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

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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

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

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

References:

[1]	T. Cazenave, "Semilinear Schödinger Equations,", Courant Institute of Mathematical Sciences, (2003). Google Scholar
[2]	S. Cohn, Global existence for the nonresonant Schrödinger equation in two space dimensions,, Canad. Appl. Math. Quart., 2 (1994), 247. Google Scholar
[3]	M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions,, Differential Integral Equations, 17 (2004), 297. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. Google Scholar
[22]	T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations,, Funkcial. Ekvac., 38 (1995), 217. Google Scholar
[23]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations,, Calc. Var. Partial Differential Equations, 25 (2006), 403. 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. 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. Google Scholar
[26]	A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities,, Comm. Partial Differential Equations, 31 (2006), 1407. 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. Google Scholar
[28]	J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Comm. Pure Appl. Math., 38 (1985), 685. doi: 10.1002/cpa.3160380516. Google Scholar
[29]	W. Strauss, Nonlinear scattering at low energy,, J. Funct. Anal., 43 (1981), 281. 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. 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. Google Scholar

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References:

[1]	T. Cazenave, "Semilinear Schödinger Equations,", Courant Institute of Mathematical Sciences, (2003). Google Scholar
[2]	S. Cohn, Global existence for the nonresonant Schrödinger equation in two space dimensions,, Canad. Appl. Math. Quart., 2 (1994), 247. Google Scholar
[3]	M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions,, Differential Integral Equations, 17 (2004), 297. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. Google Scholar
[22]	T. Ozawa, Remarks on quadratic nonlinear Schrödinger equations,, Funkcial. Ekvac., 38 (1995), 217. Google Scholar
[23]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations,, Calc. Var. Partial Differential Equations, 25 (2006), 403. 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. 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. Google Scholar
[26]	A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities,, Comm. Partial Differential Equations, 31 (2006), 1407. 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. Google Scholar
[28]	J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Comm. Pure Appl. Math., 38 (1985), 685. doi: 10.1002/cpa.3160380516. Google Scholar
[29]	W. Strauss, Nonlinear scattering at low energy,, J. Funct. Anal., 43 (1981), 281. 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. 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. Google Scholar

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