A remark on the korteweg-de vries equation

Summary

Let u(x, t) be a solution of the KdV equation u_t — u_xxx— 6u_xu = 0, z(x,t) be a solution of the first-order PDE z_t- 2uz_x = 0 and v(z) be defined in terms of u and z by the formula r[z(x,i),t] = [z_x(x,t)]^-2 u(x,t) — [z_x(x,t)]^-1/2[z_x{x,t)^-1/2}_xx. Then v_t(z,t) = 0, namely v(z,t) does not change with time (for fixed z).