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Nash-Moser methods for the solution of quasilinear Schrödinger equations. (English) Zbl 0935.35153
The authors study the nonlinear Schrödinger equation $iu_t=-\Delta u+K(\Delta|u|^2)u,\quad u(0,x)=\phi(x),\quad (x\in\mathbb{R}^n).$ Applying some new techniques by M. Poppenberg on Nash-Moser type implicit function theorems for Fréchet spaces, the local existence, uniqueness and continuous dependence of smooth solutions in space dimension $$n=1$$, is proved. The basic function space $$H^\infty$$ is used.
The method consists in finding an appropriate linearization of the equation, and proving that this linear Schrödinger equation admits a strongly continuous evolution operator which provides the necessary a priori estimates for any derivative.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35F25 Initial value problems for nonlinear first-order PDEs
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##### References:
 [1] DOI: 10.1016/0370-1573(90)90093-H [2] DOI: 10.1007/s002200050191 · Zbl 0948.81025 [3] DOI: 10.1002/mma.1670080136 · Zbl 0626.35083 [4] Brüll L, Expositiones Math 4 pp 279– (1986) [5] DOI: 10.1063/1.527552 · Zbl 0639.35014 [6] DOI: 10.1090/S0273-0979-1982-15004-2 · Zbl 0499.58003 [7] Hartman, P. 1964. ”Ordinary differential equations”. J. Wiley. [8] DOI: 10.1007/BF01325508 [9] DOI: 10.1016/0370-1573(90)90130-T [10] DOI: 10.1143/JPSJ.50.3262 [11] DOI: 10.1063/1.525675 · Zbl 0548.35101 [12] Lange H, Lecture Notes Pure Applied Mathematics 155 pp 307– (1994) [13] Lange H, Mathematische Modellierung pp 114– (1986) [14] Lange H, Proc. Trends in Applications of Mathematics to Mechanics pp 98– (1988) [15] DOI: 10.1063/1.531120 · Zbl 0823.35159 [16] Litvak A.G, JETP Letters 27 pp 517– (1978) [17] DOI: 10.1016/0022-1236(79)90109-5 · Zbl 0431.46032 [18] DOI: 10.1143/JPSJ.42.1824 [19] DOI: 10.1006/jfan.1996.0147 · Zbl 0884.46001 [20] Poppenberg M., Math. Nachr [21] Poppenberg M., The Proceedings of the London Math. Soc · Zbl 0663.46003 [22] DOI: 10.1063/1.861553 [23] Quispel G.R.W, Physica 110 pp 41– (1982) [24] DOI: 10.1143/PTP.65.172 [25] Tanabe, H. 1979. ”Equations of Evolution”. London Melbourne San Francisco: Pitman. · Zbl 0417.35003
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