On the Bäcklund transform and the stability of the line soliton of the KP-II equation on the plane

Abstract

We provide a detailed study of the Bäcklund transform (or soliton addition map) of the KP-II equation on the Euclidean plane at critical regularity. The transform has the effect of superimposing a modulated line soliton onto a given KP-II solution. Our analysis is motivated by the problem of stability of KP-II multisolitons in <em>L</em>²(&reals;²), currently solved only for the single soliton, and by the broader question of understanding the connections between integrability and soliton stability in PDEs admitting line solitons. <br/> We show that the transform is well-defined in a critical Sobolev space, establish appropriate <em>L</em>² estimates, and prove that it commutes with the KP-II flow. The mapping properties of the transform are more complicated than those of KdV and other 1D models. In this direction, we show that its range, when intersected with a small ball in a mildly weighted space, forms exactly a codimension-1 manifold. This codimension-1 condition is an intrinsic property of the Bäcklund transform, and we conjecture that it corresponds to a known long time behavior of perturbed line solitons. <br/>The <em>L</em>²-stability of the line soliton in the aforementioned manifold follows as an immediate corollary, providing a first partial stability result at sharp regularity. Using this map for the full stability of solitons will require a follow-up study. <br/>Finally, we show the construction of a multisoliton addition map for a certain class of KP-II multisolitons.