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Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices
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Existence of solitary waves in nonlinear equations of Schrödinger type
1. | Departamento de Matemáticas, E.T.S.I Industriales & Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Avda. de Camilo José Cela, 3 Universidad de Castilla-La Mancha, 13071 Ciudad Real |
2. | Departamento de Matemáticas, E. T. S. de Ingenieros Industriales, and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha 13071 Ciudad Real, Spain |
References:
[1] |
F. Kh. Abdullaev, A. Gammal, L. Tomio and T. Frederico, Stability of trapped Bose-Einstein condensates, Phys. Rev. A, 63 (2001), 043604.
doi: doi:10.1103/PhysRevA.63.043604. |
[2] |
A. Ambrosetti, V Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: doi:10.4171/JEMS/24. |
[3] |
A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. |
[4] |
A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Adv. Math., 104, Cambridge University Press, Cambridge, 2007. |
[5] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: doi:10.1016/0022-1236(73)90051-7. |
[6] |
A. Bahri and P. L. Lions, On the existence of positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincaré, Analyse Nonlinéaire, 14 (1997), 365-413. |
[7] |
I. V. Barashenkov and V. G. Makhankov, Soliton-like "bubbles" in the system of interacting bosons, Phys. Lett. A, 128 (1988), 52-56.
doi: doi:10.1016/0375-9601(88)91042-0. |
[8] |
T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N2$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741.
doi: doi:10.1080/03605309508821149. |
[9] |
J. Belmonte-Beitia, On the existence of bright solitons in cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity, Mathematical Problems in Engineering, 2008 (2008), Article ID 935390.
doi: doi:10.1155/2008/935390. |
[10] |
J. Belmonte-Beitia, Symmetric and asymmetric bound states for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, J. Phys. A: Math. Theor., 42 (2009), 035208.
doi: doi:10.1088/1751-8113/42/3/035208. |
[11] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and V. V. Konotop, Localized nonlinear waves in systems with time and space modulated nonlinearities, Phys. Rev. Lett., 100 (2008), 164102.
doi: doi:10.1103/PhysRevLett.100.164102. |
[12] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett., 98 (2007), 064102.
doi: doi:10.1103/PhysRevLett.98.064102. |
[13] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems - Series B, 9 (2008), 221-233. |
[14] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I and II, Arch. Rat. Mech. Anal., 82 (1983), 313-345; 347-379 (MR0695536). |
[15] |
G. Boudeps, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun., 219 (2003), 427-433.
doi: doi:10.1016/S0030-4018(03)01341-5. |
[16] |
M. Centurion, M. A. Porter, P. G. Kevrekidis and D. Psaltis, Nonlinearity management in optics: Experiment, theory and simulation, Phys. Rev. Lett., 97 (2006), 033903.
doi: doi:10.1103/PhysRevLett.97.033903. |
[17] |
C. Chin, T. Kraemer, M. Mark, J. Herbig, P. Waldburger, H.-C. Nägeri and R. Grim, Observation of Feshbach-like resonances in collisions between ultracold molecules, Phys. Rev. Lett., 94 (2005), 123201.
doi: doi:10.1103/PhysRevLett.94.123201. |
[18] |
S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C. E. Wieman, Stable 85Rb Bose-Einstein condensates with widely tunable interactions}, Phys. Rev. Lett., 85 (2000), 1795.
doi: doi:10.1103/PhysRevLett.85.1795. |
[19] |
F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.
doi: doi:10.1103/RevModPhys.71.463. |
[20] |
A. S. Davydov, "Solitons in Molecular Systems," Translated from the Russian by Eugene S. Kryachko. Mathematics and its Applications (Soviet Series), 4. D. Reidel Publishing Co., Dordrecht, 1985. |
[21] |
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. |
[22] |
A. Hasegawa, "Optical solitons in Fibers," Springer-Verlag, Berlin, 1989. |
[23] |
P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Nonlinearity management in higher dimensions, J. Phys. A: Math. Gen., 39 (2006), 479-488.
doi: doi:10.1088/0305-4470/39/3/002. |
[24] |
Y. Kivshar and G. P. Agrawal, "Optical Solitons: From Fibers to Photonic Crystals," Academic Press, 2003. |
[25] |
P. L. Lions, The concentration-compactness principle in the calculus of varitions, the locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[26] |
B. A. Malomed, "Soliton Management in Periodic Systems," Springer, New York, 2006. |
[27] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: doi:10.1007/s00032-005-0047-8. |
[28] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Averaging for solitons with nonlinearity management, Phys. Rev. Lett., 91 (2003), 240201.
doi: doi:10.1103/PhysRevLett.91.240201. |
[29] |
C. Sulem and P. Sulem, "The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse," Springer, Berlin, 2000. |
[30] |
P. Torres, Guided waves in an multi-layered optical structure, Nonlinearity, 19 (2006), 2103-2113.
doi: doi:10.1088/0951-7715/19/9/006. |
[31] |
M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. |
[32] |
C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao and Y. Nie, Third- and fifth- order optical nonlinearities in a new stilbazolium derivative, J. Opt. Soc. Am. B, 19 (2002), 369-375.
