Positive periodic solutions for the Korteweg-de Vries equation

Abstract

In this paper we prove that the Korteweg-de Vries equation ∂tu + ∂3xu + u∂xu = 0 has unique positive solution u(t, x) which is ⍵-periodic with respect to the time variable t and u(0, x) ∈ Ḃγp,q ([α, b]), γ > 0, γ ∉ {1, 2,...}, p > 1, q ≥ 1, α < b are fixed constants, x ∈ [α, b]. The period ⍵ > 0 is arbitrary chosen and fixed.