| 1. |
- Lenells, Jonatan, 1981-
(författare)
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Classification of all travelling-wave solutions for some nonlinear dispersive equations
- 2007
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Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 365:1858, s. 2291-2298
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Tidskriftsartikel (refereegranskat)abstract
- We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa-Holm or the Degasperis-Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractallike wave profiles.
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| 2. |
- Khesin, B., et al.
(författare)
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Curvatures of Sobolev metrics on diffeomorphism groups
- 2013
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Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 9:2, s. 291-332
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Tidskriftsartikel (refereegranskat)abstract
- Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev H0 and H1 energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.
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| 3. |
- Lenells, Jonatan, 1981-
(författare)
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Spheres, Kähler geometry and the Hunter-Saxton system
- 2013
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Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : The Royal Society. - 1364-5021 .- 1471-2946. ; 469:2154
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Tidskriftsartikel (refereegranskat)abstract
- Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter- Saxton (2HS) system, which displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold, which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulae for the solutions of 2HS.We also show thatwhen restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold, which admits a Kähler structure. We demonstrate that this manifold is in fact isometric to a subset of complex projective space, and that the above constructions provide an example of an infinite-dimensional Hopf fibration.
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| 4. |
- Lenells, Jonatan, 1981-
(författare)
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The Degasperis-Procesi equation on the half-line
- 2013
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Ingår i: Nonlinear Analysis. - : Elsevier BV. - 0362-546X .- 1873-5215. ; 76:1, s. 122-139
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Tidskriftsartikel (refereegranskat)abstract
- We analyze a class of initial-boundary value problems for the Degasperis-Procesi equation on the half-line. Assuming that the solution u(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter k.
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| 5. |
- Lenells, Jonatan, 1981-, et al.
(författare)
-
The Hunter-Saxton System and the Geodesics on a Pseudosphere
- 2013
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 38:5, s. 860-881
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Tidskriftsartikel (refereegranskat)abstract
- We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.
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