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October  2011, 4(5): 1007-1017. doi: 10.3934/dcdss.2011.4.1007

Existence of solitary waves in nonlinear equations of Schrödinger type

Juan Belmonte-Beitia 1, and  Vladyslav Prytula 2,

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

Received  July 2009 Revised  October 2009 Published  December 2010

In this work we study the existence of solitary waves in nonlinear equations of Schrödinger type. We prove the existence of the positive solution and using the bifurcation theory show that the norm of the given solution tends to zero as the coefficient corresponding to the linear term vanishes.
Keywords: bifurcation theory., Nonlinear Schrödinger equation, Mountain-Pass theorem.
Mathematics Subject Classification: Primary: 35Q51, 35Q55; Secondary: 53C3.
Citation: Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007
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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[4]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Adv. Math., 104, Cambridge University Press, Cambridge, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741. doi: doi:10.1080/03605309508821149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Hasegawa, "Optical solitons in Fibers," Springer-Verlag, Berlin, 1989. Google Scholar

[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.  Google Scholar

[24]

Y. Kivshar and G. P. Agrawal, "Optical Solitons: From Fibers to Photonic Crystals," Academic Press, 2003. Google Scholar

[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.  Google Scholar

[26]

B. A. Malomed, "Soliton Management in Periodic Systems," Springer, New York, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

C. Sulem and P. Sulem, "The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse," Springer, Berlin, 2000. Google Scholar

[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.  Google Scholar

[31]

M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[4]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Adv. Math., 104, Cambridge University Press, Cambridge, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741. doi: doi:10.1080/03605309508821149.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Hasegawa, "Optical solitons in Fibers," Springer-Verlag, Berlin, 1989. Google Scholar

[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.  Google Scholar

[24]

Y. Kivshar and G. P. Agrawal, "Optical Solitons: From Fibers to Photonic Crystals," Academic Press, 2003. Google Scholar

[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.  Google Scholar

[26]

B. A. Malomed, "Soliton Management in Periodic Systems," Springer, New York, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

C. Sulem and P. Sulem, "The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse," Springer, Berlin, 2000. Google Scholar

[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.  Google Scholar

[31]

M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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