# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 327-346. doi: 10.1017/S0308210500017182. [2] T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003. [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), 121-132. [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), 619-645. doi: 10.1007/BF02097031. [5] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. doi: 10.1006/jfan.1995.1119. [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), 530-534. [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), 369-389. doi: 10.1353/ajm.1998.0011. [8] N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222. [9] N. Hayashi and T. Ozawa, Schrödinger equations with nonlinearity of integral type, Discrete and Continuous Dynamical Systems, 1 (1995), 475-484. [10] T. Kato, "Nonlinear Schrödinger Equations," in "Schrödinger Operators" (eds. H. Holden and A. Jensen), Lecture Notes in Physics, Springer, 345, 1989. [11] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. [12] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493. doi: 10.1007/BF02101876. [13] 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. [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), 681-703. [15] C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave-Collapse," Springer-Verlag, 1999. [16] Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188. doi: 10.1090/S0273-0979-1984-15263-7. [17] K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426. doi: 10.1007/BF01212420. [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), 2806-2811. doi: 10.1103/PhysRevA.49.2806.

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##### References:
 [1] T. Cazenave, Equations de Schrödinger non linéaires en dimension deux, Proc. Roy. Soc. Edinburgh, 84 (1979), 327-346. doi: 10.1017/S0308210500017182. [2] T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003. [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), 121-132. [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), 619-645. doi: 10.1007/BF02097031. [5] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. doi: 10.1006/jfan.1995.1119. [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), 530-534. [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), 369-389. doi: 10.1353/ajm.1998.0011. [8] N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222. [9] N. Hayashi and T. Ozawa, Schrödinger equations with nonlinearity of integral type, Discrete and Continuous Dynamical Systems, 1 (1995), 475-484. [10] T. Kato, "Nonlinear Schrödinger Equations," in "Schrödinger Operators" (eds. H. Holden and A. Jensen), Lecture Notes in Physics, Springer, 345, 1989. [11] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. [12] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479-493. doi: 10.1007/BF02101876. [13] 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. [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), 681-703. [15] C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-Focusing and Wave-Collapse," Springer-Verlag, 1999. [16] Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188. doi: 10.1090/S0273-0979-1984-15263-7. [17] K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys., 110 (1987), 415-426. doi: 10.1007/BF01212420. [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), 2806-2811. doi: 10.1103/PhysRevA.49.2806.
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