Soliton Solutions to the BKP Equations. II. The Integral Equation

Abstract

We propose the following integral equation for solving the BKP equations
K(x, z)+F(x, z)−∫_−∞^xD_yK(x, y)·F(y, z) dy=0
where F(x, y) satisfies the linear differential equation
(Remark: Graphics omitted.),
with the conditions F(x, y)=−F(y, x) and F(x, −∞)=0.
The τ-function, solution to the BKP equations in the bilinear form is related to the solution K(x, y) of the integral equation through the relation
(Remark: Graphics omitted.).
The integral equation plays the same role as that of the Gel’fand-Levitan integral equation used in solving the KP (Kadomtsev-Petviashvili) equation.