doi: doi:10.1364/JOSAB.19.000369. |
[33] |
C. T. Zhou and X. T. He, Stochastic diffusion of electrons in evolution Langmuir fields, Phys. Scr., 50 (1994), 415.
doi: doi:10.1088/0031-8949/50/4/015. |
show all references
References:
[1] |
F. Kh. Abdullaev, A. Gammal, L. Tomio and T. Frederico, Stability of trapped Bose-Einstein condensates, Phys. Rev. A, 63 (2001), 043604.
doi: doi:10.1103/PhysRevA.63.043604. |
[2] |
A. Ambrosetti, V Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: doi:10.4171/JEMS/24. |
[3] |
A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. |
[4] |
A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Adv. Math., 104, Cambridge University Press, Cambridge, 2007. |
[5] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: doi:10.1016/0022-1236(73)90051-7. |
[6] |
A. Bahri and P. L. Lions, On the existence of positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincaré, Analyse Nonlinéaire, 14 (1997), 365-413. |
[7] |
I. V. Barashenkov and V. G. Makhankov, Soliton-like "bubbles" in the system of interacting bosons, Phys. Lett. A, 128 (1988), 52-56.
doi: doi:10.1016/0375-9601(88)91042-0. |
[8] |
T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N2$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741.
doi: doi:10.1080/03605309508821149. |
[9] |
J. Belmonte-Beitia, On the existence of bright solitons in cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity, Mathematical Problems in Engineering, 2008 (2008), Article ID 935390.
doi: doi:10.1155/2008/935390. |
[10] |
J. Belmonte-Beitia, Symmetric and asymmetric bound states for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, J. Phys. A: Math. Theor., 42 (2009), 035208.
doi: doi:10.1088/1751-8113/42/3/035208. |
[11] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and V. V. Konotop, Localized nonlinear waves in systems with time and space modulated nonlinearities, Phys. Rev. Lett., 100 (2008), 164102.
doi: doi:10.1103/PhysRevLett.100.164102. |
[12] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett., 98 (2007), 064102.
doi: doi:10.1103/PhysRevLett.98.064102. |
[13] |
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems - Series B, 9 (2008), 221-233. |
[14] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I and II, Arch. Rat. Mech. Anal., 82 (1983), 313-345; 347-379 (MR0695536). |
[15] |
G. Boudeps, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun., 219 (2003), 427-433.
doi: doi:10.1016/S0030-4018(03)01341-5. |
[16] |
M. Centurion, M. A. Porter, P. G. Kevrekidis and D. Psaltis, Nonlinearity management in optics: Experiment, theory and simulation, Phys. Rev. Lett., 97 (2006), 033903.
doi: doi:10.1103/PhysRevLett.97.033903. |
[17] |
C. Chin, T. Kraemer, M. Mark, J. Herbig, P. Waldburger, H.-C. Nägeri and R. Grim, Observation of Feshbach-like resonances in collisions between ultracold molecules, Phys. Rev. Lett., 94 (2005), 123201.
doi: doi:10.1103/PhysRevLett.94.123201. |
[18] |
S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C. E. Wieman, Stable 85Rb Bose-Einstein condensates with widely tunable interactions}, Phys. Rev. Lett., 85 (2000), 1795.
doi: doi:10.1103/PhysRevLett.85.1795. |
[19] |
F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.
doi: doi:10.1103/RevModPhys.71.463. |
[20] |
A. S. Davydov, "Solitons in Molecular Systems," Translated from the Russian by Eugene S. Kryachko. Mathematics and its Applications (Soviet Series), 4. D. Reidel Publishing Co., Dordrecht, 1985. |
[21] |
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. |
[22] |
A. Hasegawa, "Optical solitons in Fibers," Springer-Verlag, Berlin, 1989. |
[23] |
P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Nonlinearity management in higher dimensions, J. Phys. A: Math. Gen., 39 (2006), 479-488.
doi: doi:10.1088/0305-4470/39/3/002. |
[24] |
Y. Kivshar and G. P. Agrawal, "Optical Solitons: From Fibers to Photonic Crystals," Academic Press, 2003. |
[25] |
P. L. Lions, The concentration-compactness principle in the calculus of varitions, the locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[26] |
B. A. Malomed, "Soliton Management in Periodic Systems," Springer, New York, 2006. |
[27] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
doi: doi:10.1007/s00032-005-0047-8. |
[28] |
D. E. Pelinovsky, P. G. Kevrekidis and D. J. Frantzeskakis, Averaging for solitons with nonlinearity management, Phys. Rev. Lett., 91 (2003), 240201.
doi: doi:10.1103/PhysRevLett.91.240201. |
[29] |
C. Sulem and P. Sulem, "The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse," Springer, Berlin, 2000. |
[30] |
P. Torres, Guided waves in an multi-layered optical structure, Nonlinearity, 19 (2006), 2103-2113.
doi: doi:10.1088/0951-7715/19/9/006. |
[31] |
M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. |
[32] |
C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao and Y. Nie, Third- and fifth- order optical nonlinearities in a new stilbazolium derivative, J. Opt. Soc. Am. B, 19 (2002), 369-375.
doi: doi:10.1364/JOSAB.19.000369. |
[33] |
C. T. Zhou and X. T. He, Stochastic diffusion of electrons in evolution Langmuir fields, Phys. Scr., 50 (1994), 415.
doi: doi:10.1088/0031-8949/50/4/015. |
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