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The Perturbed Nonlinear Schrödinger Equation

  • Charles Li
  • Stephen Wiggins
Part of the Applied Mathematical Sciences book series (AMS, volume 128)

Abstract

Consider the perturbatively damped and driven nonlinear Schrödinger equation (PNLS)
$$i{{q}_{t}} = {{q}_{{xx}}} + 2\left[ {|q{{|}^{2}} - {{\omega }^{2}}} \right]q + i\epsilon \left[ { - \alpha q + {{{\hat{D}}}^{2}}q + \Gamma } \right]$$
(2.1.1)
under the even and periodic boundary condition
$$\begin{array}{*{20}{c}} {q( - x) = q(x),} & {q(x + 1) = q(x),} \\ \end{array}$$
where \(\omega \in (\pi ,2\pi ),\epsilon \in ( - {{\epsilon }_{0}},{{\epsilon }_{0}})\) is the perturbation parameter, α(> 0) and are real constants. The operator \({{\hat{D}}^{2}}\) is a regularized Laplacian, specifically given by
$${{\hat{D}}^{2}}q \equiv - \sum\limits_{{j = 1}}^{\infty } {{{\beta }_{j}}k_{j}^{2}{{{\hat{q}}}_{j}}\cos {{k}_{j}}x,}$$
where \({{\hat{q}}_{j}}\) is the Fourier transform of q and \({{k}_{j}} \equiv 2\pi j\) The regularizing coefficient βj is defined by
$${{\beta }_{j}} \equiv \left\{ {\begin{array}{*{20}{c}} \beta \hfill & {for j \leqslant N,} \hfill \\ {{{\alpha }_{*}}k_{j}^{{ - 2}}} \hfill & {for j > N,} \hfill \\ \end{array} } \right.$$
where α*, and β are positive constants and N is a large fixed positive integer. When, the terms and are perturbatively damping terms; the former is a linear damping, and the latter is a diffusion term. Hence, this regularized Laplacian acts in such a way that it smooths the dissipation at short wavelengths. The reason for this choice is that we will need the flow generated by this infinite dimensional dynamical system to be defined for all time. We will see that the condition ω ∈ (π, 2π) implies that for an appropriate linearization of the unperturbed nonlinear Schrödinger equation (to be discussed shortly), there is precisely one exponentially growing and one exponentially decaying mode (more exponentially growing and decaying modes can be treated without difficulty). 1 The term is a perturbatively driving term.

Keywords

Invariant Manifold Perturbation Parameter Invariant Plane Bump Function Transversal Bundle 
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 Science+Business Media New York 1997

Authors and Affiliations

  • Charles Li
    • 1
  • Stephen Wiggins
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Applied MechanicsCalifornia Institute of TechnologyPasadenaUSA

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