On a class of compact perturbations of the special pole-free joint solution of KdV and $P_I^2.$

Abstract

We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+u{u}_x+\frac{1}{12}{u}_{xxx}=0$ and $P_I^2$ equation $u_{xxxx}+10u_x^2+20u{u}_{xx}+40(u^3-6tu+6x)=0$ 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 $a(\lambda ),r(\lambda ).$

57 pages, 7 figures

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