Inverse scattering for the heat operator and evolutions in 2+1 variables

Mladen Victor Wickerhauser

doi:10.1007/BF01210703

Inverse scattering for the heat operator and evolutions in 2+1 variables

Mladen Victor Wickerhauser

Department of Mathematics

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59 Scopus citations

Abstract

The asymptotic behavior of functions in the kernel of the perturbed heat operator δ₁²-δ₂-u(x) suffice to determine u(x). An explicit formula is derived using the {Mathematical expression} method of inverse scattering, complete with estimates for small and moderately regular potentials u. If u evolves so as to satisfy the Kadomtsev-Petviashvili (KP II) equation, the asymptotic data evolve linearly and boundedly. Thus the KP II equation has solutions bounded for all time. A method for calculating nonlinear evolutions related to KP II is presented. The related evolutions include the so-called "KP II Hierarchy" and many others.

Original language	English
Pages (from-to)	67-89
Number of pages	23
Journal	Communications in Mathematical Physics
Volume	108
Issue number	1
DOIs	https://doi.org/10.1007/BF01210703
State	Published - Mar 1987

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title = "Inverse scattering for the heat operator and evolutions in 2+1 variables",

abstract = "The asymptotic behavior of functions in the kernel of the perturbed heat operator δ12-δ2-u(x) suffice to determine u(x). An explicit formula is derived using the \{Mathematical expression\} method of inverse scattering, complete with estimates for small and moderately regular potentials u. If u evolves so as to satisfy the Kadomtsev-Petviashvili (KP II) equation, the asymptotic data evolve linearly and boundedly. Thus the KP II equation has solutions bounded for all time. A method for calculating nonlinear evolutions related to KP II is presented. The related evolutions include the so-called {"}KP II Hierarchy{"} and many others.",

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AU - Wickerhauser, Mladen Victor

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N2 - The asymptotic behavior of functions in the kernel of the perturbed heat operator δ12-δ2-u(x) suffice to determine u(x). An explicit formula is derived using the {Mathematical expression} method of inverse scattering, complete with estimates for small and moderately regular potentials u. If u evolves so as to satisfy the Kadomtsev-Petviashvili (KP II) equation, the asymptotic data evolve linearly and boundedly. Thus the KP II equation has solutions bounded for all time. A method for calculating nonlinear evolutions related to KP II is presented. The related evolutions include the so-called "KP II Hierarchy" and many others.

AB - The asymptotic behavior of functions in the kernel of the perturbed heat operator δ12-δ2-u(x) suffice to determine u(x). An explicit formula is derived using the {Mathematical expression} method of inverse scattering, complete with estimates for small and moderately regular potentials u. If u evolves so as to satisfy the Kadomtsev-Petviashvili (KP II) equation, the asymptotic data evolve linearly and boundedly. Thus the KP II equation has solutions bounded for all time. A method for calculating nonlinear evolutions related to KP II is presented. The related evolutions include the so-called "KP II Hierarchy" and many others.

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U2 - 10.1007/BF01210703

DO - 10.1007/BF01210703

M3 - Article

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VL - 108

SP - 67

EP - 89

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -

Inverse scattering for the heat operator and evolutions in 2+1 variables

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