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Title of Journal: Acta Appl Math

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Abbravation: Acta Applicandae Mathematicae

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Springer Netherlands

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DOI

10.1002/chin.200038284

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1572-9036

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An InitialBoundary Value Problem for the Korteweg

Authors: Pham Loi Vu
Publish Date: 2013/06/06
Volume: 129, Issue: 1, Pages: 41-59
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Abstract

We consider the initialboundary value problem IBVP for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems SPs on two different spectral planes The first SP is associated with the KdV equation The second SP is selfconjugate and its scattering function is found in terms of entries of the scattering matrix sk for the first SP Knowing the scattering function we solve the second inverse SP for finding the potential selfconjugate matrix Consequently the unknown object entering coefficients in the system of evolution equations for skt is found Then the timedependent scattering matrix skt is expressed in terms of sk=sk0 and of solutions of the selfconjugate SP Knowing skt we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP

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