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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 (ut + 6uux + uxxx)x = 3uyy 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
