American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability analysis for linear heat conduction with memory kernels described by Gamma functions
  • DCDS Home
  • This Issue
  • Next Article
    Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source
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 & 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.  Google Scholar

[2]

M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J., 134 (2006), 359-419. 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), 377-399. doi: 10.1016/j.anihpc.2009.11.010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 170406, 4pp. 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), 1827-1832. doi: 10.1364/OPEX.13.001827.  Google Scholar

[9]

J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. 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), 139-145. 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), 475-506. 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), 1523-1538. 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), 745-770. 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), 1824-1831. 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), 133-139 (1985); translated from Prikl. Mat. Mekh. 48 (1984), 197-204. (Russian) 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, Springer, 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), 190-194. doi: 10.1007/BF00913182.  Google Scholar

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.  Google Scholar

[2]

M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J., 134 (2006), 359-419. 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), 377-399. doi: 10.1016/j.anihpc.2009.11.010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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), 170406, 4pp. 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), 1827-1832. doi: 10.1364/OPEX.13.001827.  Google Scholar

[9]

J. M. Ghidaglia and J. C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. 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), 139-145. 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), 475-506. 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), 1523-1538. 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), 745-770. 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), 1824-1831. 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), 133-139 (1985); translated from Prikl. Mat. Mekh. 48 (1984), 197-204. (Russian) 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, Springer, 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), 190-194. doi: 10.1007/BF00913182.  Google Scholar

[1]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[2]

Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[3]

Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31
[4]

Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20
[5]

François Genoud. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evolution Equations & Control Theory, 2013, 2 (1) : 81-100. doi: 10.3934/eect.2013.2.81
[6]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[7]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[8]

Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113
[9]

Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246
[10]

Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165
[11]

Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177
[12]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
[13]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[14]

Riccardo Adami, Diego Noja, Cecilia Ortoleva. Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5837-5879. doi: 10.3934/dcds.2016057
[15]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[16]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123
[17]

Alessandro Selvitella. Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2663-2677. doi: 10.3934/cpaa.2019118
[18]

Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
[19]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459
[20]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences