American Institute of Mathematical Sciences

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December  2012, 32(12): 4265-4285. doi: 10.3934/dcds.2012.32.4265

Decay of solutions for a system of nonlinear Schrödinger equations in 2D

Chunhua Li 1,

1. 

Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan

Received  June 2011 Revised  August 2011 Published  August 2012

We deal with a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove $\mathbf{L}^{\infty }-$time decay estimates of small solutions. We also discuss existence and nonexistence of wave operators.
Keywords: wave operators., A system of nonlinear Schrödinger equations, $L^{\infty }$-time decay estimates.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B4.
Citation: Chunhua Li. Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4265-4285. doi: 10.3934/dcds.2012.32.4265
References:
[1]

T. Cazenave, "Semilinear Schödinger Equations," Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.  Google Scholar

[2]

S. Cohn, Global existence for the nonresonant Schrödinger equation in two space dimensions, Canad. Appl. Math. Quart., 2 (1994), 247-282.  Google Scholar

[3]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.  Google Scholar

[4]

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

[5]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698. doi: 10.1007/BF02570764.  Google Scholar

[6]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\geq 2$, Commun. Math. Phys., 151 (1993), 619-645. doi: 10.1007/BF02097031.  Google Scholar

[7]

N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J., 60 (1990), 717-727. doi: 10.1215/S0012-7094-90-06029-6.  Google Scholar

[8]

N. Hayashi, Global and almost global solutions to quadratic nonlinear Schrödinger equations with small initial data, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 109-129.  Google Scholar

[9]

N. Hayashi, C. Li and P. I. Naumkin, On a system of nonlinear Schrödinger equations in 2d, Differential Integral Equations, 24 (2011), 417-434.  Google Scholar

[10]

N. Hayashi, C. Li and P. I. Naumkin, Modified wave operator for a system of nonlinear Schrödinger equations in 2d, Comm. Partial Differential Equations, 37 (2012), 947-968. Google Scholar

[11]

N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differential Equations and Applications - DEA, 3 (2011), 415-426.  Google Scholar

[12]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.  Google Scholar

[13]

N. Hayashi and P. I. Naumkin, On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices, 3 (2000), 115-132.  Google Scholar

[14]

N. Hayashi and P. I. Naumkin, Global existence of small solutions to the quadratic nonlinear Schrödinger equations in two space dimensions, SIAM J. Math. Anal., 32 (2001), 1390-1403. doi: 10.1137/S0036141000372532.  Google Scholar

[15]

N. Hayashi and P. I. Naumkin, Asymptotics in time of solutions to nonlinear Schrödinger equations in two space dimensions, Funkcial. Ekvac., 49 (2006), 415-425. doi: 10.1619/fesi.49.415.  Google Scholar

[16]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified wave operators for nonlinear Schrödinger equations in one and two dimensions, Electron. J. Differential Equations, (2004), 62, 16pp.  Google Scholar

[17]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $\mathbfL^2$( Rn) spaces for some Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 17-37.  Google Scholar

[18]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance , J. Differential Equations, 251 (2011), 2549-2567. doi: 10.1016/j.jde.2011.04.001.  Google Scholar

[19]

S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions,, to appear in Comm. Pure Appl. Math., ().   Google Scholar

[20]

S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal., 78 (1982), 73-98. doi: 10.1007/BF00253225.  Google Scholar

[21]

K. Moriyama, S. Tonegawa and Y. Tsutsumi, Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996.  Google Scholar

[22]

T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232.  Google Scholar

[23]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.  Google Scholar

[24]

J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), 409-425. doi: 10.1016/0022-0396(82)90102-4.  Google Scholar

[25]

A. Shimomura, Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.  Google Scholar

[26]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423. doi: 10.1080/03605300600910316.  Google Scholar

[27]

A. Shimomura and S. Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions Differential integral Equations, 17 (2004),127-150.  Google Scholar

[28]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.  Google Scholar

[29]

W. Strauss, Nonlinear scattering at low energy, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.  Google Scholar

[30]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325. doi: 10.1016/S0022-0396(03)00125-6.  Google Scholar

[31]

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

show all references

References:
[1]

T. Cazenave, "Semilinear Schödinger Equations," Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.  Google Scholar

[2]

S. Cohn, Global existence for the nonresonant Schrödinger equation in two space dimensions, Canad. Appl. Math. Quart., 2 (1994), 247-282.  Google Scholar

[3]

M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations, 17 (2004), 297-330.  Google Scholar

[4]

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

[5]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698. doi: 10.1007/BF02570764.  Google Scholar

[6]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n\geq 2$, Commun. Math. Phys., 151 (1993), 619-645. doi: 10.1007/BF02097031.  Google Scholar

[7]

N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J., 60 (1990), 717-727. doi: 10.1215/S0012-7094-90-06029-6.  Google Scholar

[8]

N. Hayashi, Global and almost global solutions to quadratic nonlinear Schrödinger equations with small initial data, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 109-129.  Google Scholar

[9]

N. Hayashi, C. Li and P. I. Naumkin, On a system of nonlinear Schrödinger equations in 2d, Differential Integral Equations, 24 (2011), 417-434.  Google Scholar

[10]

N. Hayashi, C. Li and P. I. Naumkin, Modified wave operator for a system of nonlinear Schrödinger equations in 2d, Comm. Partial Differential Equations, 37 (2012), 947-968. Google Scholar

[11]

N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differential Equations and Applications - DEA, 3 (2011), 415-426.  Google Scholar

[12]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.  Google Scholar

[13]

N. Hayashi and P. I. Naumkin, On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices, 3 (2000), 115-132.  Google Scholar

[14]

N. Hayashi and P. I. Naumkin, Global existence of small solutions to the quadratic nonlinear Schrödinger equations in two space dimensions, SIAM J. Math. Anal., 32 (2001), 1390-1403. doi: 10.1137/S0036141000372532.  Google Scholar

[15]

N. Hayashi and P. I. Naumkin, Asymptotics in time of solutions to nonlinear Schrödinger equations in two space dimensions, Funkcial. Ekvac., 49 (2006), 415-425. doi: 10.1619/fesi.49.415.  Google Scholar

[16]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified wave operators for nonlinear Schrödinger equations in one and two dimensions, Electron. J. Differential Equations, (2004), 62, 16pp.  Google Scholar

[17]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $\mathbfL^2$( Rn) spaces for some Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 17-37.  Google Scholar

[18]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance , J. Differential Equations, 251 (2011), 2549-2567. doi: 10.1016/j.jde.2011.04.001.  Google Scholar

[19]

S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions,, to appear in Comm. Pure Appl. Math., ().   Google Scholar

[20]

S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal., 78 (1982), 73-98. doi: 10.1007/BF00253225.  Google Scholar

[21]

K. Moriyama, S. Tonegawa and Y. Tsutsumi, Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996.  Google Scholar

[22]

T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232.  Google Scholar

[23]

T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408. doi: 10.1007/s00526-005-0349-2.  Google Scholar

[24]

J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), 409-425. doi: 10.1016/0022-0396(82)90102-4.  Google Scholar

[25]

A. Shimomura, Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.  Google Scholar

[26]

A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423. doi: 10.1080/03605300600910316.  Google Scholar

[27]

A. Shimomura and S. Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions Differential integral Equations, 17 (2004),127-150.  Google Scholar

[28]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.  Google Scholar

[29]

W. Strauss, Nonlinear scattering at low energy, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.  Google Scholar

[30]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325. doi: 10.1016/S0022-0396(03)00125-6.  Google Scholar

[31]

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

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