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Periodic solutions of a singular equation with indefinite weight. (English) Zbl 1232.34064

The authors study the existence and uniqueness of \(T\)-periodic solutions for the equation \[ x''= \frac{a(t)}{x^3}, \] where \(a\) is a \(T\)-periodic function given by \[ a(t) = a_+ \;\;\text{if} \;0 \leq t < t_+, \;\;a(t) = -a_- \;\;\text{if} \;t_+ \leq t < T \] with \(a_+,a_- > 0.\) These problems arise in different physical situations such as in the stabilization of matter-wave breathers in Bose-Einstein condensates, in the propagation of guided waves in optical fibers and in the electromagnetic trapping of a neutral atom near a charged wire. If the parameters \(a_+, a_-\) are fixed, and \(T := t_+ + t_-,\) an interesting question is how to control the switching times \(t_-,t_+\) in order to get periodic states with a particular amplitude. This question is studied in the paper as well as the stability properties (in the linear sense) of the \(T\)-periodic solutions.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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[1] King, sniewski Periodic motion of atoms near a charged wire Letters in, Mathematical Physics pp 367– (1997)
[2] Montesinos, rez - Garc ıa - body dynamics of stabilized vector solitons, Chaos 15 pp 033501– (2005) · Zbl 1144.37387 · doi:10.1063/1.1984807
[3] Abdullaev, Dynamical stabilization of solitons in cubic - quintic nonlinear Schro dinger model, Phys Rev pp 035603– (2005)
[4] Carretero, Gonza lez Nonlinear waves in Einstein condensates : physical relevance and mathematical techniques Nonlin - earity, and 21 pp 139– (2008)
[5] Papini, A topological approach to superlinear indefinite boundary value problems Nonlinear no, Topol Methods Anal 15 pp 203– (2000) · Zbl 0990.34019
[6] Cornish, Dynamics of collapsing and exploding Bose - Einstein condensates, Nature pp 412– (2001)
[7] Cornish, Stable Einstein Condensates with widely tunable interactions, Phys Rev Lett 85 pp 85– (2000) · doi:10.1103/PhysRevLett.85.1795
[8] Lei, criteria for stability of periodic solutions of a newtonian equation no, Math Proc Cambridge Phil Soc 140 pp 1– (2006) · Zbl 1097.34041 · doi:10.1017/S0305004105008959
[9] Montesinos, rez - Garc ıa The method of moments for non - linear Schro dinger equations : theory and applications, SIAM Appl Math 4 pp 67– (2007)
[10] Saito, Stabilization of a Bose - Einstein droplet by hyperfine Rabi oscillations, Phys Rev pp 053619– (2007) · doi:10.1103/PhysRevA.76.053619
[11] Papini, On the periodic boundary value problem and chaotic - like dy - namics for nonlinear Hill s equations no, Nonlinear Stud 4 pp 71– (2004)
[12] Lei, Twist Property of Periodic Motion of an Atom Near a Charged Wire in rez - Garc ıa Stabilization of solitons of the multidimensional nonlinear Schro dinger equation : matter - wave breathers, Letters Mathematical Physics pp 9– (2002) · Zbl 1002.78006 · doi:10.1023/A:1015797310039
[13] Saito, Dynamically Stabilized Bright Solitons in a Two - Dimensional Einstein Condensate, Phys Rev Lett 4 pp 040403– (2003) · doi:10.1103/PhysRevLett.90.040403
[14] Itin, Reexamination of dynamical stabilization of matter - wave solitons, Phys Rev pp 033613– (2006) · doi:10.1103/PhysRevA.74.033613
[15] Torres, rez - Garc ıa ıa Moment analysis of paraxial propagation in a nonlinear graded index fibre Quantum Semiclass Opt, Opt pp 353– (2000)
[16] Rodrigues, Matter - Wave Solitons with a Periodic Piecewise - Constant Scattering Length No, Physical Review A pp 013611– (2008) · doi:10.1103/PhysRevA.78.013611
[17] Centurion, Ye Pu Modulational Instability in a Layered Kerr Medium : Theory and Experiment, Phys Rev Lett 97 pp 234101– (2006) · doi:10.1103/PhysRevLett.97.234101
[18] Hau, Bound states of guided matter waves : An atom and a charged wire Physical no, Rev 45 pp 6468– (1996)
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