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Weakly nonlinear dispersive waves under parametric resonance perturbation. (English) Zbl 1180.35457

Summary: We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35B20 Perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
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[1] Kima, Parametric resonance in an intensity-modulated magneto-optical trap, Opt. Commun. 236 (46) pp 349– (2004)
[2] Hernandez-Tenorio, Parametric resonance for solitons in the nonlinear Schr̤dinger equation model with time-dependent harmonic oscillator potential, Phys. B: Phys. Condens. Matter 398 (2) pp 460Р(2007)
[3] Kolesov, Multifrequency parametric resonance in a nonlinear wave equation, Izv. Math. 66 pp 1131– (2002) · Zbl 1084.35007
[4] Chiang, Parametric resonance of a spherical bubble, J. Fluid Mech. 229 pp 29– (1991) · Zbl 0850.76759
[5] Il’in, Transl. Math. Monographs (1992)
[6] Glebov, Slow passage through resonance for a weakly nonlinear dispersive wave, SIAM J. Appl. Math. 65 (6) pp 2158– (2005) · Zbl 1081.35113
[7] Kelley, Self-focusing of optical beams, Phys. Rev. Lett. 15 pp 1005– (1965)
[8] Talanov, On self-focusing of small bunches in nonlinear media, Letters to ZhETF 2 pp 218– (1965)
[9] Zakharov, Stability of periodic waves of finite amplitude on the surface of deep liquid, Zh. Prikladnoi Mekh. Tekh. Fiz. 2 pp 86– (1968)
[10] Kevorkyan, Passage through resonance for a one-dimensional oscillator with slowly varying frequency, SIAM J. Appl. Math. 20 pp 364– (1971)
[11] Rubenfeld, The passage of weakly coupled nonlinear oscillators through internal resonance, Stud. Appl. Math. 57 pp 77– (1977) · Zbl 0366.70019
[12] Ablowitz, Semi-resonant interactions and frequency dividers, Stud. Appl. Math. 52 pp 51– (1972) · Zbl 0257.70017
[13] Fajans, Autoresonant (nonstationary) excitation of pendulums, plutinos, plasmas, and other nonlinear oscillators, Am. J. Phys. 69 pp 1096– (2001)
[14] Yaakobi, Driven autoresonant three-oscillator interactions, Phys. Rev. E 76 (026205) pp 1– (2007)
[15] Kalyakin, Asymptotics for the solutions of the principal resonance equations, Teor. Matem. Fiz. 137 (1) pp 142– (2003) · Zbl 1129.34324
[16] Kiselev, The capture into parametric autoresonance, Nonlinear Dyn. 48 (12) pp 217– (2007) · Zbl 1180.70032
[17] Neu, Resonantly interacting waves, SIAM J. Appl. Math. 43 pp 141– (1983) · Zbl 0599.76025
[18] Kalyakin, Local resonance in weakly linear problems, Mat. Zametki 44 pp 697– (1988) · Zbl 0702.35226
[19] Glebov, On a weakly linear problem with local resonance, Diff. Eq. 31 pp 1402– (1995) · Zbl 0865.35120
[20] Assaf, Parametric autoresonance in Faraday waves, Phys. Rev. E 72 (016310) pp 1– (2005)
[21] Buslaev, Adiabatic perturbation of a periodic potential. II, Theor. Mat. Fiz 73 (3) pp 430– (1987) · Zbl 0643.34068
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