American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect
  • DCDS-B Home
  • This Issue
  • Next Article
    Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders
March  2014, 19(2): 565-588. doi: 10.3934/dcdsb.2014.19.565

Transverse instability for a system of nonlinear Schrödinger equations

Yohei Yamazaki 1,

1. 

Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan

Received  November 2012 Revised  November 2013 Published  February 2014

In this paper, we consider the transverse instability for a system of nonlinear Schrödinger equations on $\mathbb{R} \times \mathbb{T}_L $. Here, $\mathbb{T}_L$ means the torus with a $2\pi L$ period. It was shown by Colin-Ohta [11] that this system on $\mathbb{R}$ has a stable standing wave. In this paper, we regard this standing wave as the standing wave of this system on $\mathbb{R} \times \mathbb{T}_L$. Then, we show that there exists the critical period $L_{\omega}$ which is the boundary between the stability and the instability of the standing wave on $\mathbb{R} \times \mathbb{T}_L$.
Keywords: Schrödinger equation., standing wave, Stability.
Mathematics Subject Classification: Primary: 35B35, 35C08, 35Q5.
Citation: Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565
References:
[1]

J. C. Alexander, R. L. Pego and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A, 226 (1997), 187-192. doi: 10.1016/S0375-9601(96)00921-8.  Google Scholar

[2]

T. Benjamin, The stability of solitary waves, Proc. London Math. Soc., (3) 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Commun. Math. Phys., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[4]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy problem for the non linear Schrödinger equation on a compact manifold, J. Nonlinear Math., 10 (2003), 12-27. doi: 10.2991/jnmp.2003.10.s1.2.  Google Scholar

[5]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), no. 3, 569-605. doi: 10.1353/ajm.2004.0016.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003.  Google Scholar

[7]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.  Google Scholar

[8]

M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562. doi: 10.1016/j.cam.2005.05.031.  Google Scholar

[9]

M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schödinger equations with three wave interaction, Ann. I. Poincaré-AN, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.  Google Scholar

[10]

M. Colin, T. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380. doi: 10.1619/fesi.52.371.  Google Scholar

[11]

M. Colin and M. Ohta, Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations, SIAM J. Math. Anal., 44 no. 1 (2012), 206-223. doi: 10.1137/110823808.  Google Scholar

[12]

B. Deconinck, D. E. Pelinovsky and J. D. Carter, Transverse instabilities of deep-water solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys., Eng. Sci., 462 (2006), no. 2071, 2039-2061. doi: 10.1098/rspa.2006.1670.  Google Scholar

[13]

V. Georgiev and M. Ohta, Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinge equations, J. Math. Soc. Japan, 64 (2012), no. 2, 533-548. doi: 10.2969/jmsj/06420533.  Google Scholar

[14]

F. Gesztesy, C. K. R. T. Jones, Y. Latushkin and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J., 49 (2000), 221-243. doi: 10.1512/iumj.2000.49.1838.  Google Scholar

[15]

J. Ginibre and G. Velo, On a class of nonlinear Schödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[16]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. doi: 10.1007/BF01218585.  Google Scholar

[17]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[18]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[19]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[20]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[21]

T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition,Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

H. Kikuchi, Orbital stability of semitrivial standing waves for the Klein-Gordon-Schrödinger system, Ann. I. Poincaré-AN, 28 (2011), 315-323. doi: 10.1016/j.anihpc.2011.02.003.  Google Scholar

[23]

F. Merle and L. Vega, $L^2$ stability of solitons for KdV equation, Int. Math. Res. Notices, 13 (2003), 735-753. doi: 10.1155/S1073792803208060.  Google Scholar

[24]

R. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. London A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055.  Google Scholar

[25]

D. E. Pelinovsky, A mysterious threshold for transverse instability of deep-water solitons. Nonlinear waves: Computation and theory (Athens, GA, 1999), Math. Comput. Simulation, 55 (2001), no. 4-6, 585-594. doi: 10.1016/S0378-4754(00)00287-1.  Google Scholar

[26]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures. Appl., 90 (2008), 550-590. doi: 10.1016/j.matpur.2008.07.004.  Google Scholar

[27]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. I. Poincaré-AN, 26 (2009), 477-496. doi: 10.1016/j.anihpc.2007.09.006.  Google Scholar

[28]

F. Rousset and N. Tzvetkov, A simple criterion of transverse linear instability for solitary waves, Math. Res. Lett., 17 (2010), no. 1, 157-169. doi: 10.4310/MRL.2010.v17.n1.a12.  Google Scholar

[29]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), no. 2, 257-388. doi: 10.1007/s00222-010-0290-7.  Google Scholar

[30]

F. Rousset and N. Tzvetkov, Stability and instability of the KdV solitary wave under the KP-I flow, Comm. Math. Phys., 313 (2012), no. 1, 155-173. doi: 10.1007/s00220-012-1495-y.  Google Scholar

[31]

J. Shatah and W. Strauss, Spectral condition for instability, Contemp. Math., 255 (2000), 189-198. doi: 10.1090/conm/255/03982.  Google Scholar

[32]

H. Takaoka and N. Tzvetkov, On 2D nonlinear Schrödinger equations with Data on $\mathbbR \times \mathbbT$, J. Funct. Anal., 182 (2001), 427-442. doi: 10.1006/jfan.2000.3732.  Google Scholar

