arXiv subjects

Joel Dahne

Publications and source records attributed to Joel Dahne.

7 recordsLinked to original sources

arXiv2026

Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations

We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schrödinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau Equation. We are also able to control their monotone index (number of monotone intervals). In particular, we prove the existence of monotone radial self-similar singular solutions for the three dimensional cubic Nonlinear Schrödinger Equation. The paper combines sharp analytic bounds of the self-similar profile at infinity with computer assisted bounds around zero and their matching at an intermediate value.

math.AP↗

arXiv2026

Non-uniqueness for the Complex Ginzburg-Landau Equation

We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.

math.AP↗

arXiv2026

Monotonicity of the first Dirichlet eigenvalue of regular polygons

In this paper we prove that the first Dirichlet eigenvalue $λ_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N \geq 3$, as well as the monotonicity of the quotients $\displaystyle \frac{λ_1^{N}}{λ_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].

math.SP↗

arXiv2023

Highest Cusped Waves for the Fractional KdV Equations

In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations $f_t + f f_x = |D|^α f_x$ for all $α\in (-1,0)$ and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.

math.AP↗

arXiv2022

Highest Cusped Waves for the Burgers-Hilbert equation

In this paper we prove the existence of a periodic highest, cusped, traveling wave solution for the Burgers-Hilbert equation $f_{t} + f f_{x} = H[f]$ and give its asymptotic behaviour at $0$. The proof combines careful asymptotic analysis and a computer-assisted approach.

math.AP↗

arXiv2021

A counterexample to Payne's nodal line conjecture with few holes

Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with many holes and raised the question of the minimum number of holes a counterexample can have. In this paper we prove it is at most 6.

math.SP↗

arXiv2020

Computation of Tight Enclosures for Laplacian Eigenvalues

Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.

math.NA↗