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Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation
1. | 14 East Packer Avenue, Christmas-Saucon Hall, Lehigh University, Bethlehem, PA 18015, United States |
References:
[1] |
M. Berti and P. Bolle, Cantor families of periodic solutions of wave equations with $C^k$ nonlinearities,, NoDEA Nonlinear Differential Equations Appl, 15 (2008), 247.
doi: 10.1007/s00030-007-7025-5. |
[2] |
M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations,, Duke Math. J., 134 (2006), 359.
doi: 10.1215/S0012-7094-06-13424-5. |
[3] |
M. Berti, P. Bolle and M. Procesi, An abstract Nash-Moser theorem with parameters and applications to PDEs,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 377.
doi: 10.1016/j.anihpc.2009.11.010. |
[4] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44.
doi: 10.1007/BFb0095239. |
[5] |
S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces,, J. Differential Equations, 74 (1988), 285.
doi: 10.1016/0022-0396(88)90007-1. |
[6] |
C. Conti and S. Trillo, Nonlinear X Waves in Localized Waves,, H. E. Hernandez-Figueroa, (2007), 243. Google Scholar |
[7] |
C. Conti, S. Trillo, P. Di Trapani, A. Piskarkas, O. Jedrkiewicz and J. Trull, Nonlinear electro-magnetic X waves,, Phys. Rev. Letter, 90 (2003). Google Scholar |
[8] |
S. Droulias, K. Hizanidis, J. Meier and D. N. Christodoulides, X-waves in nonlinear normally dispersive waveguide arrays,, Optical Express, 13 (2005), 1827.
doi: 10.1364/OPEX.13.001827. |
[9] |
J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations,, J. Nonlinear Sci., 3 (1993), 169.
doi: 10.1007/BF02429863. |
[10] |
J. M. Ghidaglia and J. C. Saut, Nonexistence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations,, J. Nonlinear Sci., 6 (1996), 139.
doi: 10.1007/BF02434051. |
[11] |
J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.
doi: 10.1088/0951-7715/3/2/010. |
[12] |
P. Kevrekidis, A. Nahmod and C. Zeng, Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D,, Nonlinearity, 24 (2011), 1523.
doi: 10.1088/0951-7715/24/5/007. |
[13] |
N. Lu, Small generalized breathers with exponentially small tails for Klein-Gordon equations,, J. Differential Equations, 256 (2014), 745.
doi: 10.1016/j.jde.2013.09.018. |
[14] |
J. Moser, A new technique for the construction of solution of nonlinear differential equations,, Proc. Nat. Acad. Sci., 47 (1961), 1824.
doi: 10.1073/pnas.47.11.1824. |
[15] |
A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase,, J. Appl. Math. Mech., 48 (1984), 133.
doi: 10.1016/0021-8928(84)90078-9. |
[16] |
C. Sulem and J. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse,, Applied Mathematical Sciences 139, (1999).
|
[17] |
L. Vuillon, D. Dutykh and F. Fedele, Some special solutions to the hyperbolic NLS equation, preprint,, , (). Google Scholar |
[18] |
V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, J. Appl. Mech. Tech. Phys., 9 (1968), 190.
doi: 10.1007/BF00913182. |
show all references
References:
[1] |
M. Berti and P. Bolle, Cantor families of periodic solutions of wave equations with $C^k$ nonlinearities,, NoDEA Nonlinear Differential Equations Appl, 15 (2008), 247.
doi: 10.1007/s00030-007-7025-5. |
[2] |
M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations,, Duke Math. J., 134 (2006), 359.
doi: 10.1215/S0012-7094-06-13424-5. |
[3] |
M. Berti, P. Bolle and M. Procesi, An abstract Nash-Moser theorem with parameters and applications to PDEs,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 377.
doi: 10.1016/j.anihpc.2009.11.010. |
[4] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44.
doi: 10.1007/BFb0095239. |
[5] |
S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces,, J. Differential Equations, 74 (1988), 285.
doi: 10.1016/0022-0396(88)90007-1. |
[6] |
C. Conti and S. Trillo, Nonlinear X Waves in Localized Waves,, H. E. Hernandez-Figueroa, (2007), 243. Google Scholar |
[7] |
C. Conti, S. Trillo, P. Di Trapani, A. Piskarkas, O. Jedrkiewicz and J. Trull, Nonlinear electro-magnetic X waves,, Phys. Rev. Letter, 90 (2003). Google Scholar |
[8] |
S. Droulias, K. Hizanidis, J. Meier and D. N. Christodoulides, X-waves in nonlinear normally dispersive waveguide arrays,, Optical Express, 13 (2005), 1827.
doi: 10.1364/OPEX.13.001827. |
[9] |
J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations,, J. Nonlinear Sci., 3 (1993), 169.
doi: 10.1007/BF02429863. |
[10] |
J. M. Ghidaglia and J. C. Saut, Nonexistence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations,, J. Nonlinear Sci., 6 (1996), 139.
doi: 10.1007/BF02434051. |
[11] |
J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.
doi: 10.1088/0951-7715/3/2/010. |
[12] |
P. Kevrekidis, A. Nahmod and C. Zeng, Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D,, Nonlinearity, 24 (2011), 1523.
doi: 10.1088/0951-7715/24/5/007. |
[13] |
N. Lu, Small generalized breathers with exponentially small tails for Klein-Gordon equations,, J. Differential Equations, 256 (2014), 745.
doi: 10.1016/j.jde.2013.09.018. |
[14] |
J. Moser, A new technique for the construction of solution of nonlinear differential equations,, Proc. Nat. Acad. Sci., 47 (1961), 1824.
doi: 10.1073/pnas.47.11.1824. |
[15] |
A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase,, J. Appl. Math. Mech., 48 (1984), 133.
doi: 10.1016/0021-8928(84)90078-9. |
[16] |
C. Sulem and J. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse,, Applied Mathematical Sciences 139, (1999).
|
[17] |
L. Vuillon, D. Dutykh and F. Fedele, Some special solutions to the hyperbolic NLS equation, preprint,, , (). Google Scholar |
[18] |
V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, J. Appl. Mech. Tech. Phys., 9 (1968), 190.
doi: 10.1007/BF00913182. |
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