Applications to Partial Differential Equations—Nonlinear Equations

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


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
$$\left\{ {_{u(x,0) = {u_0}(x)in{\mathbb{R}^2}}^{\frac{1}{i}\frac{{\partial u}}{{\partial t}} - \Delta u + k{{\left| u \right|}^2}u = 0in]0,\infty [x{\mathbb{R}^2}}} \right.$$
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
$$\left\{{_{u(0) = {u_0}}^{\frac{{du}}{{dt}} + {A_0}u + F(u) = 0fort>0}}\right.$$

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


Linear Operator Classical Solution Global Solution Mild Solution Fractional Power 
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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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