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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

Nan Lu 1,

1. 

14 East Packer Avenue, Christmas-Saucon Hall, Lehigh University, Bethlehem, PA 18015, United States

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].
Keywords: periodic orbits, standing wave, Hyperbolic NLS, Nash-Moser interation., generalized breather.
Mathematics Subject Classification: Primary: 37C27, 37C29; Secondary: 35B25, 35B3.
Citation: Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 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), 247-276. 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), 359-419. 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), 377-399. doi: 10.1016/j.anihpc.2009.11.010.

[4]

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.

[5]

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.

[6]

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.

[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), 170406, 4pp.

[8]

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.

[9]

J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. 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), 139-145. 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), 475-506. 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), 1523-1538. 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), 745-770. 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), 1824-1831. 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), 133-139 (1985); translated from Prikl. Mat. Mekh. 48 (1984), 197-204. (Russian) 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, Springer, 1999.

[17]

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

[18]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194. 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), 247-276. 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), 359-419. 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), 377-399. doi: 10.1016/j.anihpc.2009.11.010.

[4]

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.

[5]

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.

[6]

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.

[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), 170406, 4pp.

[8]

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.

[9]

J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. 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), 139-145. 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), 475-506. 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), 1523-1538. 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), 745-770. 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), 1824-1831. 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), 133-139 (1985); translated from Prikl. Mat. Mekh. 48 (1984), 197-204. (Russian) 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, Springer, 1999.

[17]

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

[18]

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

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