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# Accession Number:

## ADA114503

# Title:

## Classical Solutions of the Korteweg-deVries Equation for Non-Smooth Initial Data via Inverse Scattering.

# Descriptive Note:

## Technical summary rept.,

# Corporate Author:

## WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER

# Report Date:

## 1981-11-01

# Pagination or Media Count:

##
59.0

# Abstract:

## The Cauchy problem for the Korteweg-deVries equation KdV for short q sub t x,t q sub xxx x,t - 6qx,tq sub xx,t equal 0 qx,0 equal Qx is solved classically for t greater than 0 via the so-called inverse scattering method. This approach, originating with Gardner, Greene, Kruskal, and Miura 9, relates the KdV equation to the one-dimensional Schrodinger equation -fx,k uxfx,k equal k superscript 2 fx,k. By considering the effect on the scattering data associated to the Schrodinger equation when the potential ux evolves in t according to the KdV equation , one obtains a linear evolution equation for the scattering data. The inverse scattering method of solving consists of calculating the scattering data for the initial value Qx. letting it evolve to time t, and then recovering qx,t from the evolved scattering data. Recently, P. Deift and E. Trubowitz 7 presented a new method for solving the inverse scattering problem obtaining the potential from its scattering data. Our solution of the KdV initial value problem uses this approach to construct a classical solution.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#