Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8256325 | Physica D: Nonlinear Phenomena | 2016 | 9 Pages |
Abstract
We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.
Keywords
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Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
S. Dyachenko, D. Zakharov, V. Zakharov,