Inverse scattering transform for the KPI equation on the background of a one-line soliton*

Abstract

We study the initial value problem of the Kadomtsev-Petviashvili I (KPI) equation with initial data u(x 1 , x 2 , 0) = u 1 (x 1 )+u 2 (x 1 , x 2 ), where u 1 (x 1 ) is the one-soliton solution of the Korteweg-de Vries equation evaluated at zero time and u 2 (x 1 , x 2 ) decays sufficiently rapidly on the (x 1 , x 2 )-plane. This involves the analysis of the nonstationary Schrödinger equation (with time replaced by x 2 ) with potential u(x 1 , x 2 , 0). We introduce an appropriate sectionally analytic eigenfunction in the complex k-plane where k is the spectral parameter. This eigenfunction has the novelty that in addition to the usual jump across the real k-axis, it also has a jump across a segment of the imaginary k-axis. We show that this eigenfunction can be reconstructed through a linear integral equation uniquely defined in terms of appropriate scattering data. In turn, these scattering data are uniquely constructed in terms of u 1 (x 1 ) and u 2 (x 1 , x 2 ). This result implies that the solution of the KPI equation can be obtained through the above linear integral equation where the scattering data have a simple t-dependence.

Towards an inverse scattering theory for non-decaying potentials of the heat equation

“…Therefore, according to our previous discussion, only the discontinuities at k = ia and at k = ib are left. Thus in the case of the heat equation (at least for N = 1) the Green's function of the Jost solution has no an additional cuts in contrast with the case of the nonstationary Schrödinger equation [23,15], but, anyway, due to the special singularities at k = ia and at k = ib, the definition of the spectral data also for the case of a perturbed one soliton potential is not standard and needs a detailed analysis. This problem will be faced in a following work.…”

Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution

“…However, the Cauchy problem is not rigorously solved in [8] and it is unlikely that it could be solved for an arbitrary large data φ using IST since the Cauchy problem with purely localized data has been solved by IST techniques only for small initial data (see [35,38]). …”

Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*

“…Their methods are consistent with the zero curvature condition in modern soliton theory. Using the hyperelliptic sigma function and defining natural sigma functions of more general algebraic curves, the authors in [21][22][23][24][25][26] have been constructing deeper theories of Abelian functions and soliton equations [35][36][37][38][39][40][41][42].…”