Title: Black Soliton Linearization for the 1D Gross-Pitaevsksii Equation: Low Frequency Effects
Abstract: Consider the nonlinear Schrödinger equation$$i\partial_tu(t,x)+\partial_x^2u(t,x)-2u(t,x)(|u(t,x)|^2-1)=0$$in spatial dimension one, subject to the non-vanishing boundary conditions $|u(t,x)|\to1$ as $|x|\to\infty$.
Linearizing around the black soliton $q(x)=\tanh(x)$ yields an evolution equation
$$i\partial_t\left[\begin{array}{c}w\\\bar w\end{array}\right]=\mathcal L\left[\begin{array}{c}w\\\bar w\end{array}\right]+O(w^2)$$
where $w=w(t,x)$ is a complex valued perturbation and
$$\mathcal L=\left[\begin{array}{cc}-\partial_x^2+2&2\\-2&\partial_x^2-2\end{array}\right]+\left[\begin{array}{cc}-4&-2\\2&4\end{array}\right]\textrm{sech}^2(x).$$
In earlier work it has been shown using a direct perturbation theory that a shelf develops around the black soliton (a localized intensity dip in intensity that goes to zero). Here we take a different approach by carrying out a spectral and Fourier analysis of $\mathcal L$ to derive a formula for the propagator $e^{-it\mathcal L}$. This is performed via a natural distorted Fourier transform obtained from explicit squared Jost solutions $f$ to $\mathcal Lf=Ef$ (in the defocusing regime we diagonalize $\mathcal L$).
The long-time asymptotics are described by separating (in the form $e^{-it\mathcal L}P_s$) the projection onto the zero eigenvalue embedded in the continuous spectrum (identified by computing the resolvent kernel explicitly), from the temporal evolution formula, as well as subtracting off an explicit `spreading shelf' operator $A(t)$. In the potential-free case a resonance is present instead of an embedded eigenvalue.
Finally we prove dispersive and Strichartz estimates for $e^{-it\mathcal L}P_s-A(t)$. In particular we have time decay of rate $t^{-1/2}$ away from from zero energy. However at zero energy one gets a weaker rate of $t^{-1/3}$ via the method of stationary phase.