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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

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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

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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

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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

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