arXiv subjects

Andreas Stahel

Publications and source records attributed to Andreas Stahel.

3 recordsLinked to original sources

arXiv2013

The Novikov-Veselov Equation: Theory and Computation

Recent progress in the theory and computation for the Novikov-Veselov (NV) equation is reviewed with initial potentials decaying at infinity, focusing mainly on the zero-energy case. The inverse scattering method for the zero-energy NV equation is presented in the context of Manakov triples, treating initial data of conductivity type rigorously. Special closed-form solutions are presented, including multisolitons, ring solitons, and breathers. The computational inverse scattering method is used to study zero-energy exceptional points and the relationship between supercritical, critical, and subcritical potentials.

math.AP↗

arXiv2013

Transverse instability of plane wave soliton solutions of the Novikov-Veselov equation

The Novikov-Veselov (NV) equation is a dispersive (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. This paper considers the stability of plane wave soliton solutions of the NV equation to transverse perturbations. To investigate the behavior of the perturbations, a hybrid semi-implicit/spectral numerical scheme was developed, applicable to other nonlinear PDE systems. Numerical simulations of the evolution of transversely perturbed plane wave solutions are presented. In particular, it is established that plane wave soliton solutions are not stable for transverse perturbations.

math-ph↗

arXiv2011

The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis

The Novikov-Veselov (NV) equation is a (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. Solution of the NV equation using the inverse scattering method has been discussed in the literature, but only formally (or with smallness assumptions in case of nonzero energy) because of the possibility of exceptional points, or singularities in the scattering data. In this work, absence of exceptional points is proved at zero energy for evolutions with compactly supported, smooth and rotationally symmetric initial data of the conductivity type: $q_0=γ^{-1/2}Δγ^{1/2}$ with a strictly positive function $γ$. The inverse scattering evolution is shown to be well-defined, real-valued, and preserving conductivity-type. There is no smallness assumption on the initial data.

math.AP↗