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The unified method: I. Nonlinearizable problems on the half-line
2012 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 45, no 19, 195201Article in journal (Refereed) Published
Abstract [en]

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix RiemannHilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions. © 2012 IOP Publishing Ltd.

Place, publisher, year, edition, pages
2012. Vol. 45, no 19, 195201
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URN: urn:nbn:se:kth:diva-163805DOI: 10.1088/1751-8113/45/19/195201ISI: 000303615100005ScopusID: 2-s2.0-84860369246OAI: diva2:802388

QC 20150417

Available from: 2015-04-13 Created: 2015-04-12 Last updated: 2015-04-17Bibliographically approved

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Lenells, Jonatan
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