March  2013, 2(1): 81-100. doi: 10.3934/eect.2013.2.81

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

Received  September 2012 Revised  December 2012 Published  January 2013

In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refractive index.
Citation: François Genoud. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evolution Equations and Control Theory, 2013, 2 (1) : 81-100. doi: 10.3934/eect.2013.2.81
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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