[33]

S. Terracini, N. Tzvetkov and N. Visciglia, The nonlinear Schrödinger equation ground state on product spaces,, preprint, ().   Google Scholar

[34]

Y. Tsutsumi, $L^2$-solution for nonlinear Schrödinger equatoion and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar

[35]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar

show all references

References:
[1]

J. C. Alexander, R. L. Pego and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A, 226 (1997), 187-192. doi: 10.1016/S0375-9601(96)00921-8.  Google Scholar

[2]

T. Benjamin, The stability of solitary waves, Proc. London Math. Soc., (3) 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Commun. Math. Phys., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[4]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy problem for the non linear Schrödinger equation on a compact manifold, J. Nonlinear Math., 10 (2003), 12-27. doi: 10.2991/jnmp.2003.10.s1.2.  Google Scholar

[5]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), no. 3, 569-605. doi: 10.1353/ajm.2004.0016.  Google Scholar

[6]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, Amer. Math. Soc., 2003.  Google Scholar

[7]

T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.  Google Scholar

[8]

M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562. doi: 10.1016/j.cam.2005.05.031.  Google Scholar

[9]

M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schödinger equations with three wave interaction, Ann. I. Poincaré-AN, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011.  Google Scholar

[10]

M. Colin, T. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcial. Ekvac., 52 (2009), 371-380. doi: 10.1619/fesi.52.371.  Google Scholar

[11]

M. Colin and M. Ohta, Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations, SIAM J. Math. Anal., 44 no. 1 (2012), 206-223. doi: 10.1137/110823808.  Google Scholar

[12]

B. Deconinck, D. E. Pelinovsky and J. D. Carter, Transverse instabilities of deep-water solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys., Eng. Sci., 462 (2006), no. 2071, 2039-2061. doi: 10.1098/rspa.2006.1670.  Google Scholar

[13]

V. Georgiev and M. Ohta, Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinge equations, J. Math. Soc. Japan, 64 (2012), no. 2, 533-548. doi: 10.2969/jmsj/06420533.  Google Scholar

[14]

F. Gesztesy, C. K. R. T. Jones, Y. Latushkin and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J., 49 (2000), 221-243. doi: 10.1512/iumj.2000.49.1838.  Google Scholar

[15]

J. Ginibre and G. Velo, On a class of nonlinear Schödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[16]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. doi: 10.1007/BF01218585.  Google Scholar

[17]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[18]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[19]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[20]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[21]

T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition,Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

H. Kikuchi, Orbital stability of semitrivial standing waves for the Klein-Gordon-Schrödinger system, Ann. I. Poincaré-AN, 28 (2011), 315-323. doi: 10.1016/j.anihpc.2011.02.003.  Google Scholar

[23]

F. Merle and L. Vega, $L^2$ stability of solitons for KdV equation, Int. Math. Res. Notices, 13 (2003), 735-753. doi: 10.1155/S1073792803208060.  Google Scholar

[24]

R. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. London A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055.  Google Scholar

[25]

D. E. Pelinovsky, A mysterious threshold for transverse instability of deep-water solitons. Nonlinear waves: Computation and theory (Athens, GA, 1999), Math. Comput. Simulation, 55 (2001), no. 4-6, 585-594. doi: 10.1016/S0378-4754(00)00287-1.  Google Scholar

[26]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures. Appl., 90 (2008), 550-590. doi: 10.1016/j.matpur.2008.07.004.  Google Scholar

[27]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. I. Poincaré-AN, 26 (2009), 477-496. doi: 10.1016/j.anihpc.2007.09.006.  Google Scholar

[28]

F. Rousset and N. Tzvetkov, A simple criterion of transverse linear instability for solitary waves, Math. Res. Lett., 17 (2010), no. 1, 157-169. doi: 10.4310/MRL.2010.v17.n1.a12.  Google Scholar

[29]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), no. 2, 257-388. doi: 10.1007/s00222-010-0290-7.  Google Scholar

[30]

F. Rousset and N. Tzvetkov, Stability and instability of the KdV solitary wave under the KP-I flow, Comm. Math. Phys., 313 (2012), no. 1, 155-173. doi: 10.1007/s00220-012-1495-y.  Google Scholar

[31]

J. Shatah and W. Strauss, Spectral condition for instability, Contemp. Math., 255 (2000), 189-198. doi: 10.1090/conm/255/03982.  Google Scholar

[32]

H. Takaoka and N. Tzvetkov, On 2D nonlinear Schrödinger equations with Data on $\mathbbR \times \mathbbT$, J. Funct. Anal., 182 (2001), 427-442. doi: 10.1006/jfan.2000.3732.  Google Scholar

[33]

S. Terracini, N. Tzvetkov and N. Visciglia, The nonlinear Schrödinger equation ground state on product spaces,, preprint, ().   Google Scholar

[34]

Y. Tsutsumi, $L^2$-solution for nonlinear Schrödinger equatoion and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar

[35]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar

[1]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[2]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[3]

Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
[4]

Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[5]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[6]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825
[7]

François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[8]

Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193
[9]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
[10]

Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure & Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351
[11]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
[12]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
[13]

Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
[14]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[15]

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
[16]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[17]

Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080
[18]

Huifang Jia, Gongbao Li, Xiao Luo. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2739-2766. doi: 10.3934/dcds.2020148
[19]

Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073
[20]

Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences