The Davey-Stewartson I Equation on the Quarter Plane with Homogeneous Dirichlet Boundary Conditions

2004/12/01 by A. S. Fokas, Fokas, A. S.
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0412005

arxiv created 2004/12/01 · openalex publication_date 2004/12/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Dromions are exponentially localised coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions.In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey-Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of non-zero Dirichlet boundary conditions is also discussed.

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