Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator $L = - \partial_{\theta} - \epsilon \partial_{\theta} (\sin \theta \partial_{\theta})$ subject to the periodic boundary conditions on $\theta \in [-\pi,\pi]$. We prove that the operator is closed in $L^2([-\pi,\pi])$ with the domain in $H^1_{\rm per}([-\pi,\pi])$ for $|\epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{\rm per}([-\pi,\pi])$.Comment: 22 pages, 10 figure

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes.Comment: 28 pages, 3 figures. Main result strengthened by removing a smallness condition. Limiting case of the black soliton now postponed to a companion pape

Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.Comment: 15 pages, no figure

Orbital stability in the cubic defocusing NLS equation: II. The black soliton

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.Comment: 19 pages, no figur