On the B\"acklund transform and the stability of the line soliton of the KP-II equation on $\mathbb R^2$

Abstract

We study the Miura map of the KP-II equation on $\mathbb R^2$ and the resulting B\"acklund transform, which adds a line soliton to a given solution. This work aims to develop a complementary approach to T. Mizumachi's method for the $L^2$-stability of the line soliton, which the potential for generalization to multisolitons. We construct the B\"acklund transform by classifying solutions of the Miura map equation close to a modulated kink; this translates into studying eternal solutions of the forced viscous Burgers' equation under distinct boundary conditions at $\pm\infty$. We then show that its range, when intersected with a small ball in $|D_x|^{1/2} L^2(\mathbb R^2)\cap L^2(\mathbb R^2)\cap \langle{y}\rangle^{0-}\!L^1(\mathbb R^2)$, forms a codimension-1 manifold. We prove codimension-1 $L^2$-stability of the line soliton in the aforementioned weighted space as a corollary, providing the first stability result at sharp regularity. The codimension-1 condition in the range of the B\"acklund transform is an intrinsic property, and we conjecture that it corresponds to a known long time behavior of perturbed line solitons. The stability is expected to hold without this condition, as in Mizumachi's works. Finally, we show the construction of a multisoliton addition map for $(k,1)$-multisolitons, $k\geq 1$.