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Table of Contents
                            Square Integrability and Uniqueness of the Solutions of the Kadomtsev−Petviashvili-I Equation
Soliton Asymptotics of Solutions of the Sine-Gordon Equation
On the Davey−Stewartson and Ishimori Systems
Stochastic Isometries in Quantum Mechanics
Complex Star Algebras
Momentum Tunneling between Tori and the Splitting of Eigenvalues of the Laplace−Beltrami Operator on Liouville Surfaces
Nonclassical Thermomechanics of Granular Materials
Random Operators and Crossed Products
Schrödinger Operators with Empty Singularly Continuous Spectra
An Asymptotic Expansion for Bloch Functions on Riemann Surfaces of Infinite Genus and Almost Periodicity of the Kadomcev−Petviashvilli Flow
Lifshitz Asymptotics via Linear Coupling of Disorder
Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients
Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I
Contents of Volume 2
Document Text Contents
Page 1

Mathematical Physics, Analysis and Geometry2: 1–24, 1999.
© 1999Kluwer Academic Publishers. Printed in the Netherlands.


Square Integrability and Uniqueness of the
Solutions of the Kadomtsev–Petviashvili-I Equation

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

(Received: 20 February 1998; in final form: 27 November 1998)

Abstract. We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I
Equation obtained by the inverse spectral method belongs to the Sobolev spaceHk(R2) for k > 0,
under the assumption that the initial datum is a small Schwartz function. This solution is shown to be
the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.

Mathematics Subject Classification (1991):35Q53.

Key words: Kadomtsev–Petviashvili-I equation, inverse spectral method, Cauchy problem, unique-
ness of solutions.

1. Introduction

The Cauchy problem

(qt − 6qqx + qxxx)x = 3qyy, (1.1a)
q(x, y,0) = q0(x, y), (1.1b)

for the Kadomtsev–Petviashvili-I (KPI) equation (cf. [8]) is solved in [6] by the
inverse spectral method, under the assumption thatq0 is a small Schwartz function.
It is shown in [6] that the solutionq(x, y, t) obtained by the inverse spectral method
is aC∞

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