2024/12/17 by Lorenzo Pompili, Pompili, Lorenzo · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.2412.12530
We study the Miura map of the KP-II equation on \mathbb R2 and the resulting Bäcklund 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 L2-stability of the line soliton, which the potential for generalization to multisolitons. We construct the Bäcklund 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 ±∞. We then show that its range, when intersected with a small ball in |Dx|1/2 L2(\mathbb R2)∩ L2(\mathbb R2)∩ ⟨y⟩0- L1(\mathbb R2), forms a codimension-1 manifold. We prove codimension-1 L2-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 Bäcklund 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≥ 1.