Long-Time Asymptotics for the Kadomtsev–Petviashvili I Equation with Small Initial Data

Author

Samir Donmazov, University of KentuckyFollow

Author ORCID Identifier

https://orcid.org/0000-0001-8814-3572

Date Available

7-19-2025

Year of Publication

2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Faculty

Peter Perry

Faculty

Bertrand Guillou

Abstract

We study the initial value problem for the Kadomtsev--Petviashvili I (KP I) equation (u_t + 6uu_x + u_xxx)_x = 3u_yy with small initial data belonging to a subspace of the energy space for the KP I equation. We establish the long-time asymptotics for solutions of the KP I equation using the inverse scattering transform formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann--Hilbert problem (RHP) in two spatial dimensions. As part of the asymptotic analysis, we determine the long-time behavior of the solution to the nonlocal RHP, along with its x-derivative. To our knowledge, this is the first rigorous analysis of the long-time behavior for the KP I equation in the small-data regime. Our work complements earlier formal studies by Manakov, Santini, and Takhtajan based on a classical IST formalism involving a Gelfand--Levitan--Marchenko integral equation.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2025.297

Recommended Citation

Donmazov, Samir, "Long-Time Asymptotics for the Kadomtsev–Petviashvili I Equation with Small Initial Data" (2025). Theses and Dissertations--Mathematics. 122.
https://uknowledge.uky.edu/math_etds/122