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Applications to Partial Differential Equations—Nonlinear Equations

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Part of the Applied Mathematical Sciences book series (AMS, volume 44)

Abstract

In this section we consider a simple application of the results of Section 6.1 to the initial value problem for the following nonlinear Schrödinger equation in ∝2
$$\left\{ {_{u(x,0) = {u_0}(x)in{\mathbb{R}^2}}^{\frac{1}{i}\frac{{\partial u}}{{\partial t}} - \Delta u + k{{\left| u \right|}^2}u = 0in]0,\infty [x{\mathbb{R}^2}}} \right.$$
(1.1)
where u is a complex valued function and k a real constant. The space in which this problem will be considered is L2(R2). Defining the linear operator A 0 by D(A 0 ) = H 2 (R2)and A 0 u = — i & u for u ϵ D(A0) the initial value problem (1.1) can be rewritten as
$$\left\{{_{u(0) = {u_0}}^{\frac{{du}}{{dt}} + {A_0}u + F(u) = 0fort>0}}\right.$$
(1.2)

where F(u) = ik\u\ 2 u.

Keywords

Linear Operator Classical Solution Global Solution Mild Solution Fractional Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1983

Authors and Affiliations

  • A. Pazy
    • 1
  1. 1.Planning and Budgeting CommitteeCouncil for Higher EducationJerusalemIsrael

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