We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+uu_x+\frac{ 1}{ 12}u_{ xxx}=0$ and $P_I^2$ equation $u_{ xxxx}+10u_x^2+20uu_{ xx}+40(u^3-6tu+6x)=0$ under the action of the KdV flow. We show that if the perturbation is compact and of bounded variation, then the initial value problem for the KdV equation has a classical solution. Our method is the inverse scattering transform method in the form of the Riemann-Hilbert problem method. Namely, we construct the corresponding spectral functions $a(\lambda), r(\lambda),$ and give characterization of the compact perturbations in terms of $a(\lambda), r(\lambda).$
Categories: math-ph math.MP