February  2013, 33(2): 789-801. doi: 10.3934/dcds.2013.33.789

Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension

1. 

Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

2. 

Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received  September 2011 Revised  March 2012 Published  September 2012

We study the Cauchy problem for cubic Schrödinger equations modelling ultra-short laser pulses propagating along the line. The global existence, blow-up, and scattering of solutions is described exclusively in the charge space $L^2({\bf R})$ without any approximating arguments.
Citation: Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 789-801. doi: 10.3934/dcds.2013.33.789
References:
[1]

T. Cazenave, Equations de Schrödinger non linéaires en dimension deux,, Proc. Roy. Soc. Edinburgh, 84 (1979), 327. doi: 10.1017/S0308210500017182. Google Scholar

[2]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics 10, (2003). Google Scholar

[3]

R. Cipolatti and O. Kavian, On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompactglobal attractor,, Discrete and Continuous Dynamical Systems, 17 (2007), 121. Google Scholar

[4]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations on space dimension $n\geq2$,, Commun. Math. Phys., 151 (1993), 619. doi: 10.1007/BF02097031. Google Scholar

[5]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119. Google Scholar

[6]

V. S. Grigoryan, A. I. Maimistov and Y. M. Sklyarov, Evolution of light pulses in a nonlinear amplifying medium,, Sov. Phys. JETP, 67 (1988), 530. Google Scholar

[7]

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

[8]

N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, SIAM J. Math. Anal., 29 (1998), 1256. doi: 10.1137/S0036141096312222. Google Scholar

[9]

N. Hayashi and T. Ozawa, Schrödinger equations with nonlinearity of integral type,, Discrete and Continuous Dynamical Systems, 1 (1995), 475. Google Scholar

[10]

T. Kato, "Nonlinear Schrödinger Equations,", in, 345 (1989). Google Scholar

[11]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar

[12]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension,, Commun. Math. Phys., 139 (1991), 479. doi: 10.1007/BF02101876. Google Scholar

[13]

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

[14]

T. Ozawa and Y. Yamazaki, Smoothing effect and large time behavior of solutions to Schrödinger equations with nonlinearity of integral type,, Commun. Contemp. Math., 6 (2004), 681. Google Scholar

[15]

C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave-Collapse,", Springer-Verlag, (1999). Google Scholar

[16]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations,, Bull. Amer. Math. Soc., 11 (1984), 186. doi: 10.1090/S0273-0979-1984-15263-7. Google Scholar

[17]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, Commun. Math. Phys., 110 (1987), 415. doi: 10.1007/BF01212420. Google Scholar

[18]

E. V. Vamin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak and L. Vázquez, Dissipative optical solitons,, Physical Review A, 49 (1994), 2806. doi: 10.1103/PhysRevA.49.2806. Google Scholar

show all references

References:
[1]

T. Cazenave, Equations de Schrödinger non linéaires en dimension deux,, Proc. Roy. Soc. Edinburgh, 84 (1979), 327. doi: 10.1017/S0308210500017182. Google Scholar

[2]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics 10, (2003). Google Scholar

[3]

R. Cipolatti and O. Kavian, On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompactglobal attractor,, Discrete and Continuous Dynamical Systems, 17 (2007), 121. Google Scholar

[4]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations on space dimension $n\geq2$,, Commun. Math. Phys., 151 (1993), 619. doi: 10.1007/BF02097031. Google Scholar

[5]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119. Google Scholar

[6]

V. S. Grigoryan, A. I. Maimistov and Y. M. Sklyarov, Evolution of light pulses in a nonlinear amplifying medium,, Sov. Phys. JETP, 67 (1988), 530. Google Scholar

[7]

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

[8]

N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, SIAM J. Math. Anal., 29 (1998), 1256. doi: 10.1137/S0036141096312222. Google Scholar

[9]

N. Hayashi and T. Ozawa, Schrödinger equations with nonlinearity of integral type,, Discrete and Continuous Dynamical Systems, 1 (1995), 475. Google Scholar

[10]

T. Kato, "Nonlinear Schrödinger Equations,", in, 345 (1989). Google Scholar

[11]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar

[12]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension,, Commun. Math. Phys., 139 (1991), 479. doi: 10.1007/BF02101876. Google Scholar

[13]

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

[14]

T. Ozawa and Y. Yamazaki, Smoothing effect and large time behavior of solutions to Schrödinger equations with nonlinearity of integral type,, Commun. Contemp. Math., 6 (2004), 681. Google Scholar

[15]

C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave-Collapse,", Springer-Verlag, (1999). Google Scholar

[16]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations,, Bull. Amer. Math. Soc., 11 (1984), 186. doi: 10.1090/S0273-0979-1984-15263-7. Google Scholar

[17]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, Commun. Math. Phys., 110 (1987), 415. doi: 10.1007/BF01212420. Google Scholar

[18]

E. V. Vamin, A. I. Korytin, A. M. Sergeev, D. Anderson, M. Lisak and L. Vázquez, Dissipative optical solitons,, Physical Review A, 49 (1994), 2806. doi: 10.1103/PhysRevA.49.2806. Google Scholar

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