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Solitary gravity-capillary water waves with point vortices
A new method for the boundedness of semilinear Duffing equations at resonance
1. | School of Mathematical Sciences, Soochow University, Suzhou 215006 |
2. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
3. | School of Mathematical Sciences, Ocean University of China, Qingdao 266100 |
References:
[1] |
J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111.
doi: 10.1088/0951-7715/9/5/003. |
[2] |
J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220.
doi: 10.1006/jdeq.1997.3367. |
[3] |
V. I. Arnold, On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian, Sov. Math. Dokl., 3 (1962), 136-140. |
[4] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95. |
[5] |
T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931. |
[6] |
R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977. |
[7] |
L. Jiao, D. Piao and Y. Wang, Boundedness for general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113.
doi: 10.1016/j.jde.2011.09.019. |
[8] |
A. M. Krssnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Math. Comput. Model. 32 (2000), 1445-1455.
doi: 10.1016/S0895-7177(00)00216-8. |
[9] |
A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68.
doi: 10.1007/BF02410787. |
[10] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83.
doi: 10.1007/BF02100285. |
[11] |
B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174.
doi: 10.1006/jdeq.1998.3553. |
[12] |
B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.
doi: 10.1006/jmaa.1998.6219. |
[13] |
B. Liu, Quasi-periodic solutions of a semilinear Liénard equation at resonance, Sci. China Ser. A: Mathematics, 48 (2005), 1234-1244.
doi: 10.1360/04ys0019. |
[14] |
B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance, J. Differential Equations, 246 (2009), 3471-3495.
doi: 10.1016/j.jde.2009.02.015. |
[15] |
J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214.
doi: 10.1007/s11253-007-0016-1. |
[16] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[17] |
J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20. |
[18] |
R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.
doi: 10.1112/jlms/53.2.325. |
[19] |
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., 79 (1999), 381-413.
doi: 10.1112/S0024611599012034. |
[20] |
C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version). |
[21] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer-Verlag, Berlin, 1007 (1983), 677-718.
doi: 10.1007/BFb0061441. |
[22] |
Y. Wang, Boundedness of solutions in a class of Duffing equations with oscillating potentials, Nonlinear Anal.TAM, 71 (2009), 2906-2917.
doi: 10.1016/j.na.2009.01.172. |
[23] |
X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(Engl. Ser.),19 (2003), 765-782.
doi: 10.1007/s10114-003-0249-3. |
[24] |
X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949. |
[25] |
X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012. |
[26] |
J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's nonresonance condition, J. Math. Pures. Appl., 80 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
show all references
References:
[1] |
J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111.
doi: 10.1088/0951-7715/9/5/003. |
[2] |
J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220.
doi: 10.1006/jdeq.1997.3367. |
[3] |
V. I. Arnold, On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian, Sov. Math. Dokl., 3 (1962), 136-140. |
[4] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95. |
[5] |
T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931. |
[6] |
R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977. |
[7] |
L. Jiao, D. Piao and Y. Wang, Boundedness for general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113.
doi: 10.1016/j.jde.2011.09.019. |
[8] |
A. M. Krssnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Math. Comput. Model. 32 (2000), 1445-1455.
doi: 10.1016/S0895-7177(00)00216-8. |
[9] |
A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68.
doi: 10.1007/BF02410787. |
[10] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83.
doi: 10.1007/BF02100285. |
[11] |
B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174.
doi: 10.1006/jdeq.1998.3553. |
[12] |
B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.
doi: 10.1006/jmaa.1998.6219. |
[13] |
B. Liu, Quasi-periodic solutions of a semilinear Liénard equation at resonance, Sci. China Ser. A: Mathematics, 48 (2005), 1234-1244.
doi: 10.1360/04ys0019. |
[14] |
B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance, J. Differential Equations, 246 (2009), 3471-3495.
doi: 10.1016/j.jde.2009.02.015. |
[15] |
J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214.
doi: 10.1007/s11253-007-0016-1. |
[16] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2061-7. |
[17] |
J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20. |
[18] |
R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.
doi: 10.1112/jlms/53.2.325. |
[19] |
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., 79 (1999), 381-413.
doi: 10.1112/S0024611599012034. |
[20] |
C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version). |
[21] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer-Verlag, Berlin, 1007 (1983), 677-718.
doi: 10.1007/BFb0061441. |
[22] |
Y. Wang, Boundedness of solutions in a class of Duffing equations with oscillating potentials, Nonlinear Anal.TAM, 71 (2009), 2906-2917.
doi: 10.1016/j.na.2009.01.172. |
[23] |
X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(Engl. Ser.),19 (2003), 765-782.
doi: 10.1007/s10114-003-0249-3. |
[24] |
X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949. |
[25] |
X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012. |
[26] |
J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's nonresonance condition, J. Math. Pures. Appl., 80 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
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