The Novikov-Veselov Equation:Theory and Computation

Abstract

Abstract. We review recent progress in theory and computation for the Novikov-Veselov (NV) equation with potentials decaying at infinity, focusing mainly on the zero-energy case. The inverse scattering method for the zeroenergy 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.

IST Versus PDE: A Comparative Study

“…We refer to [19] for a systematic study of the dispersive properties of general third order (local) operators in twodimensions. We refer to the excellent survey [54] for a rather complete account of what is known in this case and which comprises some interesting numerics and a rich bibliography.…”

“…The Novikov-Veselov equation with zero energy has a global solution for critical and subcritical initial data, but its solution may blow up in finite time for supercritical initial data. 12 We refer to [54] for some partial results toward its resolution.…”

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

“…In fact, a huge and ever-growing body of works related to the study of NV equations has been established (see, for example, Refs. [15][16][17][18] for a rather extensive review of a recent literature on the subject). In particular, a lot of spotlight has been focused on the solutions of (14) and (15).…”

“…Many articles were dedicated to unusual and fascinating properties of the multi-dimensional solutions, including those for seemingly ordinary flat waves. In particular, in [24] it has been shown that plane wave soliton solutions of NV equation are not stable for transverse perturbations; the paper [18] demonstrates that NV equation permits such interesting solutions as multi-solitons, ring solitons, and the breathers; while the authors of [25] construct a Mach-type soliton of the NV equation. One of the most effective mathematical tools for studying the NV equation is the inverse scattering method.…”

Nonlinear Fourier analysis for discontinuous conductivities: Computational results

“…It is a nonlinear evolution equation generalizing the celebrated Korteweg-de Vries (KdV) equation into dimension (2+1). There has been significant recent progress in linearizing the NV equation using inverse scattering methods, see [38,37,44,42,14].…”