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The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs

2005; Wiley; Volume: 58; Issue: 5 Linguagem: Inglês

10.1002/cpa.20076

1097-0312

A. S. Fokas,

Numerical methods for differential equations

Communications on Pure and Applied MathematicsVolume 58, Issue 5 p. 639-670 The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs A. S. Fokas, A. S. Fokas t.fokas@damtp.cam.ac.uk University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB30WA, United KingdomSearch for more papers by this author A. S. Fokas, A. S. Fokas t.fokas@damtp.cam.ac.uk University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB30WA, United KingdomSearch for more papers by this author First published: 24 February 2005 https://doi.org/10.1002/cpa.20076Citations: 75AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n−1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized. © 2005 Wiley Periodicals, Inc. Citing Literature Volume58, Issue5May 2005Pages 639-670 RelatedInformation