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Orbitally stable standing waves for the asymptotically linear one-dimensional NLS
1. | Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland |
References:
[1] |
S. A. Akhmanov, R. V. Khokhlov and A. P. Sukhorukov, Self-Focusing, self-defocusing, and self-modulation of laser beams, in "Laser Handbook" (eds. F. T. Arecchi and E. O. Schulz-Dubois), North-Holland, New York (1972), 1151-1228. |
[2] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, American Mathematical Society, Providence, Rhode Island, 2003. |
[3] |
R. Y. Chiao, E. Garmire and C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett., 13 (1964), 479-482. |
[4] |
D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $\mathbb R^N$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[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 and C. A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.
doi: 10.3934/dcds.2008.21.137. |
[7] |
F. Genoud, Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case, J. Differential Equations, 246 (2009), 1921-1943.
doi: 10.1016/j.jde.2008.10.029. |
[8] |
F. Genoud, Bifurcation and stability of travelling waves in self- focusing planar waveguides, Adv. Nonlinear Stud., 10 (2010), 357-400. |
[9] |
F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line, Nonlinear Anal., 74 (2011), 4533-4543.
doi: 10.1016/j.na.2011.04.019. |
[10] |
F. Genoud, Global bifurcation for asymptotically linear Schrödinger equations, to appear in NoDEA Nonlinear Differential Equations Appl.
doi: 10.1007/s00030-012-0152-7. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," $2^{nd}$ Edition, Springer, 2001. |
[12] |
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.
doi: 10.1016/0022-1236(87)90044-9. |
[13] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[14] |
L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N2$ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614.
doi: 10.1051/cocv:2002068. |
[15] |
H. Jeanjean and C. A. Stuart, Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states, Adv. Differential Equations, 4 (1999), 639-670. |
[16] |
J. B. McLeod, C. A. Stuart and W. C. Troy, Stability of standing waves for some nonlinear Schrödinger equations, Differential Integral Equations, 16 (2003), 1025-1038. |
[17] |
P. J. Rabier and C. A. Stuart, Application of elliptic regularity to bifurcation in stationary nonlinear Schrödinger equations, Nonlinear Anal., 52 (2003), 869-890.
doi: 10.1016/S0362-546X(02)00138-4. |
[18] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators," Academic Press, London, 2003. |
[19] |
B. E. A. Saleh and M. C. Teich, "Fundamentals of Photonics," Wiley, New York, 1991. |
[20] |
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.
doi: 10.1103/PhysRevE.78.046602. |
[21] |
G. I. Stegeman and M. Segev, Optical spatial solitons and their interactions: universality and diversity, Science, 286 (1999), 1518-1523. |
[22] |
C. A. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rational Mech. Anal., 125 (1993), 145-200.
doi: 10.1007/BF00376812. |
[23] |
C. A. Stuart, An introduction to elliptic equations on $R^N$, in "Nonlinear Functional Analysis and Applications to Differential Equations" (Trieste, 1997), World Sci. Publ., River Edge, NJ (1998), 237-285. |
[24] |
C. A. Stuart, Uniqueness and stability of ground states for some nonlinear Schrödinger equations, J. Eur. Math. Soc., 8 (2006), 399-414.
doi: 10.4171/JEMS/60. |
[25] |
C. A. Stuart, Existence and stability of TE modes in a stratified non-linear dielectric, IMA J. Appl. Math., 72 (2007), 659-679.
doi: 10.1093/imamat/hxm033. |
[26] |
C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
doi: 10.1007/s00032-008-0089-9. |
[27] |
C. A. Stuart and H.-S. Zhou, A variational problem related to self-trapping of an electromagnetic field, Math. Methods Appl. Sci., 19 (1996), 1397-1407.
doi: 10.1002/(SICI)1099-1476(19961125)19:17<1397::AID-MMA833>3.0.CO;2-B. |
[28] |
C. A. Stuart and H.-S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on $R^N$, Comm. Partial Differential Equations, 24 (1999), 1731-1758.
doi: 10.1080/03605309908821481. |
[29] |
C. A. Stuart and H.-S. Zhou, Axisymmetric TE-modes in a self-focusing dielectric, SIAM J. Math. Anal., 37 (2005), 218-237.
doi: 10.1137/S0036141004441751. |
[30] |
C. A. Stuart and H.-S. Zhou, Global branch of solutions for nonlinear Schrödinger equations with deepening potential well, Proc. London Math. Soc., 92 (2006), 655-681.
doi: 10.1017/S0024611505015637. |
[31] |
O. Svelto, Self-focusing, self-trapping, and self-phase modulation of laser beams, in "Prog. Opt." 12, North-Holland, Amsterdam (1974), 1-51. |
[32] |
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.
doi: 10.1017/S0308210500032042. |
[33] |
N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783-789. |
[34] |
H.-S. Zhou and H. Zhu, Asymptotically linear elliptic problem on $\mathbb R^N$, Q. J. Math., 59 (2008), 523-541.
