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

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  • 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].
    Mathematics Subject Classification: Primary: 37C27, 37C29; Secondary: 35B25, 35B36.


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  • [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), 247-276.doi: 10.1007/s00030-007-7025-5.


    M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J., 134 (2006), 359-419.doi: 10.1215/S0012-7094-06-13424-5.


    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), 377-399.doi: 10.1016/j.anihpc.2009.11.010.


    C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, (Montecatini Terme, 1994), in Lecture Notes in Math., 1609, Springer-Verlag, Berlin, (1995), 44-118.doi: 10.1007/BFb0095239.


    S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.doi: 10.1016/0022-0396(88)90007-1.


    C. Conti and S. Trillo, Nonlinear X Waves in Localized Waves, H. E. Hernandez-Figueroa, M. Zamboni-Rached and E. Recomi (Eds.), Hoboken, NJ: John Wiley & Sons Inc. (2007), 243-272.


    C. Conti, S. Trillo, P. Di Trapani, A. Piskarkas, O. Jedrkiewicz and J. Trull, Nonlinear electro-magnetic X waves, Phys. Rev. Letter, 90 (2003), 170406, 4pp.


    S. Droulias, K. Hizanidis, J. Meier and D. N. Christodoulides, X-waves in nonlinear normally dispersive waveguide arrays, Optical Express, 13 (2005), 1827-1832.doi: 10.1364/OPEX.13.001827.


    J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195.doi: 10.1007/BF02429863.


    J. M. Ghidaglia and J. C. Saut, Nonexistence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145.doi: 10.1007/BF02434051.


    J. M. Ghidaglia and J. C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506.doi: 10.1088/0951-7715/3/2/010.


    P. Kevrekidis, A. Nahmod and C. Zeng, Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D, Nonlinearity, 24 (2011), 1523-1538.doi: 10.1088/0951-7715/24/5/007.


    N. Lu, Small generalized breathers with exponentially small tails for Klein-Gordon equations, J. Differential Equations, 256 (2014), 745-770.doi: 10.1016/j.jde.2013.09.018.


    J. Moser, A new technique for the construction of solution of nonlinear differential equations, Proc. Nat. Acad. Sci., 47 (1961), 1824-1831.doi: 10.1073/pnas.47.11.1824.


    A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139 (1985); translated from Prikl. Mat. Mekh. 48 (1984), 197-204. (Russian)doi: 10.1016/0021-8928(84)90078-9.


    C. Sulem and J. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139, Springer, 1999.


    L. Vuillon, D. Dutykh and F. Fedele, Some special solutions to the hyperbolic NLS equation, preprint, arXiv:1307.5507.


    V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.doi: 10.1007/BF00913182.

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