Stability for the Korteweg-de Vries equation by inverse scattering theory

The stability problems for the Korteweg-de Vries equation, where linear stability fails, are investigated by the inverse scattering method. A rather general solution u( t, x) of the K-dV equation is shown to depend, for fixed time t, continuously on the initial condition u(0, x). For a continuum sol...

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Bibliographic Details
Published in:Annals of physics 1981-01, Vol.134 (1), p.56-75
Main Authors: Scharf, G, Wreszinski, W.F
Format: Article
Language:English
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Summary:The stability problems for the Korteweg-de Vries equation, where linear stability fails, are investigated by the inverse scattering method. A rather general solution u( t, x) of the K-dV equation is shown to depend, for fixed time t, continuously on the initial condition u(0, x). For a continuum solution u c( t, x), this continuity holds uniformly in t (stability), but for a soliton solution this is not true. A soliton solution can be uniquely decomposed into a continuum and discrete (soliton) part: u( t, x) = u e( t, x) + u d( t, x). Then the perturbed solution u is close to u after a suitable t-dependent “shift” of the soliton part (form stability).
ISSN:0003-4916
1096-035X
DOI:10.1016/0003-4916(81)90004-X