TY - JOUR

T1 - Primitive potentials and bounded solutions of the KdV equation

AU - Dyachenko, S.

AU - Zakharov, D.

AU - Zakharov, V.

N1 - Funding Information:
The authors would like to thank Harry Braden, Percy Deift, Igor Krichever and Thomas Trogdon for insightful discussions. The third author gratefully acknowledges support of the Russian Science Foundation Grant No. 14-22-00174 .

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

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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 -