A simple and direct periodic solution of the Korteweg-de vries equations

Abstract

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.