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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.
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.
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.
|
[4] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem,, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79.
|
[5] |
T. Ding, Nonlinear oscillations at a point of resonance,, Sci. Sin., 25 (1982), 918.
|
[6] |
R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, Lecture Notes in Math 568, (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.
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), 32 (2000), 1445.
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.
doi: 10.1007/BF02410787. |
[10] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.
doi: 10.1007/BF02100285. |
[11] |
B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.
doi: 10.1006/jdeq.1998.3553. |
[12] |
B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.
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.
doi: 10.1360/04ys0019. |
[14] |
B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance,, J. Differential Equations, 246 (2009), 3471.
doi: 10.1016/j.jde.2009.02.015. |
[15] |
J. Mawhin, Resonance and nonlinearity: A survey,, Ukrainian Math. J., 59 (2007), 197.
doi: 10.1007/s11253-007-0016-1. |
[16] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences 74, (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.
|
[18] |
R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc., 53 (1996), 325.
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.
doi: 10.1112/S0024611599012034. |
[20] |
C. Pan and X. Yu, Magnitude Estimates,, Shandong Science and Technology Press, (1983). Google Scholar |
[21] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes in Math., 1007 (1983), 677.
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.
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.
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.
|
[25] |
X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations,, Ph.D. thesis, (2012). Google Scholar |
[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.
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.
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.
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.
|
[4] |
R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem,, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79.
|
[5] |
T. Ding, Nonlinear oscillations at a point of resonance,, Sci. Sin., 25 (1982), 918.
|
[6] |
R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, Lecture Notes in Math 568, (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.
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), 32 (2000), 1445.
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.
doi: 10.1007/BF02410787. |
[10] |
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.
doi: 10.1007/BF02100285. |
[11] |
B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.
doi: 10.1006/jdeq.1998.3553. |
[12] |
B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.
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.
doi: 10.1360/04ys0019. |
[14] |
B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance,, J. Differential Equations, 246 (2009), 3471.
doi: 10.1016/j.jde.2009.02.015. |
[15] |
J. Mawhin, Resonance and nonlinearity: A survey,, Ukrainian Math. J., 59 (2007), 197.
doi: 10.1007/s11253-007-0016-1. |
[16] |
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences 74, (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.
|
[18] |
R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc., 53 (1996), 325.
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.
doi: 10.1112/S0024611599012034. |
[20] |
C. Pan and X. Yu, Magnitude Estimates,, Shandong Science and Technology Press, (1983). Google Scholar |
[21] |
H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes in Math., 1007 (1983), 677.
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.
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.
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.
|
[25] |
X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations,, Ph.D. thesis, (2012). Google Scholar |
[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.
doi: 10.1016/S0021-7824(01)01221-1. |
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