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1. A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

   https://doi.org/10.1093/imrn/rnz156  

   T-R McLaughlin, Kenneth; Nabelek, Patrik V. (November 2019, International Mathematics Research Notices)

   Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation. 

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2. Finite time blow-up in a 1D model of the incompressible porous media equation

   https://doi.org/10.1088/1361-6544/adcdb8  

   Kiselev, Alexander; Sarsam, Naji A (April 2025, Nonlinearity)

   Abstract We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba–Córdoba–Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data. 

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3. Stability of periodic waves for the fractional KdV and NLS equations

   https://doi.org/10.1017/prm.2020.54  

   Hakkaev, Sevdzhan; Stefanov, Atanas G. (August 2021, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L 2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $$\lambda > 0$$ , there is a travelling wave solution to fKdV and fNLS $$\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $$ , which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in H α/2 [ − T , T ] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers. 

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4. Trace formulas revisited and a new representation of KdV solutions with short-range initial data

   https://doi.org/10.1088/1361-6544/ad7c10  

   Rybkin, Alexei (September 2024, Nonlinearity)

   Abstract We put forward a new approach to Deift-Trubowitz type trace formulas for the 1D Schrodinger operator with potentials that are summable with the first moment (short-range potentials). We prove that these formulas are preserved under the KdV flow whereas the class of short-range potentials is not. Finally, we show that our formulas are well-suited to study the dispersive smoothing effect. 

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5. Extremal spectral gaps for periodic Schrödinger operators

   https://doi.org/10.1051/cocv/2018029  

   Kao, Chiu-Yen; Osting, Braxton (January 2019, ESAIM: Control, Optimisation and Calculus of Variations)

   The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m , we consider the problem of maximizing the gap-to-midgap ratio for the m th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices. 

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