TY - JOUR

T1 - Primitive potentials and bounded solutions of the KdV equation

AU - Dyachenko, S.

AU - Zakharov, D.

AU - Zakharov, V.

PY - 2016/10/15

Y1 - 2016/10/15

N2 - 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.

AB - 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.

KW - Integrability

KW - Schrödinger operator

KW - Solitonic gas

UR - http://www.scopus.com/inward/record.url?scp=84992312368&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2016.04.002

DO - 10.1016/j.physd.2016.04.002

M3 - Article

AN - SCOPUS:84992312368

VL - 333

SP - 148

EP - 156

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -