearity and nonlinear eigenvalue problems, Quart. J. Math. Oxford 24 (1973), 241–250.
[29] J.F. Toland, Global bifurcation for Neumann problems without eigenvalues, J. Differential Equations 44 (1982), 82–110.
[30] J.F. Toland, Uniqueness of positive solutions of some semilinear Sturm-Liouville problems on the half line, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 259–263.
[31] M.G. Vakhitov and A.A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16 (1973), 783–785.
[32] M.I. Weinstein, Modulational stability of ground states of nonlinear Schr ̈odinger equations, SIAM J. Math. Anal. 16 (1985), 472–491.