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The derivative nonlinear Schrödinger equation on the half-line
University of Cambridge, United Kingdom .ORCID iD: 0000-0001-6191-7769
2008 (English)In: Physica D: Non-linear phenomena, ISSN 0167-2789, Vol. 237, no 23, 3008-3019 p.Article in journal (Refereed) Published
Abstract [en]

We analyze the derivative nonlinear Schrödinger equation i qt + qx x = i (| q |2 q)x on the half-line using the Fokas method. Assuming that the solution q (x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x, t dependence and is given in terms of the spectral functions a (ζ), b (ζ) (obtained from the initial data q0 (x) = q (x, 0)) as well as A (ζ), B (ζ) (obtained from the boundary values g0 (t) = q (0, t) and g1 (t) = qx (0, t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q0 (x), g0 (t), g1 (t)} such that there exist spectral functions satisfying the global relation, we show that the function q (x, t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.

Place, publisher, year, edition, pages
2008. Vol. 237, no 23, 3008-3019 p.
Keyword [en]
DNLS equation, Riemann-Hilbert problem, Boundary value problems, Nonlinear equations, Probability density function, Spectrum analysis, Boundary values, Compatibility conditions, Dinger equations, Hilbert problems, Nonlinear, Spectral functions, Spectral parameters, T dependences, Problem solving
National Category
Physical Sciences Mathematics
URN: urn:nbn:se:kth:diva-163826DOI: 10.1016/j.physd.2008.07.005ISI: 000261463000003ScopusID: 2-s2.0-54149106278OAI: diva2:802367

QC 20150427

Available from: 2015-04-12 Created: 2015-04-12 Last updated: 2015-04-27Bibliographically approved

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Lenells, Jonatan
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