The Perturbed Nonlinear Schrödinger Equation

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 β_j 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.

Article Footnote

From the point of view of existence and differentiability of invariant manifolds, and their persistence under perturbation, neutrally stable modes (in the linear approximation) pose more analytical difficulties than exponentially growing and decaying modes. In the language of dynamical systems theory, elliptic perturbation problems are generally more difficult than hyperbolic perturbation problems (compare the proof of the stable and unstable manifold theorem for a hyperbolic invariant set with that of the KAM theorem).