# Features of the nonlinear Fourier transform for the dNLS equation

Molnar, Jan. Features of the nonlinear Fourier transform for the dNLS equation. 2016, University of Zurich, Faculty of Science.

## Abstract

In the first part of this thesis we extend the well-known spectral theory of the Zakharov-Shabat
operator $\mathit{L}(\phi)$ = $\begin{pmatrix} i & 0\\0 & -i\end{pmatrix}$$\partialx + \begin{pmatrix} 0 & \phi-\\\phi+ & 0\end{pmatrix}, acting on the interval [.0; 1]., to the case where the potential \phi = (\phi_{-},\phi_{+}). is a complex, 1-periodic element of the Fourier Lebesgue space \mathit{FL}^{\mathit{p}}, 1 \leqslant \mathit{p} < \infty, and prove asymptotic estimates for its periodic and Dirichlet eigenvalues in terms of the Fourier coefficients of \phi. The spectral theory is then used to extend the definition of the actions (\mathit{I}_{\mathit{n}})_{\mathit{n}\in\mathbb{Z}} and the canonically conjugated angles (\theta_{\mathit{n}})_{\mathit{n}\in\mathbb{Z}} from \mathit{L}^{2} to \mathit{FL}^{\mathit{p}}, \mathit{p} > 2, which, in turn, are used to construct real analytic Birkhoff coordinates on \mathit{FL}^{\mathit{p}}. In the second part of this thesis we derive, using the Birkhoff coordinates, a novel formula for the dNLS frequencies which allows to extend them analytically to \mathit{FL}^{\mathit{p}}, \mathit{p} > 2, and to characterize their asymptotic behavior. Similarly, we derive a formula for the dNLS Hamiltonian which is used to extend this Hamiltonian, after appropriate renormalization, real analytically to \mathit{FL}^{4}_{\mathit{r}}. When expressed in action variables \mathit{I} = (\mathit{I}_{\mathit{n}})_{\mathit{n}\in\mathbb{Z}}, this renormalized Hamiltonian defines a function which is real analytic and strictly concave in a neighborhood of 0 in the positive quadrant \ell^{2}_{+}(\mathbb{Z}) of \ell^{2}(\mathbb{Z}). Finally, we use our previously obtained results on the frequencies to study the initial value problem of dNLS in Birkhoff coordinates. In the final part of this thesis we investigate the Birkhoff map in Sobolev spaces of high regularity. We prove uniform tame estimates of all integer Sobolev norms \parallel\phi\parallel_{\mathit{m}}, \mathit{m} \geqslant 1, in terms of weighted \ell^{2}-norms of the Birkhoff coordinates and vice versa. ## Abstract In the first part of this thesis we extend the well-known spectral theory of the Zakharov-Shabat operator \mathit{L}(\phi) = \begin{pmatrix} i & 0\\0 & -i\end{pmatrix}$$\partial$x + $\begin{pmatrix} 0 & \phi-\\\phi+ & 0\end{pmatrix}$, acting on the interval [.0; 1]., to the case where the potential $\phi$ = ($\phi_{-},\phi_{+}$). is a complex, 1-periodic element of the Fourier Lebesgue space $\mathit{FL}^{\mathit{p}}$, 1 $\leqslant \mathit{p} < \infty$, and prove asymptotic estimates for its periodic and Dirichlet eigenvalues in terms of the Fourier coefficients of $\phi$. The spectral theory is then used to extend the definition of the actions ($\mathit{I}_{\mathit{n}}$)$_{\mathit{n}\in\mathbb{Z}}$ and the canonically conjugated angles ($\theta_{\mathit{n}}$)$_{\mathit{n}\in\mathbb{Z}}$ from $\mathit{L}^{2}$ to $\mathit{FL}^{\mathit{p}}$, $\mathit{p} >$ 2, which, in turn, are used to construct real analytic Birkhoff coordinates on $\mathit{FL}^{\mathit{p}}$.

In the second part of this thesis we derive, using the Birkhoff coordinates, a novel formula for the dNLS frequencies which allows to extend them analytically to $\mathit{FL}^{\mathit{p}}$, $\mathit{p} >$ 2, and to characterize their asymptotic behavior. Similarly, we derive a formula for the dNLS Hamiltonian which is used
to extend this Hamiltonian, after appropriate renormalization, real analytically to $\mathit{FL}^{4}_{\mathit{r}}$. When expressed in action variables $\mathit{I}$ = ($\mathit{I}_{\mathit{n}}$)$_{\mathit{n}\in\mathbb{Z}}$, this renormalized Hamiltonian defines a function which is real analytic and strictly concave in a neighborhood of 0 in the positive quadrant $\ell^{2}_{+}$($\mathbb{Z}$) of $\ell^{2}$($\mathbb{Z}$). Finally, we use our previously obtained results on the frequencies to study the initial value problem of dNLS in Birkhoff coordinates.
In the final part of this thesis we investigate the Birkhoff map in Sobolev spaces of high regularity. We prove uniform tame estimates of all integer Sobolev norms $\parallel\phi\parallel_{\mathit{m}}$, $\mathit{m} \geqslant$ 1, in terms of weighted $\ell^{2}$-norms of the Birkhoff coordinates and vice versa.