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August  2015, 35(8): 3533-3567. doi: 10.3934/dcds.2015.35.3533

## Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

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Received  April 2014 Revised  January 2015 Published  February 2015

We construct two families of non-localized standing waves for the hyperbolic cubic nonlinear Schrödinger equation $iu_t+u_{xx}-u_{yy}+|u|^2u=0.$ The first family of standing waves consists of solutions which correspond to some generalized breathers for each fixed time $t$, while solutions in the second family are periodic both in $x$ and $y$. The second family of solutions were numerically observed by Vuillon, Dutykh and Fedele in a recent preprint [17].
Citation: Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44. doi: 10.1007/BFb0095239. Google Scholar [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. Google Scholar [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. Google Scholar [9] J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations,, J. Nonlinear Sci., 3 (1993), 169. doi: 10.1007/BF02429863. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] C. Sulem and J. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse,, Applied Mathematical Sciences 139, (1999). Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44. doi: 10.1007/BFb0095239. Google Scholar [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. Google Scholar [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. Google Scholar [9] J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations,, J. Nonlinear Sci., 3 (1993), 169. doi: 10.1007/BF02429863. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] C. Sulem and J. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse,, Applied Mathematical Sciences 139, (1999). Google Scholar [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. Google Scholar
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