Real soliton lattices of the Kadomtsev-Petviashvili II equation and desingularization of spectral curves: the \(\mathrm{Gr}^{\mathrm{TP}}(2, 4)\) case (Q2280432)

The aim of this paper is to apply the general construction developed by the authors in [Commun. Math. Phys. 361, No. 3, 1029--1081 (2018; Zbl 1400.14094); KP theory, plabic networks in the disk and rational degenerations of \(M\)-curves, Preprint, \url{arXiv: 1801.00208}; Sel. Math., New Ser. 25, No. 3, Paper No. 43, 64 p. (2019; Zbl 1426.37052)] to the first nontrivial case of \(\mbox{Gr}^{\mbox{TP}}(2,4)\). Here \(\mbox{Gr}^{\mbox{TP}}(2,4)\) is the main cell in the real Grassmannians \(\mbox{Gr}^{\mbox{TNNP}}(2,4)\). In particular, the authors construct finite-gap real quasi-periodic solutions of the Kadomtsev-Petviashvili (KP)-II equation in the form of a soliton lattice corresponding to a smooth \(M\)-curve of genus \(4\) which is a desingularization of a reducible rational \(M\)-curve for soliton data in the main cell \(\mbox{Gr}^{\mbox{TP}}(2,4)\). They describe bases of cycles and differentials, and check numerically the consistency of the construction. This paper is organized as follows. Section 1 is an introduction to the subject. Section 2 deals with finite-gap solutions of the KP-II equation. The authors recall how to construct quasi-periodic solutions of the KP-II equation using the finite-gap approach, which was first introduced for the Korteweg-de Vries equation by \textit{S. P. Novikov} [Funct. Anal. Appl. 8, 236--246 (1974; Zbl 0299.35017); translation from Funkts. Anal. Prilozh. 8, No. 3, 54--66 (1974)]. Section 3 deals with real regular bounded multi-line soliton solutions of the KP-II equation. Section 4 concerns the construction of the reducible curve and of the KP-II divisor for soliton data in \(\mbox{Gr}^{\mbox{TP}}(k,n)\). In this section the authors briefly summarize their construction in [Commun. Math. Phys. 361, No. 3, 1029--1081 (2018; Zbl 1400.14094)]. Section 5 deals with KP-II divisor for soliton data in \(\mbox{Gr}^{\mbox{TP}}(2,4)\). Section 6 deals with the spectral curve for \(\mbox{Gr}^{\mbox{TP}}(2,4)\) and its desingularization. In this section the authors construct the rational curve associated to soliton data in \(\mbox{Gr}^{\mbox{TP}}(2,4)\). Section 7 is devoted to numerical simulations. Here the authors provide results of numerical calculations in order to illustrate the construction above.