The Miura map on the line
By Thomas Kappeler, Peter Perry, M. Shubin, P. Topalov. 2005
59 citations recorded by the source corpus.
Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrodinger operators Lq = −d2 /dx2 + q where r and q are real-valued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Korteweg-de Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/66277488bd79738135abc2fda6e896b0786d3239
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In Mathematical Physics
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In Statistical and Nonlinear Physics
Citations in the corpus
- Eigenvalue asymptotics for Sturm–Liouville operators with singular potentials (2006, result): “Our aim here is to show that (under some not very restrictive assumption) lσ coincides with mτ for a suitable choice of τ. See also [ 19 ] for similar results on the whole axis.”
- KdV is well-posed in H–1 (2018, background): “is hopeless without first knowing uniqueness.) For a restricted class of H−1 initial data (namely, that in the range of the traditional Miura map), the existence of weak solutions was shown earlier in [28]; see also [67]. In Section 7 we will give a new derivation of the a priori local smoothing boun”
- Transformation Operators for Sturm-Liouville Operators with Singular Potentials (2004, result): “See also [19] for similar results on the whole axis.”