A SIMPLE AND DIRECT PERIODIC SOLUTION OF THE KORTEWEG-DE VRIES EQUATIONS


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Abstract

The solution q(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 relates wt(x, t, λ) = ϕt(x+c, x, t, λ) with qt(x, t). The function ϕ(x, x0, t, λ) obeys the Schrödinger equation and the boundary conditions ϕ(x0, x0, t, λ) = 0, ϕx(x0, x0; t, λ) = 1. The shifting c is equal to the period. We differentiate wt(x, t, λ) three times with respect to the x coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect to x allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions of w(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replaces qt(x, t) by an expression of the KdV hiearchy in the relation between qt(x, t) and wt(x, t, λ) and transforms it. We estimated also the limit, when c → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.

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