doi: 10.1093/qmath/ham047. |
show all references
References:
[1] |
S. A. Akhmanov, R. V. Khokhlov and A. P. Sukhorukov, Self-Focusing, self-defocusing, and self-modulation of laser beams, in "Laser Handbook" (eds. F. T. Arecchi and E. O. Schulz-Dubois), North-Holland, New York (1972), 1151-1228. |
[2] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, American Mathematical Society, Providence, Rhode Island, 2003. |
[3] |
R. Y. Chiao, E. Garmire and C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett., 13 (1964), 479-482. |
[4] |
D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $\mathbb R^N$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[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 and C. A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.
doi: 10.3934/dcds.2008.21.137. |
[7] |
F. Genoud, Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case, J. Differential Equations, 246 (2009), 1921-1943.
doi: 10.1016/j.jde.2008.10.029. |
[8] |
F. Genoud, Bifurcation and stability of travelling waves in self- focusing planar waveguides, Adv. Nonlinear Stud., 10 (2010), 357-400. |
[9] |
F. Genoud, Bifurcation from infinity for an asymptotically linear problem on the half-line, Nonlinear Anal., 74 (2011), 4533-4543.
doi: 10.1016/j.na.2011.04.019. |
[10] |
F. Genoud, Global bifurcation for asymptotically linear Schrödinger equations, to appear in NoDEA Nonlinear Differential Equations Appl.
doi: 10.1007/s00030-012-0152-7. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," $2^{nd}$ Edition, Springer, 2001. |
[12] |
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.
doi: 10.1016/0022-1236(87)90044-9. |
[13] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[14] |
L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N2$ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614.
doi: 10.1051/cocv:2002068. |
[15] |
H. Jeanjean and C. A. Stuart, Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states, Adv. Differential Equations, 4 (1999), 639-670. |
[16] |
J. B. McLeod, C. A. Stuart and W. C. Troy, Stability of standing waves for some nonlinear Schrödinger equations, Differential Integral Equations, 16 (2003), 1025-1038. |
[17] |
P. J. Rabier and C. A. Stuart, Application of elliptic regularity to bifurcation in stationary nonlinear Schrödinger equations, Nonlinear Anal., 52 (2003), 869-890.
doi: 10.1016/S0362-546X(02)00138-4. |
[18] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics IV: Analysis of Operators," Academic Press, London, 2003. |
[19] |
B. E. A. Saleh and M. C. Teich, "Fundamentals of Photonics," Wiley, New York, 1991. |
[20] |
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.
doi: 10.1103/PhysRevE.78.046602. |
[21] |
G. I. Stegeman and M. Segev, Optical spatial solitons and their interactions: universality and diversity, Science, 286 (1999), 1518-1523. |
[22] |
C. A. Stuart, Guidance properties of nonlinear planar waveguides, Arch. Rational Mech. Anal., 125 (1993), 145-200.
doi: 10.1007/BF00376812. |
[23] |
C. A. Stuart, An introduction to elliptic equations on $R^N$, in "Nonlinear Functional Analysis and Applications to Differential Equations" (Trieste, 1997), World Sci. Publ., River Edge, NJ (1998), 237-285. |
[24] |
C. A. Stuart, Uniqueness and stability of ground states for some nonlinear Schrödinger equations, J. Eur. Math. Soc., 8 (2006), 399-414.
doi: 10.4171/JEMS/60. |
[25] |
C. A. Stuart, Existence and stability of TE modes in a stratified non-linear dielectric, IMA J. Appl. Math., 72 (2007), 659-679.
doi: 10.1093/imamat/hxm033. |
[26] |
C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
doi: 10.1007/s00032-008-0089-9. |
[27] |
C. A. Stuart and H.-S. Zhou, A variational problem related to self-trapping of an electromagnetic field, Math. Methods Appl. Sci., 19 (1996), 1397-1407.
doi: 10.1002/(SICI)1099-1476(19961125)19:17<1397::AID-MMA833>3.0.CO;2-B. |
[28] |
C. A. Stuart and H.-S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on $R^N$, Comm. Partial Differential Equations, 24 (1999), 1731-1758.
doi: 10.1080/03605309908821481. |
[29] |
C. A. Stuart and H.-S. Zhou, Axisymmetric TE-modes in a self-focusing dielectric, SIAM J. Math. Anal., 37 (2005), 218-237.
doi: 10.1137/S0036141004441751. |
[30] |
C. A. Stuart and H.-S. Zhou, Global branch of solutions for nonlinear Schrödinger equations with deepening potential well, Proc. London Math. Soc., 92 (2006), 655-681.
doi: 10.1017/S0024611505015637. |
[31] |
O. Svelto, Self-focusing, self-trapping, and self-phase modulation of laser beams, in "Prog. Opt." 12, North-Holland, Amsterdam (1974), 1-51. |
[32] |
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.
doi: 10.1017/S0308210500032042. |
[33] |
N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783-789. |
[34] |
H.-S. Zhou and H. Zhu, Asymptotically linear elliptic problem on $\mathbb R^N$, Q. J. Math., 59 (2008), 523-541.
doi: 10.1093/qmath/ham047. |
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