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 .
Publisher Copyright:
© 2016 Elsevier B.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
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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 -