Transverse instability for a system of nonlinear Schrödinger equations

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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

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

Received November 2012 Revised November 2013 Published February 2014

Abstract
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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. doi: 10.1016/S0375-9601(96)00921-8. Google Scholar*
[2]	T. Benjamin, The stability of solitary waves,, Proc. London Math. Soc., 328* (1972), 153. 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. 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. 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), 569. doi: 10.1353/ajm.2004.0016. Google Scholar*
[6]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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. 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. 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. 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. 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 (2012), 206. 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., 462* (2006), 2039. 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), 533. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[21]	T. Kato, Perturbation theory for linear operators,, Reprint of the 1980 edition, (1980). Google Scholar
[22]	H. Kikuchi, Orbital stability of semitrivial standing waves for the Klein-Gordon-Schrödinger system,, Ann. I. Poincaré-AN, 28 (2011), 315. 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. 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. 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), 4. 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. 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. 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), 157. 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), 257. 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), 155. doi: 10.1007/s00220-012-1495-y. Google Scholar*
[31]	J. Shatah and W. Strauss, Spectral condition for instability,, Contemp. Math., 255* (2000), 189. 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. 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. Google Scholar
[35]	M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. 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. doi: 10.1016/S0375-9601(96)00921-8. Google Scholar*
[2]	T. Benjamin, The stability of solitary waves,, Proc. London Math. Soc., 328* (1972), 153. 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. 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. 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), 569. doi: 10.1353/ajm.2004.0016. Google Scholar*
[6]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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. 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. 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. 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. 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 (2012), 206. 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., 462* (2006), 2039. 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), 533. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[21]	T. Kato, Perturbation theory for linear operators,, Reprint of the 1980 edition, (1980). Google Scholar
[22]	H. Kikuchi, Orbital stability of semitrivial standing waves for the Klein-Gordon-Schrödinger system,, Ann. I. Poincaré-AN, 28 (2011), 315. 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. 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. 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), 4. 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. 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. 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), 157. 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), 257. 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), 155. doi: 10.1007/s00220-012-1495-y. Google Scholar*
[31]	J. Shatah and W. Strauss, Spectral condition for instability,, Contemp. Math., 255* (2000), 189. 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. 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. Google Scholar
[35]	M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. doi: 10.1137/0516034. Google Scholar

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